An Engineering Method for Comparing Selectively Compliant Joints in ...

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Jun 13, 2014 - Abstract—Large displacement compliant joints can substitute traditional ... fying the joint's selective compliance by means of local and global.
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IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 19, NO. 6, DECEMBER 2014

An Engineering Method for Comparing Selectively Compliant Joints in Robotic Structures Giovanni Berselli, Alessandro Guerra, Gabriele Vassura, and Angelo Oreste Andrisano

Abstract—Large displacement compliant joints can substitute traditional kinematic pairs in robotic articulated structures for increasing ease-of-assembly, robustness, and safety. Nonetheless, besides their limited motion capabilities, compliant joints might be subjected to undesired spatial deformations which can deteriorate the system stability and performance whenever a low number of control inputs is available. In all these cases, it is convenient to select/design joint morphologies which enable a selectively compliant behavior, i.e., a low stiffness along a single desired direction. Within this context, this paper outlines an engineering method for quantifying the joint’s selective compliance by means of local and global performance indices. The approach is validated by comparing two beam-like flexures whose analytic solution is known from the literature. Finally, two joint morphologies, previously employed in the fabrication of robotic/prosthetic hands, are critically compared on the basis of the proposed criteria. Index Terms—Design criteria, large displacement compliant joints, performance evaluation.

Fig. 1. Large displacement rotational compliant joints. (a) JC , multiple-axis flexural hinge. (b) JR , corner filleted flexural hinge. (c) SPIR, spiral joint. (d) HEL, helical joint.

I. INTRODUCTION COMPLIANT joint (CJ) is a flexible connector that can provide limited displacement between two rigid parts through material’s deformation. According to [1], CJs can be used to substitute traditional kinematic pairs (like bearing couplings) in rigid-body mechanisms, thus obtaining the so-called lumped compliant mechanisms, in which compliance is concentrated in relatively small regions connected through rigid links. The benefits of CJs, when compared to traditional pairs, include the absence of wear, backlash, and friction while ensuring size and weight reduction. In addition, the advent of new materials [2], new manufacturing technologies (e.g., additive manufacturing [3]), and new fields of applications (e.g., MEMS devices [4], compliant actuators [5] and sensors [6]) largely encourages the development of this concept. On the other hand, CJs can suffer fatigue failures if poorly designed and can induce an undesired stiffness which might force to oversize the actuation system (e.g., [7], [8]).

A

Manuscript received May 26, 2012; revised October 27, 2012, April 7, 2013, September 11, 2013, and February 15, 2014; accepted March 20, 2014. Date of publication May 2, 2014; date of current version June 13, 2014. Recommended by Technical Editor E. Richer. G. Berselli and A. O. Andrisano are with the “Enzo Ferrari” Engineering Department, University of Modena and Reggio Emilia, 41125 Modena, Italy (e-mail: giovanni. [email protected]; [email protected]). A. Guerra is with the R&D Department, Comer Industries S.p.A., Italy (e-mail: [email protected]). G. Vassura is with the Industrial Engineering Department, University of Bologna, 40126 Bologna, Italy (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMECH.2014.2315508

Concerning robotic/mechatronic applications, the introduction of CJs in serial articulated chains, like anthropomorphic hands and prosthesis, seems promising as it can allow the generation of very slender mechanisms which are possibly lighter and easier to assemble when compared to their rigid-body counterpart. Also, in the event of unintended contacts or impacts, the inclusion of joint compliance can enhance the system robustness and safety when interacting with unknown environments or humans [9], [10]. From a design perspective, various types of CJs have been proposed in the literature, which are classifiable in terms of amount of displacement (small [11] or large [12]), number, and type (rotational or translational) of degrees of freedom (d.o.f.) [13], adopted materials (multi- [14] or monomaterial [15]), and morphology (for instance, notch type CJ or leaf spring type CJ [16]). Concerning small-displacement CJs, the past literature mainly focuses on the optimal design and modeling of flexural hinges and, especially, notch hinges (i.e., CJs which are obtained by machining one or two cutouts in a blank material with constant width). For instance, compliance equations for circular, elliptical, and corner-filleted flexures are provided in [11], whereas a generalized model for various planar and spatial hinge profiles is reported in [17]. In parallel, empirical equations based on the finite element method (FEM) have also been reported [18]. However, the reliability of the aforementioned formulations is limited to slender, beam-like structures [see, e.g., Fig. 1(a) and (b)] and the extension to the largedisplacement range is acceptable only under particular loading conditions [19].

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BERSELLI et al.: ENGINEERING METHOD FOR COMPARING SELECTIVELY COMPLIANT JOINTS IN ROBOTIC STRUCTURES

Concerning large-displacement CJs, considerable work has been done in designing and modeling compliant mechanisms by means of FEM (e.g., [20]), analytical methods derived by the solid-mechanics domain [21], or pseudorigid body (PRB) techniques [12]. Lately, several planar multi-d.o.f. PRB models have been proposed in order to closely capture the CJ nonlinear deflection for generic static loads [22]. Nonetheless, in the simplest PRB formulation, the CJ is modeled as an ideal pin joint with a torsional spring mounted in parallel, thus reducing the joint kinematics to a single d.o.f. rotation. This PRB approximation becomes very useful for designing compliant mechanism with prescribed load-displacement profiles at one point on their structure [23], [24], for static and dynamic analysis [25], [26], and for the model-based control of complex robotic systems [27]. As for the torsional spring compliance, analytical solutions for relatively large displacements are mainly known concerning beam-like joints [12], [19], whereas nonlinear FEM is still the only viable choice for several CJ morphologies widely used in the engineering practice. Besides the modeling method, CJs are possibly subjected to undesired spatial deformations (parasitic motions) which can deteriorate the system stability and performance in robotic devices where a low number of control inputs is available (e.g., prosthetic and underactuated hands [9], [28]). In these cases, the joint is not only modeled, but also designed in order to provide one d.o.f. only. Hence, once the application constraints (e.g., range of motion, manufacturability, cost) are verified, CJ morphologies which are capable of providing a high compliance along a single desired direction should be given preference. For instance, let one consider the joint morphologies depicted in Fig. 1. In particular, the large displacement CJs drawn in Fig. 1(c) and (d) (first proposed in [29]) are characterized by: 1) the same deflection under the application of the same bending moment (i.e., the same bending compliance, Cθ y M y ); 2) the same production technology and the same material (with equal E/Y ratio, E and Y indicating the material Young modulus and yield strength, respectively); 3) similar range of motion (i.e., ±45◦ rotation) before limit stress. Analogous arguments apply to the beam-like flexures depicted in Fig. 1(a) and (b). In such a case, however, large displacements can be achieved by either using materials with very low Young modulus or very long slender beams. Supposing anyway that the desired displacement can be guaranteed, a critical comparison of these four joints allows us to state that the corner filleted flexural hinge JR [see Fig. 1(b)] can be better suited for application as a single d.o.f. joint if compared to the circular cross-section joint JC [see Fig. 1(a)] which is usually employed as a connector in spatial compliant mechanism [30]. In the same way, it can be easily stated that joints JC and JR are designed with relatively simple shapes but high overall length, whereas the spiral joint SPIR [see Fig. 1(c)] and the helical joint HEL [see Fig. 1(d)] are characterized by lower overall length but rely on nontrivial morphologies in order to achieve the desired deformation. Still, it can be hard for designers to select between the four CJs without a deeper

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study of their behavior for what concerns undesired spatial motions. In order to overcome such difficulties, this paper outlines a comparison method for evaluating the joint’s sensitivity to parasitic effects. The method is based on the evaluation of a set of configuration-dependent, normalized, compliance matrices [31], [32]. Each matrix, which is computed in the neighborhood of a configuration of interest, is firstly split into two submatrices containing the coefficients relative to either linear or angular displacements. Then, a particular norm, namely the weighted Frobenious norm, is used to assign one scalar value to each submatrix, the resulting two scalars being referred to as translational, IT k , and rotational, IR k , local performance index (LPI). In particular, IT k = IT R k = 1 in the worst-case scenario, whereas an ideal CJ is characterized by IT k = 0 and IR k ∝ Cθ y M y . Finally, the overall CJ selective compliance is described by means of two global performance indices (GPIs), either translational or rotational, which are simply defined as the average of the LPIs. Note that, similar approaches have been proposed for evaluating the stiffness performance of multibody robotic systems (see, e.g., [33]). In the same manner, an optimization routine to design selectively compliant planar trusses has been proposed in [34]. Nonetheless, to the best of the author’s knowledge, a comparison metric for evaluating the CJ selective compliance and accounting for the peculiarity of large displacement CJs for robotic application is still missing. The proposed method is first validated by analytically comparing joint JC and JR in the small displacement range. Second, the behavior of the four CJs depicted in Fig. 1 is evaluated, all joints being analyzed by means of FEM and designed to achieve an overall 90◦ rotation. According to the computed translational and rotational GPIs (respectively ranging from 0.6244 to 0.9788, and from 0.6567 to 0.9903), joint HEL provides a better selective compliance when compared to joint JR , whereas joint SPIR outperforms any other considered CJ design. At last, possible applications for the production of better behaved robotic/prosthetic hands are presented and critically discussed. II. BASIC NOTATIONS As previously stated, most practical applications still require CJs which are designed to provide one d.o.f. only. In such a case, the CJ is conceived in order to allow a principal displacement along a desired reference direction when subjected to a principal load (torque or force) acting along the same direction. For instance, Fig. 2(a) shows the joint JR regarded as an elastic structure connecting two rigid links (link A and link B) in fixed-free boundary conditions. This CJ can act as a revolute joint by generating a rotational principal displacement 0 θ = [0 0 θy 0]T [see Fig. 2(b)] under the action of a principal load 0 m = [0 0 my 0]T due to the application of a force or moment at same point of link B (e.g., point f Op ). In the following, the left superscript of a vector or a matrix will denote the coordinate frame (CF) in which its components are expressed. The axis of rotation y0 is called compliant (sensitive) axis [35] and the ratio between 0 θy and 0 my is called principal compliance. Secondary displacements (also referred

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(which assumes linear elastic material and small deflections), the kinetostatic behavior of a CJ in the 3-D space can be deduced by the analysis of its compliance matrix [31], [32]. With reference to Fig. 2(a), given an external perturbation wrench Δp w whose components are expressed with respect to some coordinate frame CFp , and acting on a point of interest of body B, the incremental displacement vector (twist) Δp s is expressed as   p   p  p Cu f p Cu m Δ f Δ u p = = p C · Δp w · Δ s= p Δp θ Δp m Cθ f p Cθ m (1)

Fig. 2. Large displacement compliant joints: basic notations. The principal load and displacement are depicted with dotted line. (a) Reference frames. (b) Principal load 0 m y and secondary loads. (c) Principal displacement 0 θy and secondary displacements.

to as parasitic effects in [11]) along the other reference directions may occur in real applications for two reasons: 1) the presence of secondary loads (also referred to as parasitic loads in [11]) acting along those directions; 2) the presence of compliant axis drift (also referred to as parasitic motion in [20]) meaning that the compliant axis might be subjected to a spatial motion during CJ deformation even in the absence of secondary loads. Any ratio between any secondary displacement and any given load is referred to as secondary compliance. It should be noted that the presence of secondary loads is usually hardly predictable, as long as the compliant mechanism might interact with an unknown environment. On the other hand, deformations due to the principal load (usually the actuation force which is assumed as given) might be evaluated a priori, by means of simulation, and taken into account in the design and/or modeling process of the robotic device. In any case, being undesired motions, secondary displacements should be avoided or, at least, minimized. To this respect, given the loads, the amount of secondary displacements is strictly dependent on the CJ morphology: joints that behave similarly as to the principal displacement can behave quite differently in terms of secondary displacements. In particular, the property of a CJ to maintain a high compliance along the compliant axis and a high stiffness in every other directions is usually referred to as selective compliance [11].

where Δp s is composed of an incremental translation Δp u = [Δp ux Δp uy Δp uz ]T and an incremental rotation Δp θ = [Δp θx Δp θy Δp θz ]T , Δp w is composed of an incremental force Δp f = [Δp fx Δp fy Δp fz ]T and an incremental torque Δp m = [Δp mx Δp my Δp mz ]T , whereas p Cu f , p Cu m , p Cθ f , p Cθ m are 3-D matrices composed of entries with dimensions [m/N], [1/N], [rad/N], and [rad/Nm], respectively. As a consequence, p C ≡ p Cij is a 6 × 6 matrix with entries of nonuniform physical dimensions, the submatrices p CT = [p Cu f p Cu m ] and p CR = [p Cθ f p Cθ m ] relating the external wrench to the resulting translations and rotations, respectively. As for the frame dependency, it is well known that 6 × 6 adjoint matrices [31], [36] relate twists and wrenches, and thus compliance matrices, at different reference frames. In particular, let p Rq denote the rotation matrix of frame CFq with respect to frame CFp (i.e., the columns of p Rq are the unit vectors of frame CFq in the coordinated of frame CFp ). The position vector p rq , expressed in the frame CFp , locates the  = C(v) denote the cross product origin of frame CFq . Let v  u = v × u for any vector matrix, i.e., the matrix such that v u. The adjoint matrix p Tq is an augmentation of the rotation matrix and of the cross product matrix defined as ⎤  ⎡  p q T Rp 0 Rq 0 p ⎦ . (2) =⎣ q Tq = p  rq · p Rq p Rq (  rp · q Rp )T q RTp Given the adjoint matrix, the following relations hold: 

In order to compare CJs in terms of secondary displacements, it is necessary to understand how the joints react to external loads. Within the validity limits of the superposition principle





Δp u Δp θ

 = Tp

 =

TTp

(3a)

Δp m 

q



Δp f

q

Δq m

III. DESCRIPTION OF THE METHOD A. Compliance Matrix for Elastically Coupled Rigid Bodies: Background Theory

Δq f

Δq u Δq θ

 .

(3b)

Following from (3a) and (3b), once the compliance matrix C at some frame CFp [see Fig. 2(a)] is known, the compliance matrix related to frame CF0 can be calculated as p

0

p 0 −1 p T p p C = 0 T−T p · C · Tp = T0 · C · T0 .

(4)

BERSELLI et al.: ENGINEERING METHOD FOR COMPARING SELECTIVELY COMPLIANT JOINTS IN ROBOTIC STRUCTURES

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Fig. 3. Graphical representation of compliance matrices for joint JC (a) and for an ideal revolute CJ, Ji d (b). Index I for rows and index J for columns. (a) Joint JC . (b) Ideal joint Ji d .

Fig. 5. COR determination for a generic CJ morphology. (a) Overall COR for a large displacement CJ. (b) Corresponding 1 d.o.f. model. Fig. 4.

Ellipsoids related to the each submatrix defined in (5).

The matrix 0 C can be, once again, split into four submatrices  0 Cu f 0 Cu m 0 C= 0 Cθ f 0 Cθ m ⎡0

C11

⎢ 0C ⎢ 21 ⎢ ⎢ 0 C31 ⎢ =⎢ ⎢ 0 C41 ⎢ ⎢0 ⎢ C51 ⎣ 0 C61

0

C12

0

C13

0

C14

0

C15

0

0

C22

0

C23

0

C24

0

C25

0

0

C32

0

C33

0

C34

0

C35

0

C42

0

C43

0

C44

0

C45

0

C52

0

C53

0

C54

0

C55

C62

0

C63

0

C64

0

C65

0

C16



C26 ⎥ ⎥ ⎥ 0 C36 ⎥ ⎥ ⎥ (5) 0 C46 ⎥ ⎥ ⎥ 0 C56 ⎥ ⎦ 0

C66

which are related to a reference frame located on the point envisaged as the CJ center of rotation, whose y-axis is envisaged as the CJ compliant axis. Concerning this particular frame, recalling that the term 0 C55 = Cθ y M y relates the rotation around the principal axis y0 to the action of the principal load 0 my , an ideal revolute CJ, which is expected of pure rotation along its compliant axis even in the presence of secondary loads, will present a compliance matrix 0 C where only the term 0 C55 is finite, all the other terms being null. This situation is perfectly modeled by means of a 1 d.o.f. pseudorigid approximation [12]. On the contrary, a real joint will exhibit finite values of compliance coefficients also along the other directions. A graphical 3-D bar representation of compliance matrices 0 Cu f , 0 Cu m , 0 Cθ f , 0 Cθ m for joint JC and for an ideal revolute CJ, Jid , is shown in Fig. 3. Note that, for comparison purposes, JC is sized in order to provide the same principal compliance as Jid . It is also interesting to notethat each of the aforementioned matrices 0 Cu f , 0 Cu m , 0 Cθ f , 0 Cθ m can be interpreted as an operator that maps a unit force (or torque) sphere (e.g., [0 fx 0 fy 0 fz ]T  = 1) into an ellipsoid of the corresponding incremental displacement (either translational or rotational). The direction of each ellipsoid axis may be found as the eigen-

vector of the related submatrix whereas the application of a unit force (or torque) directed along one of those axis results in a displacement along the same direction equaling the corresponding eigenvalue [37]. For the purpose of the following sections, let one denote as P C u f , P C u m , P C θ f , and P C θ m , the direction of the ellipsoid’s major axis related to each of the submatrices previously defined in (5). In any case, being a differential operator [32], the compliance matrix (and, consequently, the related ellipsoids) measures a local property which depends on the joint’s configuration. Therefore, it is clear that ◦ C cannot be directly used as a comparison metric. First of all, because a unique compliance matrix is not enough to describe the behavior of a large displacement joint within its whole workspace. Second, because the elevate number of elements (namely, 36 elements) composing 0 C does not allow an easy and synthetic comparison of different CJs. Therefore, in order to define a comparison method, the following steps are proposed: 1) Definition and discretization of the CJ workspace. 2) Subsequent evaluation and normalization of the compliance matrix in each joint configuration. 3) Definition and evaluation of LPIs which characterize the CJ’s performance in terms of selective compliance. 4) Definition and evaluation of GPIs which summarize the overall joint performance. B. Definition of the Joint Workspace With reference to Fig. 5, let one suppose, without loss of generality, that the rigid link B is loaded with a bending moment mN at the free end until some limit stress is reached at some point of a generic CJ structure (e.g., until the Von Mises stress equals the material yield or fatigue stress [38]). This condition will be referred to as the maximum CJ deflection. In order to model the CJ as 1 d.o.f. kinematic pair [see Fig. 5(b)], it is necessary to locate a point which approximates

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Fig. 6.

IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 19, NO. 6, DECEMBER 2014

Workspace discretization. (a) JC , multiple-axis flexural hinge. (b) JR , corner filleted flexural hinge. (c) SPIR, spiral joint. (d) HEL, helical joint.

the center of rotation (COR) of link B with respect to link A (fixed end), hereafter referred to as effective COR. Among the different approaches presented in the past literature [11], [12], a possible method is to assume that the rigid link B undergoes a planar motion under the application of a principal load only. Such condition generally holds for well-designed 1 d.o.f. CJs mainly for symmetry reasons (e.g., the CJs depicted in Fig. 1 are symmetric along the xz plane). Under these assumptions, it is possible to identify the motion of the instantaneous COR (i.e., the herpolhode) by intersecting the normal to the trajectories of two points belonging to link B. With reference to Fig. 5(a) and similarly to [39], let one consider the positions of point f P = [f Px f Py f Pz ]T and point f Q = [f Qx f Qy f Qz ]T for a CJ in its uninflected and maximally deflected conditions (the points being indicated as f P1 , f Q1 or f PN , f QN , respectively). In addition, let one consider the corresponding displacements Δf ux (P) = Δf ux (L, 0, H/2), Δf uz (P) = Δf ux (L, 0, H/2), Δf ux (Q) = Δf ux (L, 0, −H/2) known, for instance, via FEM analysis. The position of the effective COR, f O0 , will be given by f

O0 ⎡

Ws . Such workspace is discretized into a finite number N of angular positions 0 θy k , where k = 1, . . . , N . As an example, the Ws discretization of the CJ depicted in Fig. 1 is shown in Fig. 6. Obviously, this spatial subdivision involves a corresponding loading subdivision, so that the initial position (0 θy 1 = 0), an intermediate position (0 θy k ), and the final position (0 θy N ) are reached for 0 m1 = 0, 0 < 0 mk < 0 m and 0 mN = 0 mm ax , respectively. Note that, in case the CJ is loaded by an actuation force other than pure bending moment, the CJ COR and the considered configurations will change accordingly (see, e.g., [22]), the rest of the evaluation method proposed in the following remaining unaltered. Finally, it should be underlined that the overall motion of link B is practically limited by the strength of the employed CJ material and by the possible self-collision of the joint or links surfaces. Therefore, Ws could also be defined as that portion of space reachable by some point of interest of link B, the CJ deformation being constrained by surface collision avoidance and/or maximum bearable stress. Nonetheless, this definition involves the evaluation of a very large number of CJ configurations which are reached only in case of improper functioning.

⎤ C. Evaluation of Normalized Compliance Matrices Δf ux (P) + Δf ux (Q) HΔf uz (P) +L ⎥ 2 ⎢ Δf ux (P)−Δf uz (P) + Once the workspace discretization is defined, it is proposed ⎢ ⎥ ⎥ to evaluate a local compliance matrix 0 Ck for each joint’s con=⎢ 0 ⎢ ⎥ ⎣ HΔf u (P) Δf ux (P) · Δf ux (Q) Δf uy (P) ⎦ figuration via the following steps: x − + f 2 1) Referring to a frame CFk attached to body B [CFk ≡ Δf ux (Q )−Δ f u x (P) 2 Δ uz (P) CFp , f Op ≡ f Ok in Fig. 2(a)], a fraction of the maximum (6) principal load 0 m is applied to link B (on point f Ok ) in where L is the CJ uninflected length [see Fig. 1 and 5(a)] and H order to reach the kth configuration (k = 1, . . . , N ). is the distance between points P and Q which remains unaltered 2) Referring to the same frame CFk , a small variation of one secondary load component is applied on point f Ok , during CJ deflection. After the definition of an effective COR, while maintaining the principal load previously set. Then, the CJ maximum workspace Wm ax is defined as the principal displacement range [0 θy ,m in ÷ 0 θy ,m ax ] which can be achieved the generated displacements k Δs can be measured and by the joint before limit stress, when subjected to a principal used for the calculation of the six components of k Ck , which are related to this particular loading condition (i.e., load of adequate intensity. kth CJ configuration) and this particular frame CFk . A Naturally, the CJ operative workspace Ws might be a subset of Wm ax , the position of point f O0 varying accordingly. load variation is said to be small if it generates a small Let then suppose that the analysis of a discrete number of joint’s displacement [32]. As a general rule of thumb, recalling configurations within its workspace is sufficient to obtain an estithat all CJs share the same principal compliance, the ormation of the joint’s compliance behavior. Hence, the workspace der of magnitude of any disturbance moment might be is divided into a finite set N of CJ configurations to be further supposed equal to the order of magnitude of the principal analyzed. For instance, with reference to the previous definiloads which generates a principal rotation of about 5% tion, if the joint can achieve a rotation of 0 θy = ±45◦ at the the principal displacement range (similarly to [40]). In the free end before yielding, when subjected to a proper moment same manner, any disturbance force applied on the point 0 mm ax , then 0 θy ∈ [±45◦ ] coincides with the CJ workspace Ok (at a distance d from O0 ) generates on point O0 a

BERSELLI et al.: ENGINEERING METHOD FOR COMPARING SELECTIVELY COMPLIANT JOINTS IN ROBOTIC STRUCTURES

From (10) and (11), it follows that a dimensionless compliance matrix can be defined as ⎤ ⎡ 0  um Cu f 0 C 0 ⎦ C = ⎣0  θm Cθ f 0 C

moment of the same order of magnitude of any other disturbance moments. For comparison purposes, referring to (3a), the perturbation forces and torques are chosen such that Δ0 f  and Δ0 m are equal for each considered CJ and each considered configuration. 3) The procedure is repeated for every secondary load component, allowing to calculate the whole joint’s compliance matrix in the kth configuration as (7), shown at the bottom of the page, where for instance, Δk ux (Δk fy ) denotes the displacement Δk ux caused by the applied force Δk fy whereas all the other matrix entries are similarly defined with obvious notation of symbols. 4) The compliance matrix related to CF0 are finally derived by applying (4). As previously stated, each 0 Ck has entries of nonuniform physical dimensions. Therefore, in order to reason on dimensionless quantities, the conditions of maximum possible incremental displacement and rotation within the analyzed CJ configurations are evaluated and used as a comparison metric. Resorting to the ellipsoids concerning each of the four submatrices defined in (5), let one first suppose that an incremental force and an incremental torque, with moduli Δ0 f  and Δ0 m, are applied along the directions P C u f and P C u m . The modulus of the resulting incremental displacement will be Δ0 sm ax  = 0 λu f Δ0 f  + 0 λu m Δ0 m

⎡ ⎢ =⎢ ⎣

where 0 λθ f and 0 λθ m are the maximum eigenvalues of matrices Cθ f and 0 Cθ m , respectively. By using these maximum parasitic effects as a comparison metric, the following vectors of dimensionless displacements and rotations can be computed:

(11)



Δk ux (Δk fx )/Δk fx

⎢ k ⎢ Δ uy (Δk fx )/Δk fx ⎢ k Ck = ⎢ .. ⎢ . ⎣ Δk θy (Δk fx )/Δk fx

Cu f

Δ0 f  0 Cθ f Δ0 θm ax 

⎤ Δ0 m 0 C um ⎥ Δ0 sm ax  ⎥. ⎦ 0 Δ m 0 C θ m Δ0 θm ax 

(12)

0 0

 Tk = 0 C i,j , i = 1, 3; j = 1, 6 C

 Rk = 0 C i,j , i = 4, 6; j = 1, 6. C

Naturally, an increased accuracy of the method can be obtained by increasing the number N of analyzed joint configurations. At last, note that due to the small magnitude of secondary loads, it is supposed that the CJ is subjected to large principal displacements but small secondary displacements. This hypothesis seems reasonable for a number of reasons. First of all, CJs are usually designed to withstand a large principal displacement but would simply fail in case of large secondary displacements. In practice, the designer should contemplate the presence of proper mechanical stops (or CJ surfaces’ self-collision) if large secondary displacements are expected. In addition, it is reasonable to think that the compliance trend (assumed constant through the entries of the compliance matrix) would also at least “approximate” the CJ behavior for rather large secondary displacements. If all these hypotheses do not hold, recalling the possible definition of CJ workspace outlined in Section III C, the overall portion of space practically reachable by some point of interest of link B should be checked, the proposed evaluation method simply becoming more onerous in terms of computational time.

0

Δ0 θ Δ0 f Δ0 m 0 0 = + . C C θ f θ m Δ0 θm ax  Δ0 f  Δ0 m

m ax 

0

(13)

(9)

(10)

Δ 0 s

0  CTk 0 , where Ck = 0  CRk

where 0 λu f and 0 λu m are the maximum eigenvalues of matrices 0 Cu f and 0 Cu m as computed over the k configurations. Note that, due to the small perturbation assumption, the effect superposition principle can be applied. Similarly, let one then suppose that an incremental force and an incremental torque, with moduli Δ0 f  and Δ0 m, are applied along the directions P C θ f and P C θ m . The resulting incremental rotation will be

Δ0 f Δ0 m Δ0 s 0 0 = + C C u f u m Δ0 sm ax  Δ0 f  Δ0 m

Δ0 f 

 k can be evalTherefore, a set of normalized local matrices 0 C uated for each CJ configuration and subsequently split into two  Tk and 0 C  Rk , containing respectively the cosubmatrices, 0 C efficients relative to linear and angular displacements along the reference directions of CF0 :

(8)

Δ0 θm ax  = 0 λθ f Δ0 f  + 0 λθ m Δ0 m

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D. Definition and Evaluation of Local and Global Performance Indices  k has been evaluated for Once a local compliance matrix 0 C the configurations of interest, a set of LPIs which are useful to compare different CJs can be defined. The proposed indices are ⎤

Δk ux (Δk fy )/Δk fy

···

Δk uy (Δk fy )/Δk fy

···

.. .

···

⎥ Δk uy (Δk mz )/Δk mz ⎥ ⎥ ⎥ .. ⎥ . ⎦

Δk θy (Δk fy )/Δk fy

···

Δk θy (Δk mz )/Δk mz

Δk ux (Δk mz )/Δk mz

(7)

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based on the weighted Frobenius norm, that is   m 1 n  1 tr(AAT ) |aij |2 = AF = n i=1 j =1 n

as follows: IT k , i d = IT g , i d = 0 

(14)

IR k , i d = IR g , i d =

where A is a generic n × m matrix (with n < m). Specifically, a translational LPI, IT and a rotational LPI, IR are defined as  Tk F IT k = 0 C

(15a)

 Rk F . IR k = 0 C

(15b)

A smaller LPI indicates a higher local selective compliance, i.e., lower secondary displacements for a given perturbation wrench. As it is known, the weighted Frobenius norm yields the rms value of the matrix eigenvalues and it has been suggested as the preferred norm also in [33], when quantifying the elastostatic performance of serial and parallel manipulators. Obviously, once the compliance matrix is normalized, one could possibly adopt any norm to intuitively evaluate how large the matrix entries are. Nonetheless, as discussed in the same reference [33], some other norm would be misleading. For example, by using the 2-norm (defined as the maximum matrix eigenvalue) the CJ would be evaluated on the basis of the worst-case scenario only. Also, if the CJ compliance matrix is known in its analytic form (see, e.g., Section IV A), (15a) and (15b) return an analytic function. At last, when compared to other possible norms, the Frobenius norm has the useful property of being invariant under rotations. The final step of the comparison approach consists in the definition and evaluation of GPIs, which summarize the overall joint’s performance in terms of selective compliance, over the whole workspace Ws N IT g =

i=1

IT k

N

N

i=1

IR k

(16a)

. (16b) N A smaller GPI indicates a higher global selective compliance. Note that 1) small secondary rotations at joint level can be dramatically amplified at the end of serial articulated chains, hence the evaluation of IR g is usually more significant; 2) the definition of global indices is appreciable in terms of synthesis. Still, the analysis of the LPI’s trend within the workspace can capture dangerous effects (e.g., load stiffening), which might be hidden when analyzing the GPIs only. Naturally, the proposed method considers one among many CJ figure of merits [20], that is selective compliance. If several CJs with different geometries show identical (or very similar) LPI’s trend, the designer should choose on the basis of other figures of merit, such as cost-effectiveness, ease of manufacturing, available manufacturing technology. Finally, note that, according to (15a), (15b), (16a), and (16b), the local and global performance indices concerning an ideal CJ in any configuration, IT k , i d , IR k , i d , IT g , i d , and IR g , i d , respectively, can be computed IR g =

(17) 1 Δ0 m Cθ y M y . 3 Δ0 θ m ax 

(18)

IV. EVALUATION AND PLAUSIBILITY CHECK A. Beam-Like Flexures in the Small Displacement Range Let one first consider the joints JC and JR . The compliance matrix concerning beam-like flexures within the small displacement assumption is known from the literature [38], [41] ⎤ ⎡ L 0 0 0 0 0 ⎥ ⎢ AE ⎥ ⎢ ⎢ 3 2 ⎥ L L ⎥ ⎢ 0 0 0 0 ⎥ ⎢ 3EIz 2EIz ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ L3 L2 ⎢ 0 0 0 − 0 ⎥ ⎥ ⎢ 3EIy 2EIy ⎥ ⎢ 1 ⎥ ⎢ C=⎢ ⎥ L ⎥ ⎢ 0 0 ⎥ 0 0 ⎢ 0 GJ ⎥ ⎢ ⎥ ⎢ 2 ⎥ ⎢ L L ⎢ 0 0 − 0 0 ⎥ ⎥ ⎢ 2EIy EIy ⎥ ⎢ ⎥ ⎢ ⎣ 2 L ⎦ L 0 0 0 0 2EIz EIz (19) where concerning the examined beams, E, G, L, A, Iz , Iy , J are, respectively, modulus of elasticity, shear modulus, length, cross-sectional area, moment of inertias about the z- and yaxis, and polar moment of inertia about the yz plane. Within the assumption of small displacements, the effective COR coincides with the geometric center of the flexible hinge, i.e., f O0 = [L/2, 0, 0]T [refer to Fig. 1(c) and (d)]. By noticing that 1 r0 = [−L/2 0 0]T and that 1 R0 = I, the adjoint matrix 1 T0 is given by ⎡ ⎤ 1 0 0 0 0 0 1 0 0 0 0⎥ ⎢0 ⎢ ⎥ 0 1 0 0 0⎥ ⎢0 1 T0 = ⎢ (20) ⎥. 0 0 1 0 0⎥ ⎢0 ⎣ ⎦ 0 0 L/2 0 1 0 0 −L/2 0 0 0 1 Hence, the compliance matrix related to CF0 is given by 0

C = 1 TT0 · 1 C · 1 T0 =   L3 L L L3 L L , , = diag (21) , , , AE 12EIz 12EIy GJ EIy EIz

which highlights how the compliance matrix becomes diagonal if expressed in this particular reference frame. In the small displacement range, the point O0 is also usually denoted as the center of stiffness (see, e.g., [31]). Let one then consider a joint JC with cross-section radius r (i.e., A = πr2 , Iy = Iz = πr4 /4, J = πr4 /2). The rotational and translational performance indices, IT J C and IR J C can be

BERSELLI et al.: ENGINEERING METHOD FOR COMPARING SELECTIVELY COMPLIANT JOINTS IN ROBOTIC STRUCTURES

calculated by means of (21), (15a), and (15b) Δ0 f  L  4 9r + 2L4 IT J C = Δ0 sm ax  9πr4 E 2 Δ0 m L  2 8G + E 2 . IR J C = 3π Δ0 θm ax  Gr4 E

(22) (23)

In the same manner, let one consider a joint JR with cross section thickness h and width b (i.e., A = bh, Iy = bh3 /12, Iz = b3 h/12, J = (h2 + b2 )bh/12). The rotational and translational performance indices, IT J R and IR J R , are given by 1 Δ0 f  L  4 4 b h + (b4 + h4 )L4 0 3 3 Δ sm ax  b h3 E  Δ0 m L b4 + h 4 1 =4 0 + . Δ θ m ax  hb G2 (h2 + b2 )2 E 2 b4 h 4

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IT J R =

(24)

IR J R

(25)

The ratio of those indices, which can be easily used for comparison purposes, is given by  IT J R 3πr4 b4 h4 + (b4 + h4 )L4 = 3 3 (26) IT J C b h 2L4 + 9r4  IR J R 6πr4 GE G−2 (b2 + h2 )−2 + E −2 (b−4 + h−4 ) = . IR J C bh 8G2 + E 2

Fig. 7. h.

Values of IT J /IT J R

C

and IR J /IR J R

C

as function of JR thickness,

Fig. 8. Model meshes. (a) JC , multiple-axis flexural hinge. (b) JR , corner filleted flexural hinge. (c) SPIR, spiral joint. (d) HEL, helical joint.

(27) In particular, beam-like flexures can be designed on the basis of desired principal compliance Cθ y M y , and desired principal rotation 0 θy N . Defining Y as the material yield strength and resorting to basic beam equations [38], the following relations hold: L (28) Iy = ECθ y M y  0 θy N cE c=r for JC where (29) L≥ c = h/2 for JR . Y As an example, if the adopted material is Nylon (E = 1300 MPa, G = 500 MPa, Y = 45 MPa), a JC joint having 0 θy = ±45◦ and Cθ y M y = [8.70] rad/Nm can be uniquely sized by means of (28) and (29), the resulting joint dimensions being L = 30mm and r = 1.355 mm. On the other hand, (28) and (29) are not sufficient to unambiguously determine the geometry of a JR joint, which is fully defined once the beam length L, and the rectangular cross section thickness h and width b are chosen. Such JR joint can then be sized resorting to the index ratios of (26) and (27). In particular, a useful relation between the beam thickness h and the index ratios can be found by imposing beam length and desired principal compliance (i.e., L = 30 mm and Cθ y M y = [8.70] rad/Nm as in the previous case) and by introducing the beam width b found via (28) (i.e., b = 12L/(Eh3 Cθ y M y )) into (26) and (27). The resulting values of IT J R /IT J C and IR J R /IR J C as function of h are reported in Fig. 7, which trivially shows that a smaller beam height and a larger beam width are preferable in terms of selective compliance. For instance, if the beam thickness is h = 1.80 mm, and consequently the beam width is b = 5.40 mm, application of

(26) and (27) return a value IT J R /IT J C = 0.7157 and a value IR J R /IR J C = 0.5457, confirming that a well-designed JR outperforms the JC selective compliance. B. Joint Evaluation in the Large Displacement Range The method is finally tested by comparing the four CJs depicted in Fig. 1 within the large displacement range. A commercial FEM software (ANSYS release 13.0) is used for both optimally sizing the joint and for the numerical evaluation of the various compliance matrices. In particular, all the CJs are designed in order to withstand a principal displacement 0 θy = ±45◦ and to provide the same principal compliance Cθ y M y = [8.70] rad/Nm. For what concerns the FEM model, the ANSYS meshing tool has been used for mesh generation, only the strictly deformable part of each CJ being modeled so as to reduce the computational cost (see Fig. 8). The elements chosen for the analysis are SOLID186 for JC and JR , and SOLID187 for SPIR and HEL [42], i.e., higher order elements exhibiting a quadratic displacement behavior and having either 10 or 20 nodes. Both SOLID186 and SOLID187 are well suited to modeling irregular meshes and implement large deflection and large strain capabilities. With regards to solution controls, a large displacements static analysis is performed, employing the sparse direct equation solver, and a mesh convergence control is carried out concerning the configuration of maximum deflection. As for joints JC and JR , they can be sized by means of (28) and (29). The resulting dimensions (and employed material) are the same as the ones reported in the previous section (i.e., L = 30 mm, r = 1.355 mm, h = 1.80 mm, b = 5.40 mm).

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Fig. 9. Large displacement CJs at the configuration of maximum negative deflection and under the application of perturbation wrench (negative simple torsion). Plot of total deformation (units in m). (a) JC , multiple-axis flexural hinge. (b) JR , corner filleted flexural hinge. (c) SPIR, spiral joint. (d) HEL, helical joint. (e) JC , multiple-axis flexural hinge. (f) JR , corner filleted flexural hinge. (g) SPIR, spiral joint. (h) HEL, helical joint.

As for the joints SPIR and HEL, they have been conceived for usage in articulated robotic fingers having size comparable with the human finger’s articulation. Therefore, they can fit in a cube having dimensions 14 × 10 × 10 mm3 (L = 14 mm). The spiral joint SPIR is based on a couple of planar spiral springs, whereas the helical joint HEL is based on a couple of classical helical springs with longitudinal axis parallel to the principal displacement’s direction. The adopted material is Fullcure 720, a polymeric resin suitable for rapid prototyping (E = 1214 MPa, G = 1078 MPa, Y = 60 MPa). The material Young Modulus has been taken from the literature [43], whereas the yield strength is taken from the material datasheet [44]. The joints’ cross sections have been sized with the aid of the ANSYS shape optimization routines [42], in order not to exceed a limit Von Mises stress within the CJ structure and to guarantee the desired compliance. The detailed dimensions of SPIR and HEL are reported in the Appendix section. Given the CJ dimensions, as previously stated, each CJ configuration has been reached by simply applying a principal load of adequate intensity to the initial meshed model depicted in Fig. 8, the workspace being divided into N = 5 steps: −45◦ , −22.5◦ , 0◦ (undeformed configuration), +22.5◦ , +45◦ (the positive sign refer to clockwise rotations, Fig. 6). The effective COR positions are f O0 = [10.68 0 1.38]T mm, f O0 = [10.84 0 2.21]T mm, f O0 = [6.72 0 1.99]T mm, f O0 = [5.06 0 − 4.63]T mm for JC , JR , SPIR and HEL, respectively (see Fig. 6). As an example, Fig. 9 shows the total deformation plots (units in m) for each CJ at the configuration of maximum negative deflection and when subjected to a parasitic load (negative torsion) in the uninflected configuration. With reference to Figs. 6 and 8, the perturbation wrench is set to 0.1 N, concerning each incremental force, and 1 N · mm concerning each incremental torque, and it is applied to point

f

Ok , an external node with both rotational and translational d.o.f (termed pilot node [42]) which is rigidly linked to the freeend surface of the CJ. This pilot node is located at a distance d = 35 mm from f O0 and allows a rigid transmission of the wrench to the nodes of the meshed structure. The local compliance matrices k Ck are calculated via (7) by knowing the forces/moments applied to the pilot node and by measuring the pilot node’s displacements in ANSYS. The maximum stress at 0 θy = −45◦ are σ ≈ 45 MPa, σ ≈ 31 MPa, σ ≈ 58MPa, and σ ≈ 53MPa for JC , JR , SPIR, and HEL, respectively. The compliance matrices, before and after normalization, related to 0 θy = 45◦ are reported in Fig. 10 using a bar graph representation. These graphs already allow a CJ comparison. For instance, let one consider joints JR and HEL, together with their four submatrices before normalization, 0 Cu f , 0 Cu m , 0 Cθ f , 0 Cθ m , as depicted in Fig. 10(c) and (g), respectively. A careful inspection allows us to state that joint JR should be preferred to HEL as long as each HEL secondary compliance is higher than the corresponding JR secondary compliance (at least for what concerns the maximally deflected configuration). In addition, for a given small perturbation wrench (refer to Section III-C for the definition of small wrench), the graphs concerning the normalized compliance matrices, namely Figs. 10(b), (d), (f), and (h), allow us to readily identify the major sources of parasitic motion. For instance, considering once again joint JR , Fig. 10(d) allows us to state that the highest parasitic displacement is caused by a force Δ0 fx , rather than any other load type. Similar arguments apply to the other CJ designs. Nonetheless, a comparison accounting for several configurations without the aim of concise indices seems extremely impractical. Hence, following previous discussions, the translational and rotational LPI trends are computed. The data obtained from the analysis [respectively, normalized with respect to IT J C and

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Fig. 11.

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Local performance index (LPI) trends for JC , JR , SPIR, and HEL. TABLE 1 GLOBAL INDICES FOR JC , JR , SPIR, AND HEL

Fig. 10. Graphical representation of the compliance matrices (a,c,e,g) and normalized compliance matrices (b,d,f,h) for the four joints at their maximally  - Joint JC . (c) 0 C - Joint deflected configuration. (a) 0 C - Joint JC . (b) 0 C 0 0 0   JR . (d) C - Joint JR . (e) C - Joint SPIR. (f) C - Joint SPIR. (g) 0 C - Joint  - Joint HEL. HEL. (h) 0 C

IT J C from (22) and (23)] are reported in Fig. 11, which also includes the constant trend of IR k , i d for clarity. This figure allows to clearly compare the CJs selective compliances. In addition, concerning the single joint design, Fig. 11 allows to identify the CJ configuration at which an LPI is maximum. For instance, joint SPIR is characterized by a lower selective compliance for 0 θy = 45◦ rather than for 0 θy = −45◦ , suggesting that positive deflections should be given preference if possible. Finally, the

translational and rotational GPIs computed from (16a) and (16b) are reported in Table I. The results confirm that JC is characterized by a lower selective compliance, which seems reasonable as long as any line contained in the yz plane defines a compliant axis. In fact, as stated before, joint JC is usually employed as a spherical pair in spatial compliant mechanisms. As for the overall CJ comparison, SPIR outperforms every other CJ, whereas JR should be preferred to HEL. Note that 1) being based on the computation of configurationdependent compliance matrices, the proposed method can be useful when building lumped-parameter dynamic models of compliant mechanisms with concentrated compliance (see, e.g., [45]); 2) the computation of the center of stiffness is straightforward only in the absence of a preload. Hence, differently from what happens in small displacements (see Section IV A), the mere definition of a center of stiffness is still controversial within the large displacements range (the interested reader is addressed to [32], [46], [47]). Hence, the authors simply preferred to refer the compliance matrices to the CJ’s COR which has a clear and concrete engineering interpretation. C. Application Scenario As an application scenario, let one consider the design optimization of a SPIR joint, which has been proven as the most promising CJ morphology among the considered ones. At first, for varying spiral height hs and thickness bs [see Fig. 8(c)], the ANSYS optimization tool [42] is used to automatically determine possible design solutions having the same principal compliance value (imposed by the user as Cθ y M y = [8.70] rad/Nm).

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Fig. 12.

IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 19, NO. 6, DECEMBER 2014

SPIR GPIs, IR g , and IT g , at varying spiral thickness, h s .

Fig. 14. Experimental and numerical (FEM) force–displacement profiles concerning principal p fz , and secondary p fy , bending forces, and compression force p fx (shown in modulus).

Fig. 13. Experimental setup. (a) Application of principal bending force p fz . (b) Application of compressive force p fx .

The simulations performed in Section IV-B are then carried out for each of the computed hs − bs pairs and the resulting GPI values are evaluated. In particular, similarly to Fig. 7, the plots in Fig. 12 report the SPIR joint translational IT g and rotational IR g GPIs as function of the spiral height hs . According to these graphs, provided that the desired principal rotation can be achieved without exceeding the material yield strength, solutions with smaller hs values (and consequently higher bs values) should be preferred. As a further check, in order to assess the validity of the FEM simulations, a SPIR joint prototype (whose dimensions are shown in the Appendix section) has been placed on a testing machine (see Fig. 13). A controlled displacement in the x-, y-, and z directions (see Fig. 1(c) for the axis definition) is then applied to a point located at a distance d = 35mm from f O0 (similarly to the pilot node located on point f Ok during the FEM simulation). Fig. 14 reports the experimental and numerical force–displacement curves confirming the reliability of the FEM simulations (relative errors within 10%). Finally, in order to show the practical usability of the proposed joints, SPIR and HEL have been employed in the production of the robotic fingers shown in Fig. 15(a) and (b). Also beam-like CJs with rectangular cross section (alike JR ) have been employed for the same application (see Fig. 15(c) and

Fig. 15. Robotic/prosthetic hands: 3-D model and finger prototypes. (a) Robotic hand employing SPIR joint. (b) Mono-piece finger prototype employing both SPIR and HEL joints. (c) Robotic hand employing JR -like joint. (d) Monopiece finger prototype employing JR -like joint. Joint dimensions can be found in [8].

(d) [8]). These serial compliant mechanisms are manufactured as a single piece by either additive manufacturing techniques or CNC milling. In particular, the production of well-behaved monolithic fingers seems a promising way to reduce costs and increase ease-of-assembly of robotic/prosthetic devices once the CJ morphology has been carefully considered. V. DISCUSSION Commonly adopted CJs work in the small displacements range whereas many robotic structures require large displacement capabilities. For instance, revolute CJs to be applied in robotic fingers should perform an overall 90◦ rotation, which is more than ten times the maximum range required in usual applications of compliant mechanisms. Generally, joints capable of large displacements might be obtained by increasing the extension of the region occupied by elastic material subjected to an imposed deformation (e.g., very long slender beams). Still, such solution inevitably increases the

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presence of secondary displacements which represent a critical factor against the application of any type of CJ. In fact, the presence of undesired motions at joint level can generate severe trajectory errors, thus compromising task accomplishment. In addition, small secondary displacements can be dramatically amplified at the end of serial articulated chains. In practice, the difficulties in designing/modeling/evaluating the joint behavior, together with high sensitivity to secondary loads, advise against the use of CJ in robotics as long as more reliable solutions might be easily available employing either traditional kinematic pairs or contact-aided tendon-based joints, similarly to the biological articulations. This point of view can change whenever affordability/safety are considered a value (as in the field of prosthetics) and task compatibility is demonstrated. As to task-compatibility, it means that the presence of undesired displacements can be limited and, in any case, it does not compromise the capability to satisfactorily perform the task. Considering, for instance, robotic and prosthetic hands with articulated fingers, during the approach-to-object trajectory the fingers behave as serial independent chains with very limited loads. In this situation, the trajectory errors due to secondary displacements are quite acceptable. After the application of the contacts, the whole fingers-object system can be conceived as a parallel structure [48] where fingers contribute to the overall grasp stability and robustness according to their different placement with respect to the external load, thus mitigating the effects of the single joint’s secondary displacements. These qualitative observations suggest that it is fundamental to define design methods, as the one proposed in this paper, that helps selecting the CJ morphology which can provide the highest selective compliance.

VI. CONCLUSION A novel evaluation criterium for comparing large displacement CJs has been proposed. The method allows us to quantitatively discern the best selectively compliant joint among a list of equally valuable design solutions in terms of principal compliance, range of motion, employed material, and manufacturing technology. The criterium is based on the evaluation of two local and global indices which quantify the joint selective compliance. In particular, finite element analysis has been used for the determination of the both joint kinematical behavior and compliance matrix. Then, the obtained compliance matrices have been normalized and evaluated by means of the Frobenius norm, which provides an intuitive and synthetic measure of the matrix entries’ magnitude. The plausibility of the approach has been checked by comparing two beam-like flexures whose properties are known from the literature and two novel CJs with nontrivial morphology. Finally, a case study simulation has shown how the proposed indexes can be used as a design constraint in order to optimally size a given joint morphology. Naturally, being based on the finite-element computation of configurationdependent compliance matrices, the proposed method can be employed for the comparison of generic joint structures.

Fig. 16.

SPIR and HEL dimensions. (a) Joint SPIR. (b) Joint HEL.

APPENDIX Fig. 16(a) and (b) contains a detailed description of SPIR and HEL dimensions. Please note that dimensions are given with four digits in order to fully replicate the CJ compliant behavior. ACKNOWLEDGMENT We acknowledge the contribution of Dr. M. Piccinini. REFERENCES [1] G. Ananthasuresh and S. Kota, “Designing compliant mechanisms,” Mech. Eng., vol. 117, pp. 93–6, 1995. [2] R. Gouker, S. Gupta, H. Bruck, and T. Holzschuh, “Manufacturing of multi-material compliant mechanisms using multi-material molding,” Int. J. Adv. Manuf. Tech., vol. 30, pp. 1049–1075, 2006. [3] I. Gibson, D. Rosen, and B. Stucker, Additive Manufacturing Technologies: Rapid Prototyping to Direct Digital Manufacturing. Boston, MA, USA: Springer, 2010. [4] S. Kota, J. Jinyong, S. Rogers, and J. Sniegowski, “Design of compliant mechanisms: Applications to MEMS,” Analog Integr. Circuits Signal Process., vol. 29, pp. 7–15, 2001.

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Giovanni Berselli received the Laurea degree (cum laude) in mechanical engineering from the University of Modena and Reggio Emilia, Modena, Italy, and the Ph.D. degree in mechanics of machines from the University of Bologna, Bologna, Italy, in 2004 and 2009, respectively. He is currently an Assistant Professor at the University of Modena and Reggio Emilia, where he teaches the course Design of Automatic Machinery and Industrial Robots. He was a Contract Professor at the University of Bologna and a Research Assistant with the Department of Mechanical Engineering, Monash University, Melbourne, Australia, and with CEIT, Escuela Superior de Ingenieros de la Universidad de Navarra, San Sebastian, Spain. His research activity is focused on the development of engineering methods for sustainable robotics, and on the design of compliant robotic systems.

BERSELLI et al.: ENGINEERING METHOD FOR COMPARING SELECTIVELY COMPLIANT JOINTS IN ROBOTIC STRUCTURES

Alessandro Guerra received the Laurea degree (cum laude) in mechanical engineering and the Ph.D. degree in high mechanics and automotive design and technology from the University of Modena and Reggio Emilia, Modena, Italy, in 2009 and 2013, respectively. He is currently a Senior Researcher with the R&D Department, Comer Industries S.p.A., Italy. He is or has been involved in several national projects and collaborations with industry. His main research activities concern the integrated design and simulation of mechatronic systems, the development of novel methods for multidisciplinary design optimization, and the advanced structural analysis of mechanical systems.

Gabriele Vassura received the Laurea degree (cum laude) in mechanical engineering from the University of Bologna, Bologna, Italy, in 1972. He is currently an Adjunct Lecturer in the Department of Industrial Engineering, University of Bologna, where he was previously an Associate Professor. His teaching activity is in the field of machine design, mainly focused on mechanical design of automatic machines and robots. His main research interests include the design of dexterous robotic hands, simulation of pneumatic actuation systems, and design methodologies for automated systems.

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Angelo Oreste Andrisano is the Rector of the University of Modena and Reggio Emilia, Modena, Italy, where he is also a Full Professor of Engineering Design Methods and Tools. He is also the Founder and the Former Director of INTERMECH MO.RE., an interdepartmental center for applied research in automotive design, advanced mechanics, and materials, hosted by the University of Modena and Reggio Emilia. From 2001 to 2010, he was the Director of the Department of Mechanical and Civil Engineering for three terms. From 2007 to 2008, he was the President of the National Board of Professors in Engineering Design Methods and Tools, and contributed to the foundation of the National Board in Mechanical Science, of which he was an Executive Board Member and the Head of Communications. He is the author or coauthor of more than 150 papers, and he is or has been the Coordinator of several international projects and collaborations with industry. His scientific activity is strongly related to industry-driven research, design of mechanical transmission, tribology, biomechanics, and innovative design methods for mechatronic/robotic systems.