Unified formulation for the stiffness analysis of spatial mechanisms

Mechanism and Machine Theory 105 (2016) 272–284

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Unified formulation for the stiffness analysis of spatial mechanisms Alessandro Cammarata University of Catania, Dipartimento Ingegneria Civile e Architettura, 95125, Viale A. Doria 6, Catania, Italy

A R T I C L E

I N F O

Article history: Received 4 July 2016 Received in revised form 11 July 2016 Accepted 12 July 2016 Available online xxxx Keywords: Stiffness analysis Parallel robots Partitioned matrices Condensation techniques

A B S T R A C T This paper presents a complete stiffness analysis of spatial mechanisms. Links flexibility is modeled through structural elements while joints are inherently considered by means of kinematic relations including their degree of freedoms (dofs) and degree of constraints (docs). Actuation stiffness can be included as well as flexibility of some docs can be added in a selective way leaving the remaining rigid. Preloaded joints can be also modeled including joint wrenches. The Condensed Stiffness Matrix (CSM) of an elementary kinematic chain composed of a flexible two-node element and a spatial joint is derived using a robust mathematical formulation based on partitioned matrices and condensation techniques. The CSMs are then combined to find the global stiffness matrix through techniques coming from the structural analysis. The proposed method solves some critical issues of other formulations providing possibility to work with redundant legs (or joints) of fully- and over-constrained PKMs, inherent use of joints without resorting to Lagrangian multipliers, ability to exploit positivesemidefinite joint stiffness matrices without causing a rank-deficient global stiffness matrix; selective inclusion of stiffness in joints along dofs and docs. Finally, three examples to show the potentiality of the method for different applications of robotics are described. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction The growing interest of the scientific community in robots elasticity is derived from the demand for higher performance. Moving masses reduction is one of the keys to achieve speed and high acceleration such as to shorten the overall working time. However, it must be accompanied by the choice of light and resistant materials and by a careful design to ensure the degree of precision and the ability to transmit or bear loads imposed by the actuators or the environment. Added to these are also attractive applications as soft robotics and compliant robotics that make the elasticity the core of design. In the early nineties the researchers began to analyze the stiffness of parallel manipulators considering the contribution made by individual legs. Methods based on Jacobian were the first to address this issue exploiting the duality of kinematics and statics to derive the Cartesian stiffness matrix, that is the matrix mapping the deformations of the end-effector into the wrenches applied upon it. Gosselin in Ref. [1] used the Jacobian of a Parallel Kinematic Machine (PKM) to define the Cartesian stiffness matrix considering only the stiffness of the actuators. El-Khasawneh and Ferreira in Ref. [2] investigated the problem of finding the minimum

E-mail address: [email protected] (A. Cammarata). http://dx.doi.org/10.1016/j.mechmachtheory.2016.07.011 0094-114X/© 2016 Elsevier Ltd. All rights reserved.

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and maximum stiffnesses and the directions in which they occur for a parallel manipulator at a given pose. In Refs. [3–6] Zhang and Gosselin proposed the kinetostatic modeling of a family of n-dof PKMs with passive constraining leg in which the degree of freedom (dof) of the mechanism is dependent on the passive leg’s dof. The authors used lumped models for links and joints and modeled link flexibility through virtual joints that allow the motion along the degree of constraint (doc) generated by real joints. In 2004 Zhang et al. [7] proposed an extension of the Jacobian matrix method, called Kinetostatic Modelling Method including a complete compliant model developed for the analysis of the PKMs with fixed-length legs. In particular, three types of compliance contribute to deformation of the MP, namely: actuator flexibility, leg bending and axial deformation were included. The main issue of the Kinetostatic Modelling Method seems its limited use to PKMs with fixed length links such as tripods. The VJM provided by Pashkevich et al. in Ref. [8] is general and can include preloading and loading conditions. In Ref. [9] Pashkevich et al. extended the Virtual Joint Method (VJM) to over-constrained parallel manipulators in unloaded and loaded conditions. Even if the VJM is general and can be applied to study loaded equilibrium configurations and detect buckling, some shortcomings are essentially connected to: computational complexity deriving from Jacobians calculation; simplified models to include parallelogram joints or internal loop mechanisms [10]. In Ref. [11] Kim and Lipkin used reciprocal screws to explicitly eliminate the passive joint constraints obtaining leg stiffness matrices that are inherently singular. Starting from the work of Joshi and Tsai [12] and Hong and Choi [13] the concept of Jacobian was extended by means of the screw theory to the Overall Jacobian taking into account both the actuation and constraint wrenches imposed upon the MP. In the work of Huang et al. [14] the screw theory and its concepts of reciprocity and duality between twist subspace of permissions/restrictions and wrench subspace of actuation/constraints were used to create a systematic framework based on the Generalized Jacobian. In Ref. [15] the authors applied the Generalized Jacobian to study the stiffness of lower-mobility parallel manipulators. Although these formulations are based on a sound theory, the use of the observation method to derive reciprocal screws does not seem suitable for a numerical implementation in the form of algorithm. Alternatively, they require numerical approaches such as Gram Schmidt algorithm or augmentation matrix approach to obtain the basis of the wrench subspace of constraints necessary to derive the Generalized Jacobian. Recently, Hoevenaars et al. [16] employed the Generalized Jacobian for the stiffness analysis of lower-mobility parallel manipulators including preloaded compliant joints and deformable links. The method has some limitations when zero stiffness joints are included in the stiffness analysis. Simplificative assumptions must be made to obtain non-square stiffness matrix in joint-space. Besides, only non-redundant kinematic joints in each leg can be taken into consideration. A different approach from VJM based on the strain energy of deformable systems has been used for the stiffness analysis of PKMs. In Ref. [17] Deblaise et al. exploited the Matrix Structural Analysis (MSA) to perform an analytical stiffness study of PKMs. A clear drawback arises as joint contribute is added by means of Lagrangian multipliers. This issue is overcome in the work of Taghvaeipour et al. [18] where small-amplitude displacement (SAD) screw was combined to MSA to obtain the stiffness matrix of mechanical systems in which each joint equation includes a projection operator and elastic coordinates. A similar approach is followed in Ref. [19], while in Ref. [20] the author considered also joints with compliance by introducing complementary joint-matrix and joint-array along the docs of a joint. In this paper the methodology developed in Refs. [19–21] is extended to include preload, external wrenches, flexible passive and actuated joints and deformable links inside a unified mathematical formulation based on partitioned matrices and Schur complement, typically used for static condensation in FEM. All aspects are integrated into a unique framework providing rigor and a solid base to the method. In addition, the number of cases is reduced from three to one, making the treatment more robust. Designers and analysts have the opportunity to add joint stiffness and preloading along single dof and/or doc of the same joint in a selective manner. This offers a variety of solutions that has never been provided before by other stiffness models. Below, strong points and issues of the proposed method are listed. 1.1. Pros

• Working with redundant legs or joints of fully-constrained or over-constrained PKMs; • joints are inherently added into the model without resorting to Lagrangian multipliers; • joint stiffness along dofs and/or docs can be included in a selective manner and positive-semidefinite joint stiffness matrices can be used; • parallelogram joint and internal loop mechanisms can be added without simplificative models; • same nodal partition for elastostatic and elastodynamic analyses.

1.2. Cons

• The Cartesian stiffness matrix cannot be directly derived; • Loading mode and buckling cannot be analyzed. The structure of the paper is defined as follows. In Section 2 some key-concepts as small deformation array and joint matrices are recalled. Section 3 is devoted to the derivation of the Condensed Stiffness Matrix including elasticity in joints and links

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through a mathematical approach based on partitioned matrices, described in detail in the Appendix. Finally, in Section 4 three examples are proposed to validate the method. 2. Small deformation array and joint matrices Let us briefly review the contents of small deformation array and joint matrices introduced in Ref. [19] before going deep into the novel formulation. A small deformation array is composed of three small translational displacements and three small rotations expressed in terms of Euler angles:  u = ux

uy

uz

uv

uh

ux

T

(1)

It is remarked that this array refers to a section of a deformable body rather than a point. When two bodies are coupled by a joint a kinematic relation between two consecutive small deformation arrays arises. In its most general form, it can be written as u1j = Gu2i + Hf hf + Hc hc

(2)

where, as shown in Fig. 1, the array of nodal deformations u1j of body j depends on u2i of body i and on the small joint displacements hf and hc due to deformations, both linear and angular depending on type of joint, respectively referring to dofs and docs of the joint. As explained in the next section, Eq. (2) allows us to include joints contribute without resorting to Lagrangian multipliers. The matrix G is used to consider a rigid transformation of two nodes belonging to the same rigid body, i.e.  G=

1 −D O 1

 (3)

where 1 and O, respectively, are the 3 × 3 identity- and zero-matrices and D is the Cross-Product Matrix (C.P.M.) of the distance vector between the two nodes d, [22]. Finally, the joint matrices Hf and Hc , respectively dimensioned (6 × f) and (6 × c), with f indicating the number of dofs and c ≤ 6 − f the number of docs having finite stiffness, are needed to consider the joint contribute during deformation. Below, the joint-matrices of some common kinematic pairs are reported, similar matrices can be built for other classes: Revolute joint:  Hf = 0T  Hc =

eT1

T

, hf = h1

e 1 e2 e3 0 0 0 0 0 e 2 e3

(4a)

 , hc = [s1

s2

s3

h2

h3 ]T

Fig. 1. Elementary chain of a flexible element coupled to a spatial joint.

(4b)

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where e1 is the unit vector along the axis of the revolute joint R, e2 and e3 are two not-parallel unit vectors lying onto the plane normal to e1 and 0 is the three-dimensional zero vector. The linear and angular small displacements s and h are evaluated along and about {e1 , e2 , e3 }. Cylindric joint:  Hf =

 Hc =

e1 0 0 e1

 , hf = [s1

e 2 e3 0 0 0 0 e2 e3

h1 ]T

(5a)

 , hc = [s2

s3

h2

h3 ]T

(5b)

where e1 is the unit vector parallel to the cylindric joint axis while e2 and e3 are two not-parallel unit vectors lying onto the plane normal to e1 . To clarify the role of the joint matrices and their connection with the stiffness matrices, let us consider a revolute joint with rotation axis along the unit vector e1 for which f = 1 and c ≤ 5. The equal sign stands when all constrained directions have finite stiffness. If one constraint direction is stiffer than the others it may be considered rigid and the rigid doc must be removed from Hc yielding c = 4. For example, let the translational doc along e1 be rigid, thus Hc is modified accordingly:  Hf = 0T

 Hc =

eT1

T

, hf = h1

e 2 e3 0 0 0 0 e 2 e3

(6a)

 , hc = [s2

s3

h2

h3 ]T

(6b)

In this way it is possible to assign the stiffness to the constraints in a selective way, according to the needs of the designer. It must be emphasized that this aspect has not been well addressed in the literature. Let us now consider the case of a cylindric joint in which only one of the two dofs has stiffness. The joint matrices Hf and Hc remain the same as those in Eqs. (5a) and (5b) while the stiffness matrix of the joint Khf , which refers to the dofs, exhibits zero stiffness in the diagonal entry corresponding to the passive dof without stiffness. As it will be discussed in the next section, the semi-definite positiveness of Khf does not imply compliance singularity preventing inversion of the global stiffness matrix. This aspect, of outmost importance, is not always guaranteed by other stiffness models. For instance, in Ref. [11] a leg is required to contain at least one fully elastic element as a sufficiency condition to invert the leg flexibility matrix.

3. Condensed Stiffness Matrix Starting from the work developed in Ref. [20], the theory of partitioned matrices and condensation techniques are applied to develop a robust mathematical formulation including a unique case of coupling between flexible elements and joints. In particular, it is explained how to obtain the Condensed Stiffness Matrix (CSM) of an elementary chain that includes one deformable element and a spatial joint. Referring to the system shown in Fig. 1, suppose that its energy V is composed of five terms due to: strain energy due to link flexibility, strain energy due to joint stiffness along and about its docs, strain energy deriving from the joint stiffness along and about its dofs, as in the case in which the stiffness of actuators is taken into account, the work done by preload forces and torques along/about the docs and finally the work done by preload forces and torques along/about the dofs. Considering these contributes, V assumes the form:

V=

1 T 1 1 u˜ Kj u˜ j + hTc Khc hc + hTf Khf hf + wTc hc + wTf hf 2 j 2 2

(7)

 T  T T where u˜ j = u1j , u2j is the 12-dimensional array containing the node deformations of the jth-link, hf is the f-dimensional array of joint displacements due to deformations along/about the dofs, hc is the c-dimensional array (c ≤ 6 − f) of joint displacements due to deformations along/about the docs, wf , wc are the preload wrenches applied to the joint along/about its dofs and docs, respectively. Besides, the stiffness matrix Kj refers to the jth-link while Khf and Khc refer to the joint stiffness along/about its dofs and docs, respectively.

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In order to derive the CSM of the system stationarity conditions of V w.r.t. the joint arrays hf,c must be imposed:

∂ V ∂ u˜ 2 ∂ u12 ∂V dV = 0T ⇒ + = 0T dhf ∂ hf ∂ u˜ 2 ∂ u12 ∂ hf

(8a)

dV ∂ V ∂ u˜ 2 ∂ u12 ∂V = 0T ⇒ + = 0T dhc ∂ hc ∂ u˜ 2 ∂ u12 ∂ hc

(8b)

Notice that in the previous expression dV/dhf and dV/dhc are row arrays, so via the chain rule and Eq. (2), after simplifications the following expressions are finally obtained 1 T 12 2 HTf K11 j u2 + Hf Kj u2 + Khf hf + wf = 0

(9a)

1 T 12 2 HTc K11 j u2 + Hc Kj u2 + Khc hc + wc = 0

(9b)

Introducing the kinematic bond of Eq. (2) and ordering w.r.t. hf and hc , the linear system becomes 

Dff Dfc Dcf Dcc



hf hc



 =

rf rc

 ⇒ Dh = r

(10)

where the block matrices Dff ∈ Rf, f , Dfc ∈ Rf,c , Dcf ∈ Rc, f and Dcc ∈ Rc,c are defined as T 11 Dff = HTf K11 j Hf + Khf , Dfc = Hf Kj Hc

(11a)

T 11 Dcf = HTc K11 j Hf , Dcc = Hc Kj Hc + Khc

(11b)

2 T 12 2 rf = −HTf K11 j Gui − Hf Kj u2 − wf

(12a)

2 T 12 2 rc = −HTc K11 j Gui − Hc Kj u2 − wc

(12b)

while

Notice that the matrix D ∈ Rf+c, f+c is symmetric. The reader should notice that in Eqs. (11a) and (11b) Dff and Dcc have maximum rank f and c, respectively, and can be obtained even when Khf and Khc do not have the maximum rank and are positive-semidefinite. This important feature is due to Hf and HTc K11 Hc in Eqs. (11a) and (11b). The reader should notice that Hf the presence of the positive-definite matrices HTf K11 j j and Hc have maximum rank, besides, for well-defined structural elements, the submatrix K11 is never rank-deficient. This result j of the condensation techniques adopted still remains an open problem in some formulations in which each leg must contain at least one fully elastic element as a sufficiency condition to invert the leg flexibility matrix [11]. In order to solve the linear system of Eq. (10) the theory of block matrices is used. Some relevant results of this theory are reported in Appendix for the sake of completeness. In particular, hf and hc can be obtained through the following expressions hf = (D/Dcc )−1 rf − (D/Dcc )−1 Dfc D−1 cc rc

(13a)

hc = (D/Dff )−1 rc − (D/Dff )−1 Dcf D−1 ff rf

(13b)

where D/Dii are the Schur complements of D in Dii , (i = f, c), defined as D/Dff = Dcc − Dcf D−1 ff Dfc

(14a)

D/Dcc = Dff − Dfc D−1 cc Dcf

(14b)

Considering that Dff and Dcc are nonsingular symmetric square matrices, the inverse of the Schur complements can be obtained by means of the block diagonalization forms proposed by Aitken: 

D/Dff

−1

−1 −1 −1 = D−1 Dfc D−1 cc + Dcc Dcf Dff − Dfc Dcc Dcf cc

(15a)

A. Cammarata / Mechanism and Machine Theory 105 (2016) 272–284



−1 −1 (D/Dcc )−1 = D−1 ff + Dff Dfc Dcc − Dcf Dff Dfc

−1

Dcf D−1 ff

277

(15b)

By substituting Eqs. (12a) and (12b) into Eqs. (13a) and (13b) and reordering w.r.t. u2i and u22 , it is obtained hf = Y1f u2i + Y2f u2j + yf ≡ Yf u˜ a + yf

(16a)

hc = Y1c u2i + Y2c u2j + yc ≡ Yc u˜ a + yc

(16b)

 T  T T where u˜ a = u2i , u2j and 

T 11 T 11 f ,6 Y1f = (D/Dcc )−1 Dfc D−1 cc Hc Kj G − Hf Kj G ∈ R

(17a)

 T 12 T 12 Y2f = (D/Dcc )−1 Dfc D−1 ∈ Rf ,6 cc Hc Kj − Hf Kj

(17b)

 −1

 T 11 T 11 c,6 Y1c = D/Dff Dcf D−1 ff Hf Kj G − Hc Kj G ∈ R

(17c)

 −1

 T 12 T 12 Y2c = D/Dff Dcf D−1 ∈ Rc,6 ff Hf Kj − Hc Kj

(17d)

 f ,1 yf = (D/Dcc )−1 Dfc D−1 cc wc − wf ∈ R

(17e)

 −1

 c,1 yc = D/Dff Dcf D−1 ff wf − wc ∈ R

(17f)

Eqs. (16a) and (16b) can be finally substituted into the kinematic bond of Eq. (2) to give u˜ j = Xa u˜ a + xa

(18)

standing the following expressions Xa =  xa =

G + Hf Y1f + Hc Y1c Hf Y2f + Hc Y2c O6 16 H f y f + Hc y c 06

∈ R12,12

(19a)

 ∈ R12,1

(19b)

Finally, Eqs. (16a), (16b) and (18) are substituted into the expression of V given in Eq. (7) to obtain: V=

 1 T u˜ a Au˜ a + 2aT u˜ a + a 2

(20)

where A is the 12 × 12 CSM, a is the 12-dimensional array of generalized wrenches due to the preload forces and torques and a is a constant term, respectively defined as A = XTa Kj Xa + YTf Khf Yf + YTc Khc Yc

(21a)

a = XTa Kj xa + YTf Khf yf + YTc Khc yc + YTf wf + YTc wc

(21b)

a = xTa Kj xa + yTf Khf yf + yTc Khc yc + yTf wf + yTc wc

(21c)

In Eqs. (21a), (21b) and (21c) the CSM A can be seen as the stiffness matrix of a two-node superelement, i.e. the nodes of u˜ a , that merges together a structural element and a joint. A CSM can be assembled both with other stiffness matrices or other CSMs with methods typical of the Matrix Structural Analysis or FEA. The preload wrench a is a 12-dimensional generalized array

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composed of two wrenches applied at the two nodes of the superelement identified by the CSM. In a stiffness analysis the arrays a are nodal forces to be combined with the external nodal forces. Finally, the term a can be seen as the reference level, or zero level, of the potential energy V, therefore, it does not appear explicitly in the elastostatic equations. 4. Applications The mathematical formulation developed in the previous section has different applications for spatial mechanisms. To demonstrate the effectiveness in dealing with different problems three case studies are presented in this section: a two-link frame, a spherical parallel robot and finally a pick-and-place robot. 4.1. Two-link frame In order to test the algorithm, a frame composed of two links, clamped at the extremities and coupled each other by means of a revolute joint with axis parallel to the fixed frame axis y and torsional stiffness khf , is first considered, as shown in Fig. 2. Spatial 3D Euler beams with square cross-section are used as structural elements for the two links. Geometric and structural parameters are reported in Table 1 The global stiffness matrix of the system can be found following classic methods coming from structural analysis, beyond the scope of this paper. With reference to Fig. 3 the stiffness matrix KR1 of the first beam of the right

link must take into  account

the rigid displacement d2 and then must be replaced by the CSM of Eqs. (21a), (21b) and (21c): A K2 , hRf (e), G(d1 ), kRh . f A vertical force with magnitude F = 1000 [N] along the z-axis has been applied at the joint location considering different values of khf , i.e. the torsional stiffness about the rotation axis of the revolute joint. Fig. 4 shows three deformed configurations. As expected the deflection decreases increasing the value of the joint stiffness. The comparison to Ansys results revealed good accordance, as reported in Table 2. A limit stiffness value of 1000 (Nm/rad) has been set because the frame experiences plastic deformations for higher values. 4.2. Spherical parallel robot

The Agile Eye is a pure rotational 3-dof spherical parallel robot designed by Gosselin [23, 24]. The robot has three legs of type RRR, being R a revolute joint. Each leg is composed of two links: the proximal and the distal link, as shown in Fig. 5. The axes of the revolute joints intersect at the center point O of the spherical motion. The stiffness analysis of the Agile Eye has been performed by other authors with different approaches [25,26,27] and [28]. Here, the two platforms are modeled as rigid bodies while the works of Wu and Chiang [29–31] to model the proximal and distal links by means of curved beam elements based on Timoshenko beam theory are employed. All parameters necessary for the analysis are reported in Table 3. In order to apply the formulation developed in the previous section, the following three revolute joint couplings are considered: BP-proximal link, proximal link-distal link and distal link-MP. The CSM of the generic coupling has the form: A(K, Hf , Hc , G, Khf , Khc ), in which K refers to the stiffness matrix of the curved beam element coupled to the revolute joint, Hf and Hc are the joint matrices, G yields the rigid-body displacement, Khf and Khc are the stiffness matrices of the joint and wif and wic are the preload wrenches, as defined in the previous section. The Agile Eye at the home configuration is analyzed. In this configuration, for each leg, the revolute joints of the BP and MP belong to the same vertical plane with normal vector equal to the unit vector of the intermediate revolute joint. Four different stiffness models have been investigated. The outlines of these models are reported as follows.

• I): Ideal joints In this model rigid joints and rigid body displacements due to the coupling elements of the revolute joints have been considered. The actuated joints R1,l , l = 1, 2, 3 are fixed to eliminate the mobility. The three CSMs of each coupling simplify, i.e. R1 : fixed

(22a)

Fig. 2. Layout of the two-link frame.

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Table 1 Geometric and structural parameters the two-link frame. Notation

Description

Value

Unit

n L b h E m

Number of elements for each link Links’ length Base of the cross section Height of the cross section Young modulus Poisson’s ratio

10 0.3 0.025 0.025 200 0.33

[–] [m] [m] [m] [GPa] [–]

 R2 : A Kd1 , hRf (e2 ), G(dL )

(22b)

 R3 : A Kdn , hRf (e3 ), G(dM )

(22c)

with dB = dB e1 , dL = −dL e2 and dM = dM e3 . • II): Actuation stiffness In this model the actuated joints R1,l , l = 1, 2, 3 are free to rotate and torsional springs are added to simulate proportional control. Notice that the torsional springs prevent the rank-deficiency of the global stiffness matrix. The CSM of R1 turns into

 p R1 : A K1 , hRf (e1 ), G(dB ), kRh f

Fig. 3. Revolute joint with torsional stiffness about its axis.

Fig. 4. Two-link frame partitioned in 3D Euler beams: vertical force applied at the revolute joint location along z; values of khf in (Nm/rad).

(23)

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A. Cammarata / Mechanism and Machine Theory 105 (2016) 272–284 Table 2 Frame deflection for F = [0;0;1000] (N); A algorithm outputs; B Ansys© outputs. khf (Nm/rad)

A (m)

B (m)

0 100 1000

7.335E−04 7.284E−04 6.862E−04

7.328E−04 7.278E−04 6.887E−04

• III): Flexible joints Copper alloy coupling elements, carrying out the revolute joint connection between proximal and distal links, have been added to the rigid model I). The stiffness matrix of the generic bushing element has been calculated by FEM, as shown in Fig. 6. The CSMs become

 p R1 : A K1 , hRf (e1 ), HRc (e1 ), G(dB ), kRh , KRc

(24a)

 R2 : A Kd1 , hRf (e2 ), HRc (e2 ), G(dL ), kRh , KRc

(24b)

 R3 : A Kdn , hRf (e3 ), HRc (e3 ), G(dM ), kRh , KRc

(24c)

f

f

f

• IV): Preloaded actuated joints Preload torquesare applied to the actuated joints R1,l , l = 1, 2, 3 of model II) to compensate an external force f = 10∗vM (N), √ vM =[1, 1, 1]/ 3, applied to the end-effector. Generalized preload wrenches a of Eqs. (21a), (21b) and (21c), due to the preload torques applied to R1,l , l = 1, 2, 3, have to be included in model II), i.e.

 p R1 : a K1 , hRf (e1 ), G(dB ), wif

(25)

The results of Table 4 are expressed in the form (p(m), 0(rad)) where p and 0 respectively indicate the translational and the rotational deformation of the end-effector. The numerical results are in good agreement with the results of Ansys©; some discrepancies arise because the finite element model also considers holes and fillets in three-dimensional elements. As can be seen in the model I) a torque yields a translational deformation and a rotational deformation. In the second model II), with

Fig. 5. Generic leg of the Agile Eye at the home configuration.

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Table 3 Geometric and structural parameters of the Agile Eye.

rp rd l a b m E

191.5 166.5 20.0 90 0.91 0.33 68.5

mm mm mm deg

GPa

Mean radius of proximal links Mean radius of distal links Side length of the square cross section of the curved links Opening angle of the curved links Shear correction factor Poisson’s ratio: aluminum alloy Elastic modulus of the links: aluminum alloy

torsional springs on the actuated joints, the deformation p should remain the same as the model I). The reason is due to the presence of over-constraints in the structure. In fact, the external torque yields constraint reactions that deform the structure. The remaining components act on the subspace of rotations SO(3) of the robot and are equilibrated by the reaction torques created by the torsional springs. As can be observed, Ansys© does not maintain the same p while the rotational deformation 0 increases for both models in comparison. The model III) considers a deformable coupling element between the two links of the legs. The addition of these bushings increases significantly the deformations of the end-effector, therefore confirming the importance of joints in a reliable stiffness modeling. An example of selective stiffness in flexible joints can be provided if the deformation along the Z-axis (rigid constrained direction) of the flexible joint is neglected, therefore c = 4. For this case (p, 0) = (6.212E−4 ∗vM , 4.435E−3 ∗vM ) so the error committed neglecting this doc is minimal (the error is less than 0.001%): complexity of the problem can be reduced without compromising the results. For the model IV) the preload torques acting on the actuated joints to compensate an external force applied on the MP have been calculated. A value of 7.65 (Nm) against a value of 7.82 (Nm) coming from Ansys© results has been calculated. 4.3. Pick-and-place robot Finally, the proposed method has been applied to study the elastostatics of the Ragnar robot [32, 33] a pick-and-place parallel robot with Shönflies motions. Recently, stiffness analysis of parallel robots with parallelograms has been addressed in Ref. [34] where the strain energy method and Castigliano’s theorem are used to deduce a general algebraic stiffness matrix. However, statics equations are obtained only for the case of parallelograms with spherical joints. Overconstrained parallelograms in which spherical joints are substituted by revolute joints are not discussed. Ragnar has four legs based on overconstrained parallelograms with four revolute joints (P joint). For the parameters used in the elastostatics the reader is referred to Refs. [32, 33]. Fig. 7 shows the deformations of the Ragnar robot when a vertical force F = 2000 [N] is applied at its end-effector point EE. The high value of F has been chosen to better reveal the differences between undeformed and deformed cases both displayed in the same figure. The obtained value of −0.031 [m] for the vertical deflection of point EE has been confirmed by FEM analysis with a relative error of 3% due to geometric features as holes and fillets. Finally, the elastodynamics analysis with the same mesh geometry used in the elastostatics has been implemented. In the right side of Fig. 7 the first mode shape at at 79.953 [Hz] is shown. The elastodynamic model is not further described as it is beyond the scope of this paper. 5. Conclusions In this paper, a stiffness model to study spatial mechanisms has been derived. Structural elements are combined with flexible joints inside a unique linear formulation, based on partitioned matrices, to form the Condensed Stiffness Matrices. Flexibility

Fig. 6. Flexible copper alloy coupling element and its stiffness matrix expressed in the local body-frame.

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A. Cammarata / Mechanism and Machine Theory 105 (2016) 272–284 Table 4 √ Stiffness model of the Agile Eye n = 10 ∗vM (Nm); vM = [1, 1, 1]/ 3. Model

Algorithm

Ansys©

I) II) III)

(3.368E−4 ∗vM , 2.908E−3 ∗vM ) (3.368E−4 ∗vM , 7.989E−2 ∗vM ) (6.212E−4 ∗vM , 4.435E−3 ∗vM )

(3.764E−4 ∗vM , 3.089E−3 ∗vM ) (3.581E−4 ∗vM , 7.995E−2 ∗vM ) (6.923E−4 ∗vM , 4.528E−3 ∗vM )

(a)

(b)

Fig. 7. (a) Deformations of the Ragnar robot for a vertical load F of 2000 [N] applied at the end-effector point EE; (b) first mode shape at 79.953 [Hz].

along dofs and docs of joints can be added in a selective way, besides preload wrenches can be also included in the model. The proposed method overcomes some problems of other formulations as it can work with redundant legs and/or joints of fully-constrained or over-constrained PKMs. Besides, it exploits positive-semidefinite joint stiffness matrices without causing a rank-deficient global stiffness matrix whereas other methods require a leg to contain at least one fully elastic element as a sufficiency condition to invert the leg flexibility matrix. Three examples have been used to verify the validity. A two-link frame, modeled through 3D Euler beams, has been first analyzed. Then, the spherical parallel robot Agile Eye has been modeled by means of spatial Timoshenko curved beams. Different models including: rigid connections, actuated joint stiffness, preloaded joints and bushings have been investigated. Then, the pick-and-place Ragnar robot has been studied to demonstrate that the proposed method can be used to analyze robots with parallelogram joints or internal loops without simplifying. Finally, the first mode shape has been calculated to demonstrate that the same mesh geometry can be used for both elastostatic and elastodynamic models. Acknowledgments The author declares that he has no conflict of interest. Appendix A This Appendix reports some results pertaining the theory of block matrices [35–38]. Let us consider a square matrix M partitioned into four submatrices:  M=

A B C D

 (26)

in which A and D are square matrices of dimensions n × n and m × m respectively, while the rectangular blocks B and C have dimensions n × m and m × n. If A and D are nonsingular, the Schur complements of M with respect to (often simply in) A and D are defined as M/A = D − CA−1 B

(27)

M/D = A − BD−1 C

(28)

A. Cammarata / Mechanism and Machine Theory 105 (2016) 272–284

283

Aitken proposed block diagonalization forms of M, i.e. 

A B C D



 = ≡

1 O CA−1 1 1 BD−1 O 1

 1 A−1 B O 1

   M/D O 1 O −1 O D D C 1



A O O M/A



(29)

from which it is easy to derive the following expression det(M) = det(A) ∗ det(M/A) ≡ det(D) ∗ det(M/D)

(30)

Taking the inverse of M from Eq. (29), it is obtained that  

A B C D A B C D

−1

 = 

−1 =

1 −A−1 B O 1 1 O −D−1 C 1



A−1 O O (M/A)−1



(M/D)−1 O O D−1



1 O −CA−1 1



1 −BD−1 O 1

 (31a)  (31b)

in which A and D are nonsingular. Exploiting all products of Eq. (31a), the following expressions are derived 

A B C D

−1

 =  ≡

A−1 + A−1 B(M/A)−1 CA−1 −A−1 B(M/A)−1 −(M/A)−1 CA−1 (M/A)−1 −(M/D)−1 BD−1 (M/D)−1 −D C(M/D)−1 D−1 + D−1 C(M/D)−1 BD−1

 

−1

(32)

from which the final expression of M −1 is written as M−1 =



−(M/D)−1 BD−1 (M/D)−1 −(M/A)−1 CA−1 (M/A)−1

 (33)

Besides, by comparison of block entries of Eq. (32) the Duncan’s inversion formulas are obtained as (M/D)−1 = A−1 + A−1 B(M/A)−1 CA−1

(34a)

(M/A)−1 = D−1 + D−1 C(M/D)−1 BD−1

(34b)

or their expanded versions (M/D)−1 = A−1 + A−1 B(D − CA−1 B)−1 CA−1

(35a)

(M/A)−1 = D−1 + D−1 C(A − BD−1 C)−1 BD−1

(35b)

The expression (33) for the inverse M −1 can be used to solve partitioned linear system of the form Mx = b

(36)

T T   with x = xT1 xT2 array of unknowns and b = bT1 bT2 array of given terms. The solutions for x are derived through Eq. (33), i.e. x1 = (M/D)−1 b1 − (M/D)−1 BD−1 b2

(37a)

x2 = (M/A)−1 b2 − (M/A)−1 CA−1 b1

(37b)

Appendix B. Supplementary data Supplementary data to this article can be found online at http://dx.doi.org/10.1016/j.mechmachtheory.2016.07.011.

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