Inﬂuence-Coefﬁcient Method for Identifying Maximum-Load Conﬁgurations and Variable-Load Issues in Manipulators

: The dimensioning of general-purpose machines such as manipulators involves the solution of a number of preliminary issues. The determination of reference external loads and the identiﬁcation of machine conﬁgurations that give the maximum internal load for each component are two of these issues. These two problems are commonly addressed through trial-and-error procedures based on dynamic modelling, which are implemented with the support of simulation software, since static analyses are commonly considered inadequate to solve them. Despite this, here, a technique based on inﬂuence coefﬁcients and static analyses is presented which solves them. Such technique is also able to foresee and justify dynamic issues (i.e., possible vibrations, etc.) that could heavily affect the machine behavior. The effectiveness of the proposed technique is proved by implementing it on a 3T1R parallel manipulator. The presented design method is general and applicable to any type of non-overconstrained manipulator or mechanism.


Introduction
The mechanical design of manipulators has mandatory steps summarizable as follows: type synthesis [1][2][3][4], dimensional synthesis [1,2,4], and machine-element design [5][6][7]. Type synthesis identifies the machine topologies that meet the motion requirements, dimensional synthesis determines the values of the geometric parameters that affect the motion tasks the machine has to perform, and, finally, machine-element design chooses materials and sizes of each component of the machine.
The third step needs, as input data, the reference external loads applied to the machine and the identification of the machine configurations that give the maximum internal loads for each component. In general-purpose machines, such as manipulators, these input data are not straightforwardly obtainable from the design requirements and their deduction becomes even more difficult when dealing with parallel manipulators (PMs) since PMs' reachable workspace usually includes particular configurations, named "parallel" singularities [8][9][10], where the end effector cannot carry any (even infinitesimal) load without breaking down the PM.
Various performance indices were conceived for identifying PMs' reachable-workspace regions that are free and far from singularities, where safely locating the useful workspace [11][12][13][14][15][16][17][18]. These indices well identify kinetostatic issues and can be useful to perform the dimensional synthesis and to locate the useful workspace (see, for instance, [19,20]) far from singularities. Nevertheless, they fail in relating the "distance" from singularities to the actual internal loads of the links.
Other indices take into account also robot stiffness and dynamic performances [16][17][18][21][22][23][24][25][26]. These other indices require the knowledge of data (e.g., masses, stiffness, etc.) that are available at the end of the machine design. Therefore, they are certainly suitable for comparing already-built manipulators, but their use during design is not straightforward since it requires trial-and-error procedures.
In general, all these indices can help the design procedure, but do not provide sufficient pieces of information for determining the input data of PMs' machine-element design. This determination is commonly addressed through trial-and-error procedures based on dynamic modelling, supported by multi-body simulation software, due to the preeminent dynamic nature of the loads applied to a manipulator. This widely adopted procedure is not able to justify unforeseen/unwanted machine behaviors that come out from the simulations thus making difficult to intervene for eliminating them.
Here, an algorithm, based on influence coefficients and static analyses, is presented which provides sufficient pieces of information for determining the input data of PMs' machine-element design. The proposed algorithm is also able to foresee and justify dynamic issues (i.e., possible vibrations, etc.) that could heavily affect the machine behavior. Its effectiveness is proved by implementing it on a 3T1R parallel manipulator. The presented design method is general and applicable to any type of non-overconstrained manipulator or mechanism.
The proposed algorithm is a novel way of computing and using influence coefficients in the design of general-purpose machines, mainly ideated for PMs, which aims at integrating the design methodologies based on performance indices and/or extended multibody simulations by solving their lacks. Indeed, it can overcome the fact that performance indices are not able to provide sufficient data for the machine-element design, and that extended multibody simulations often do not provide sufficient pieces of information for intervening when an unforeseen phenomenon comes out from the simulations.
The paper is organized as follows. Section 2 provides some background and presents the novel technique. Then, Section 3 applies the proposed method to a 3T1R parallel manipulator. Eventually, Section 4 discusses the results and Section 5 draws the conclusions.

Materials and Methods
Most manipulators are based on non-overconstrained architectures [27]. Non-overcon strained manipulators become isostatic structures when the actuated joints are locked. The static analysis of an isostatic structure is sufficient to relate external loads to links' internal loads and involves the solution of a problem that is linear and homogeneous in the input (i.e., external loads) and the output (i.e., links' internal loads) variables [28]. This solution does not need to know the materials the links are made of and links' cross-sections, which is an ideal condition for tools to employ during design when materials and actual sizes of the links are not known since their choice/determination is the design goal. In addition, such static analyses are based only on equilibrium equations and can take into account an esteem of the dynamic loads among the known external loads thus simulating an inverse dynamic problem [27].
Problems that involve the solution of a linear system, as the above-mentioned static analyses do, can be addressed by using the superposition principle [29]. In the case of a static analysis, the linear system to solve can be put into the following canonic form where x = (x 1 , . . . ,x r ) T is an r-tuple collecting all the scalar components, x i for i = 1, . . . ,r, of the known external loads applied to the isostatic structure, y = (y 1 , . . . ,y s ) T is an s-tuple collecting all the scalar components, y j for j = 1, . . . ,s, of links' internal loads, and A is an s × r rectangular matrix of coefficients whose entries depend on the manipulator configuration. The linearity of system (1) allows demonstrating that (superposition principle) if x is writable as follows where λ p , for p = 1, . . . ,n, are n scalar coefficients and u p = (δ 1p , . . . ,δ rp ) T (where δ ip is the Kronecker delta defined as follows: δ ip = 0 for i =p and δ ip = 1 for i = p), for p = 1, . . . ,n, are n unit r-tuple, then with v p = Au p for p = 1, . . . ,n. Hereafter, v p will be named p-th influence-coefficient vector.
During the design of a manipulator, once the dimensional synthesis has been completed, Equation (3) can be exploited for the determination of the maximum internal loads of the links by implementing the following steps: (i). according to the type of motion [30] the end effector can perform, up to three independent unit forces and up to three independent unit moments are identified for characterizing the dynamic loads applied to the end effector during motion and the other dynamic loads reduced to the end effector; (ii). the unit forces/moments determined in the previous step are transformed into u p r-tuples to introduce into Equation (2) for computing the input data of Equation (1); (iii). for each u p determined in the previous step, the corresponding influence-coefficient vector v p is computed for a discretized set of end-effector poses inside the useful workspace of the manipulator; (iv). the so-determined v p are analyzed for finding the maximum internal loads of each link due to each unit load u p together with the corresponding manipulator configurations; (v). by exploiting Equation (3), the maximum internal loads of each link determined in the previous step are combined to obtain the reference internal loads of each link to use for sizing the link in terms of material choice and cross-section.
This procedure, over providing the input data for the machine-element design, is able to highlight critical conditions due to unbalanced variable loads acting on the links, since it is a static calculus referred to dynamic loads, thus providing pieces of information that orientate the next steps of the manipulator design.
Next section exemplifies the procedure by applying it to the partially decoupled 3T1R PM shown in Figure 1. where λp, for p = 1,…,n, are n scalar coefficients and up=(δ1p,…,δrp) T (where δip is the Kronecker delta defined as follows: δip = 0 for i≠p and δip = 1 for i = p), for p = 1,…,n, are n unit r-tuple, then 1, with p p = v Au for p = 1,…,n. Hereafter, vp will be named p-th influence-coefficient vector.
During the design of a manipulator, once the dimensional synthesis has been completed, Equation (3) can be exploited for the determination of the maximum internal loads of the links by implementing the following steps: according to the type of motion [30] the end effector can perform, up to three independent unit forces and up to three independent unit moments are identified for characterizing the dynamic loads applied to the end effector during motion and the other dynamic loads reduced to the end effector; (ii).
the unit forces/moments determined in the previous step are transformed into up rtuples to introduce into Equation (2) for computing the input data of Equation (1); (iii).
for each up determined in the previous step, the corresponding influence-coefficient vector vp is computed for a discretized set of end-effector poses inside the useful workspace of the manipulator; (iv).
the so-determined vp are analyzed for finding the maximum internal loads of each link due to each unit load up together with the corresponding manipulator configurations; (v). by exploiting Equation (3), the maximum internal loads of each link determined in the previous step are combined to obtain the reference internal loads of each link to use for sizing the link in terms of material choice and cross-section.
This procedure, over providing the input data for the machine-element design, is able to highlight critical conditions due to unbalanced variable loads acting on the links, since it is a static calculus referred to dynamic loads, thus providing pieces of information that or     Figure 1 shows a single-loop PM in which the end effector (platform) is simultaneously connected to the frame (base) through two kinematic chains (limbs), one of type CRS 1 and the other of type RRC, that is, it is a PM of type CRS-RRC. The axes of the R and C pairs are all parallel. The actuated C pair of the CRS limb is obtained by means of a PR chain with the sliding direction of the P pair parallel to the axis of the R pair. Such a PM architecture has four degrees of freedom [31] and constrains the platform to perform spatial translations (3T) plus rotations around axes with one fixed direction (1R), that is, the platform can perform only motions of 3T1R type, named Scara or Schoenflies motions [30]. This Scara PM has been ideated [20,31] at the Laboratory of Mechatronics and Virtual Prototyping (LaMaViP) of the University of Ferrara and, hereafter, for the sake of brevity, it will be named "LaMaViP's CRS-RRC". In the context of the Scara PMs, LaMaViP's CRS-RRC is interesting because it can be actuated by keeping all the motors on the base and using commercial components, and it has the platform translation decoupled from the platform rotation [20,31].

Results
The following notations/definitions are introduced (see Figure 1): Cartesian reference fixed to the base with the direction of the z bcoordinate axis parallel to the R and C pair axes; -O p is a point fixed to the platform whose coordinates, (x p , y p , z p ) T , measured in O b x b y b z b , locate the position of the platform; -B 2 is a point that lies on the axis of the actuated C-pair of the CRS limb and is fixed to the output link of the C-pair; -d is the signed distance of B 2 from O b chosen as linear joint-variable of the actuated C-pair; -D 2 , A p , and A 2 are the intersection points between the plane parallel to the x b y b plane that passes through B 2 and, respectively, the axis of the passive R-pair, the axis of the passive C-pair, and the line parallel to the z b axis that passes through the center of the S-pair; -h is a geometric constant of the platform equal to the length of the segment A p O p ; -a p is a geometric constant of the platform that is equal to the length of the segment A p A 2 ; -ϕ is the angle between the segment A p A 2 and the line parallel to x b and passing through A p (the angle ϕ uniquely determines the platform orientation); -a 3 and a 4 are the lengths of the segments B 2 D 2 and D 2 A 2 , respectively; -θ 3 and θ 4 are the angular joint-variables of the actuated C-pair and of the passive R-pair, respectively; -θ 1 and θ 2 are the angular joint-variables of the two actuated R-pairs of the RRC limb; -B 1 , D 1 , and A 1 are the intersection points between the x b y b coordinate plane and, respectively, the axis of the actuated R-pair adjacent to the base, the axis of the intermediate R-pair of the RRC limb and the axis of the passive C-pair; -a 0 , a 1 and a 2 are the lengths of the segments O b B 1 , B 1 D 1 and D 1 A 1 , respectively.
From a kinematics point of view, the RRC limb constrains the platform to perform Schoenflies motions with rotation axis parallel to the z b -coordinate axis and controls only the x p and y p coordinates of O p (i.e., the platform translations parallel to the x b y b coordinate-plane). Moreover, the CRS limb controls the coordinate z p of point O p (i.e., the platform translation parallel to the z b -coordinate axis) through its linear joint-variable, d, and, independently, the platform orientation (i.e., the angle ϕ) through its angular joint-variable, θ 3 .
The dimensional synthesis of LaMaViP's CRS-RRC was addressed in [19]. In [19], a useful workspace assigned as a right circular cylinder with a radius, r uw , of 300 mm and a height of 200 mm, whose axis is coincident with the z b -coordinate axis, brought to choose the following values of the geometric constants (see Figure 1): a 0 = 800 mm, a 1 = 950 mm, a 2 = 600 mm, a 3 = 400 mm, a 4 = 400 mm, and a p = 400 mm. Figure 2 shows the scaled top view of a LaMaViP's CRS-RRC sized according to this geometric data together with the position of the chosen useful workspace (the green circle in Figure 2). The analysis of Figure 2 reveals that the translation of the platform along the direction of the z b -coordinate axis is fully decoupled from the remaining platform motion. Indeed, the motion due to the prismatic pair does not appear in the top view ( Figure 2) where the PM becomes a planar six-bar mechanism controlled by the actuated-joint variables θ 1 , θ 2 , and θ 3 . In addition, in this six-bar, θ 1 and θ 2 just control the position of the platform (i.e., the position of point A 1 ), whereas θ 3 fully controls the platform orientation (i.e., the angle ϕ) through a four-bar transmission.
Machines 2021, 9, x FOR PEER REVIEW 5 of 15 independently, the platform orientation (i.e., the angle ϕ) through its angular joint-variable, θ3. The dimensional synthesis of LaMaViP's CRS-RRC was addressed in [19]. In [19], a useful workspace assigned as a right circular cylinder with a radius, ruw, of 300 mm and a height of 200 mm, whose axis is coincident with the zb-coordinate axis, brought to choose the following values of the geometric constants (see Figure 1): a0 = 800 mm, a1 = 950 mm, a2 = 600 mm, a3 = 400 mm, a4 = 400 mm, and ap = 400 mm. Figure 2 shows the scaled top view of a LaMaViP's CRS-RRC sized according to this geometric data together with the position of the chosen useful workspace (the green circle in Figure 2). The analysis of Figure 2 reveals that the translation of the platform along the direction of the zb-coordinate axis is fully decoupled from the remaining platform motion. Indeed, the motion due to the prismatic pair does not appear in the top view ( Figure 2) where the PM becomes a planar six-bar mechanism controlled by the actuated-joint variables θ1, θ2, and θ3. In addition, in this six-bar, θ1 and θ2 just control the position of the platform (i.e., the position of point A1), whereas θ3 fully controls the platform orientation (i.e., the angle ϕ) through a four-bar transmission.

Implementation of Steps (i) and (ii)
According to the introduced notations, the 4-tuples q = (θ 1 ,θ 2 ,θ 3 ,d) T and κ = (x p ,y p ,z p ,ϕ) T collect the actuated-joint (input) variables and the platform-pose (output) variables, respectively. The inspection of Figures 1 and 2 reveals that the entries of q and κ must satisfy the following constraint equations x p = a 0 + a 1 cosθ 1 + a 2 cos(θ 1 + θ 2 ), (4a) y p = a 1 sinθ 1 + a 2 sin(θ 1 + θ 2 ), (4b) which are holonomic and time-independent. Equations (4a)-(4d) yield the following relationships between the virtual displacements δq = (δθ 1 , δθ 2 , δθ 3 , δd) T and δκ = (δx p , δy p , δz p , δϕ) T Machines 2021, 9, 27 where c w and s w stand for cos(w) and sin(w), respectively, whereas, (see Figure 2) The active-force 2 system applied to LaMaViP's CRS-RRC consists of the active external forces applied to the end effector and the generalized torques applied inside the actuated joints. The active external forces are collected into the 4-tuple γ = −(F e,x , F e,y , F e,z , M e,z ) T , where F e,x , F e,y , and F e,z are the components, along x b , y b , and z b , respectively, of their resultant force and M e,z is the component along z b of their resultant moment about O p . In addition, the generalized torques are collected into the 4-tuple τ = (τ θ 1 , τ θ 2 , τ θ 3 , τ d ) T where the right subscripts indicate the actuated joint the generalized torque refers to. The virtual work principle [32] makes it possible to write The elimination of δκ from Equation (8) by means of Equation (5) and the consideration that the resulting expression must hold for any value of δq yield −(a 1 s θ 1 + a 2 s (θ 1 +θ 2 ) ) (a 1 c θ 1 + a 2 c (θ 1 +θ 2 ) ) 0 m x (a 1 s θ 1 +a 2 s (θ 1 +θ 2 ) )−m y (a 1 c θ 1 +a 2 c (θ 1 +θ 2 ) ) The analysis of Equation (9) and of Figures 1 and 2 reveals that, over the platform, (a) F e,x and F e,y load only the RRC limb, their effects are coupled and do not depend on ϕ (i.e., on the platform orientation); (b) F e,z loads the CRS limb through the S-pair and its static effect on links 3 and 4 (see Figure 2) depends on the limb configuration. Moreover, the static equilibrium of the platform reveals that, when F e,z is applied at O p and the S-pair equilibrates it by applying a force −F e,z at A 2 , a torque, M p , with magnitude |F e,z a p | (see Figure 1) and direction perpendicular to the platform plane (i.e., the plane located by the points O p , A p and A 2 ) arises. M p must be equilibrated by the reaction in the passive C-pair of the RRC limb. As a consequence, F e,z makes the RRC limb loaded, too, by the torque M p , whose static effects on links 1 and 2 depend on the platform pose and the limb configuration; (c) M e,z loads both the limbs and all the links.
During motion, all the components of γ (= −(F e,x , F e,y , F e,z , M e,z ) T ) are generated from the inertia forces arising in the mechanism. Therefore, the static analysis must consider all the above-mentioned cases. Regarding case (a), links' internal loads, caused by a unit force (F e,x , F e,y ) = (c ψ , s ψ ) N for ψ ∈ [0 • , 360 • ], corresponding to u 1 = (c ψ , s ψ , 0, 0) T N for ψ ∈ [0 • , 360 • ], applied to the RRC limb at A 1 (see Figures 1 and 2), must be determined for a discretized set of A 1 positions inside the useful workspace. Regarding case (b), links' internal loads, caused by a unit force F e,z (= 1 N), corresponding to u 2 = (0, 0, 1, 0) T N, applied at O p (see Figures 1 and 2), must be determined for a discretized set of O p positions inside the useful workspace. Regarding case (c), links' internal loads, caused by an unit moment M e,z (=1 Nm), corresponding to u 3 = (0, 0, 0, 1) T Nm, applied to the platform at O p (see Figures 1 and 2), must be determined for a discretized set of O p positions inside the useful workspace.
3.2.2.I. Internal loads in the CRS limb: In the horizontal part of link 4 (i.e., the segment A2D2 (see Figures 1 and 2)), the unit force Fe,z (= 1 N) applied at the S-pair center generates a constant shear force Sy,4= Fe,z = 1 N, and a bending moment, Mz,4, that linearly grows from 0, at A2, to Mz,4,max = Fe,za4 = 0.4 Nm, at D2. Of course, in the small vertical part of link 4, only a constant axial force equal to Ax,4 = −Fe,z = −1 N is present. These values do not depend on the limb configuration.
Regarding link 1, at any limb configuration, the maximum M y occurs either at B 1 when the distance between f e ' s line of action and B 1 is maximum (i.e., when f e is perpendicular to the segment A 1 B 1 (see Figure 3)), where its value is M y,1 = ±|f e ||(A 1 − B 1 )|, or, for |(A 1 − B 1 )| < a 2 , at D 1 where M y,1 = M y,2,max = ±|f e |a 2 when ψ = θ 1 + θ 2 ± 90 • . Anyway, the limb configuration with the maximum M y,1 is the one where the point A 1 coincides with the point A' 1 shown in Figure 3. In this configuration, the maximum M y occurs at B 1 for ψ = ±90 • and its value is M y,1,max = ±|f e ||(A' 1 − B 1 )| = ±1.1 Nm. Moreover, whatever be the limb configuration, the maximum S z (A x ) occurs when f e is perpendicular (parallel) to the link axis, that is, when ψ = θ 1 ± 90 • (when ψ = θ 1 or ψ = θ 1 + 180 • ), and its value is S z,1,max = ±|f e | = ±1 N (A x,2,max = ±|f e | = ±1 N). Table 1 summarizes the results of case (a). Table 1. Maximum non-null influence coefficients due to the unit force f e (Figure 3) of case (a). In this case (a unit force F e,z (= 1 N) applied at O p (see Figures 1 and 2)), the equilibrium of the platform shows that the CRS limb is loaded by the same F e,z (=1 N) applied at the S-pair center. Moreover, it shows that the RRC limb is loaded by a torque, M b , lying on the motion plane (i.e., the x b y b coordinate-plane) and applied at A 1 (see Figure 2). In particular, M b has magnitude |M b | = |F e,z a p | = 0.4 Nm and direction (ϕ + 90 • ) in the motion plane, that is,

Internal Loads in the CRS Limb
In the horizontal part of link 4 (i.e., the segment A 2 D 2 (see Figures 1 and 2)), the unit force F e,z (= 1 N) applied at the S-pair center generates a constant shear force S y,4 = F e,z = 1 N, and a bending moment, M z,4 , that linearly grows from 0, at A 2 , to M z,4,max = F e,z a 4 = 0.4 Nm, at D 2 . Of course, in the small vertical part of link 4, only a constant axial force equal to A x,4 = −F e,z = −1 N is present. These values do not depend on the limb configuration.
As a consequence, in link 3, the shear force S y,3 = 1 N is constant, and the torsional moment M x,3 (=F e,z a 4 sinθ 4 = 0.4 sinθ 4 Nm) is constant, too, but depends on the limb configurations and its maximum M x,3,max = ±F e,z a 4 = ±0.4 Nm occurs when θ 4 = ±90 • . In addition, the bending moment M z,3 linearly varies from F e,z a 4 cosθ 4 , at D 2 , to (F e,z a 4 cosθ 4 + S y,3 a 3 ), at B 2 , and its maximum values occurs at B 2 when θ 4 reaches its maximum value (see Figure 4). In the studied case (Figures 2 and 4), Carnot's theorem applied to the triangle B 2 A 2 D 2 makes it possible to write 2a 3 a 4 with A 2 B 2 min = a p − r uw and A 2 B 2 max = a p + r uw (11) which gives θ 4,min = 194.36 • and θ 4,max = 302.09 • . As a consequence, the maximum bending moment is M z,3,max = F e,z a 4 cosθ 4,max + S y,3 a 3 = 0.6125 Nm.
3. Case (c): Links' Internal Loads Due to u 3 In this case (a unit moment M e,z (=1 Nm) applied at O p (see Figures 1 and 2)), the equilibrium of the platform shows that the CRS limb is loaded by the force which is applied at the S-pair center and is aligned with link 4 (see Figure 5a), and that the RRC limb is loaded by the force -f m applied at A 1 through the C-pair (see Figure 5a). The maximum and minimum magnitudes of f m occur when the angle µ reaches its minimum, µ min (see Figure 5b), and maximum, µ max (see Figure 5c), values, respectively. With reference to Figure 5b,c, in the studied case, the following relationships hold which give |f m | max = M e,z a p s µ min = 10.079 N, |f m | min = M e,z a p s µ max = 2.95084 N.
Internal Loads in the CRS Limb In link 4, the effect of the small vertical part (i.e., the one parallel to the z b -coordinate axis (Figure 1)) is minor, depends on its length, which has not been defined during the dimensional synthesis, and will be neglected in this static analysis. This assumption brings to consider the force f m directly applied at A 2 (see Figure 5a) along the horizontal part of link 4. As a consequence, this load generates only an axial force A x,4 = -|f m | whose maximum (negative) value is A x,4,max = −10.079 N (see Equation (15)).
The node equilibrium at D 2 shows that link 3, at D 2 , is loaded by the axial force, A x,3 , and the shear force, S z,3 , given by the relationships These loads, at D 2 , generate constant axial, A x,3 , and shear, S z,3 , forces along link 3 that only depend on the mechanism configuration, and a bending moment M y,3 that linearly varies from 0, at D 2 , to −S z,3 a 3 , at B 2 . Since the angles µ (see Equation (14)) and θ 4 that appear in Equations (17a) and (17b) are not independent (see system (4) and Equation (7)), the determination of the maximum/minimum values of A x,3 , S z,3 , and M y,3 must be numerically evaluated for a sufficiently high number of mechanism configurations uniformly distributed in the useful workspace and, then, compared. A mash of 3220 positions of A 1 (=(x p , y p )) uniformly distributed inside the useful-workspace circle have been chosen and, at each A 1 position, Equations (17a) and (17b) have been evaluated for 360 values of the angle ϕ spaced of 1 • . The comparison of the so-computed values yields the diagrams of Figure 6 (The MatLab program that generates Figure 6 can be seen in Supplementary Materials). The analysis of these diagrams and the ranges spanned by the angles θ 4 and µ (see Equations (11) and (15)) bring to the conclusion that, in link 3, the maximum values of the internal loads can be evaluated as follows  Figure 6 (The MatLab program that generates Figure 6 can be seen in Supplementary Materials). The analysis of these diagrams and the ranges spanned by the angles θ4 and μ (see Equations (11) and (15)) bring to the conclusion that, in link 3, the maximum values of the internal loads can be evaluated as follows    (14) reveals that this is a special case of the above-discussed case (a). Consequently, the maximum values of the internal loads of links 1 and 2 are the product of the influence coefficients of Table 1 by |fm|max (see Equation (16)). Table 3 summarizes the results of case (c).  (14) reveals that this is a special case of the above-discussed case (a). Consequently, the maximum values of the internal loads of links 1 and 2 are the product of the influence coefficients of Table 1 by |f m | max (see Equation (16)). Table 3 summarizes the results of case (c).

Implementation of Step (v)
The influence coefficients reported in Tables 1-3 must be combined by means of  Equation (3) to determine the reference internal loads to use for sizing the links. In order to do this, the values of the coefficients λ p appearing in Equation (3)  ϕ = 100 Nm. Accordingly, the influence coefficients of Tables 1 and 2 must be all multiplied by λ p = 250 and those of Table 3 by λ p = 100; then, the resulting values must be summed. Table 4 reports the reference internal loads resulting from this computation.

Discussion
The implementation reported in the previous section shows that the proposed method is simple and direct to apply and only in a few cases it really requires numerical evaluations. In particular, it only needs the results of the dimensional synthesis and, differently from the techniques based on global performance indices, it is able to relate directly the task requirements to the links' internal loads.
The analysis of Table 2 reveals that the existence of decoupled motions does not imply a static decoupling between the active forces that cause those motions. Indeed, the unit force F e,z (see Table 2) has a static effect on links 1 and 2 that belong to a limb, the RRC, that has no effect on the platform translation along the z b -coordinate axis direction. This happens without considering non-ideal phenomena (e.g., the friction in the joints). Furthermore, the found effects are difficult to compensate since the generated torsional, M x, (1,2) , and bending, M z, (1,2) , moments lie on the RRC-limb motion plane and tend to cause elastic displacements (which, in practice, are vibrations, since all the real external loads are dynamic loads that vary their magnitudes during motion) perpendicular to that plane.
Recognizing static couplings at an early stage of the design procedure with a clear identification of the causes is a peculiarity of the proposed method and is a desirable feature that can orientate the successive design choices. Other design techniques based on the extensive use of multi-body simulation software, together with a trial-and-error sizing procedure, would have found the limb vibration without giving an immediate explanation.
These results prove that the proposed algorithm integrates the design methodologies based on performance indices and/or on extended multibody simulations by solving their lacks.

Conclusions
A general technique based on influence coefficients and static analyses has been presented which allows the determination of the input data of the machine-element design (i.e., reference external loads and machine configurations that give the maximum internal load of each component) for non-overconstrained manipulators.
Differently from other techniques, the presented one uses only the results of the dimensional synthesis and does not need any tentative choice, about the materials, the links are made of, and links' cross-sections, to be adjusted during the design procedure.
The proposed technique directly relates task requirements to links' internal loads and is able to reveals dynamic phenomena (e.g., vibrations) that have a negative impact on the machine behavior.
The effectiveness of the proposed technique has been proved by implementing it on a 3T1R parallel manipulator.