Application of Mechanical Advantage and Instant Centers on Singularity Analysis of Single-Dof Planar Mechanisms 50

Application of mechanical advantage and instant centers on singularity analysis of single-dof planar mechanisms 50

APPLICATION OF MECHANICAL ADVANTAGE AND INSTANT CENTERS ON SINGULARITY ANALYSIS OF SINGLE-DOF PLANAR MECHANISMS

Soheil Zarkandi* Department of Mechanical Engineering Babol University of Technology, Mazandaran, Iran *Corresponding email: [email protected]

Abstract: Instantaneous kinematics of a mechanism becomes undetermined when it is in a singular configuration; this indeterminacy has undesirable effects on static and motional behavior of the mechanism. So these configurations must be found and avoided during the design, trajectory planning and control stages of the mechanism. This paper presents a new geometrical method to find singularities of single-dof planar mechanisms using the concepts of mechanical advantage and instant centers.

Key Words: Planar mechanisms; Singularity; Instant center; Angular velocity; Mechanical advantage.

INTRODUCTION (force), applied to the output link, need at least one The concept of instantaneous center of rotation infinite input torque (force) in the actuated joints to (instant center) was introduced by Johann Bernoulli1. be equilibrated, which in one-dof mechanisms, Instant centers are useful for velocity analysis of corresponds to a zero mechanical advantage. planar mechanisms and for determining motion Type (III) singularities (combined singularities) transmission between links2. The method has proved occur when both the inverse and the direct to be efficient in finding the input-output velocity instantaneous kinematic problems are unsolvable, i.e. relationships of complex linkages3. when two previous singularities occurs Another application of instant centers is simultaneously; In this type of singularities the singularity analysis of planar mechanisms. Different input–output instantaneous relationship, used out of approaches have been adopted in dealing with such singularities, holds no longer and the singularity analysis of planar mechanisms; mechanism behavior may change. In one-dof considering a mechanism as an input–output device, mechanisms, these singularities lead to one or more Gosselin and Angeles4 identified three types of additional uncontrollable dofs. singularities: Many articles have been presented for singularity Type (I) singularities (inverse kinematic analysis of single5-6 or multi7-10 dof planar singularities) occur when inverse instantaneous mechanisms; some of these articles have kinematic problem is unsolvable. This type of geometrically addressed the singularity analysis of singularities occurs when at least one out of the planar mechanisms using instant centers. For input-variable rates can be different from zero even instance, Daniali10 classified singularities of 3-dof though all the output-variable rates are zero. In one- planar parallel manipulators through instant centers. dof mechanisms, such singularities occur when the Di Gregorio6 presented an exhaustive analytical and output link reaches a dead center, i.e. when an output geometrical study about the singularity conditions variable reaches a border of its range; also for this occurring in single-dof planar mechanisms, which is type of mechanisms, in type (I) singularities based on the instant centers. He also9 found singular mechanical advantage becomes infinite because in configurations of multi dof planar mechanisms, this configurations, at least one component of output considering the n dof mechanisms as the union of n torque (force), applied to the output link, is one dof planar mechanisms and using the principle of equilibrated by the mechanism structure without superposition. applying any input torque (force) in the actuated There is a close relation between singularities of a joints. one-dof mechanism and its stationary configurations; Type (II) singularities (direct kinematic in this case, Yan and Wu11, 12 gave a geometric singularities) occur when direct instantaneous criterion to identify which instant centers coincide at kinematic problem is unsolvable. This type of a stationary configuration11 and developed a singularities occurs when at least one out of the geometric methodology to generate planar one-dof output-variable rates can be different from zero even mechanisms in dead center positions12. Here the though all the input-variable rates are zero. In one- author presents a method for singularity analysis of dof mechanisms, such configurations occur when the one-dof planar mechanisms using the concepts of input link reaches a dead center. In type (II) mechanical advantage and instant centers. singularities, a (finite or infinitesimal) output torque

Journal of Mechanical Engineering, Vol. ME 41, No. 1, June 2010 Transaction of the Mech. Eng. Div., The Institution of Engineers, Bangladesh Application of mechanical advantage and instant centers on singularity analysis of single-dof planar mechanisms 51

This paper is organized as follows: some notations MECHANICAL ADVANTAGE AND INSTANT are presented in section 2; section 3 shows how CENTERS instant centers can be used to calculate mechanical If we assume that a mechanism is a conservative advantage of single dof planar mechanisms; in system (i.e. energy losses due to friction, heat, etc., section 4, singularities of single dof planar are negligible compared to the total energy mechanisms are analyzed using the results obtained transmitted by the system), and if we assume that in previous section; in section 5, two illustrative there are no inertia forces, input power (Pin) is equal examples are presented to show the method; and to output power (Pout), i.e., finally section 6 presents some conclusions of this research activity. Pin = Pout

NOTATION Also, if we consider a single-dof mechanism as an As it is shown in the next section, mechanical input-output device, the input and output powers can advantage of a single dof mechanism can be be calculated in different ways according to type of computed using only the instant centers of the input and output variables. relative motions among the four links: input link (i), If the variable is rotational, then the power is output link (o), reference link (1) used to evaluate the rate of the input and output variables and link (k) P = T ω = r F ω (1a) which is an arbitrary link except the three previous ∗ ∗ ∗ ∗ ∗ ∗ links. The input (output) variable is a geometric parameter that defines the pose (position and And if the variable is translational, then the power is orientation) of link ‘‘i’’ (‘‘o’’) with respect to link ‘‘1’’. P∗ = F∗V∗ (1b) For the four links mentioned above, the different relative motions are six, and they will be denoted ‘‘1i Where (∗) can be “in” or “out” which are the ’’, ‘‘ok’’, ‘‘ik’’, ‘‘1k ’’, ‘‘oi’’ and ‘‘1o ’’ (the second summarized form of the words “input” and “output”, letter indicates the link from which the motion of the respectively and α∗, α ∈{P, T, r, F, ω, V}, denotes link, denoted by the first letter, is observed). The the input or output part of parameter “α”. P∗ is instant centers of these six motions will be denoted referred to as power. T∗ is representative of input and as Cmn, where mn ∈ {ip, ok, ik, kp, oi, op }. Cmn will output torques, ω∗ is angular velocity of the denote the position vector that locates the instant correspondent input and output links, r∗ is the center Cmn in the plane of motion (it is meant that all torque’s arm, F is the magnitude of force applied on the position vectors are defined in a unique reference ∗ the input and output links and finally V is velocity of system fixed to the plane of motion). According to ∗ input and output links. the Aronhold–Kennedy theorem, these instant centers It is worth noting that in the case, the variable is must lie on the straight lines shown in Fig. 1. translational (i.e. Eq. (1b)), the direction of force F In this paper, we just consider a single input-single ∗ is considered to be parallel to the direction of motion output mechanism (SISO mechanism), however a of the correspondent link. one dof single input-multiple output mechanism According to the type of input or output variables, (SIMO mechanism) can be considered as n we can classify single-dof mechanisms in four independent SISO mechanisms working in parallel. groups: mechanisms in which (i) both the input and

the output variables are rotational (rot–rot mechanisms), (ii) the input variable is a rotational Cik angle and the output variable is a translational (rot– tra mechanisms), (iii) the input variable is a C1k translational and the output variable is a rotational angle (tra–rot mechanisms), and, finally, (iv) both Cko input and output variables are translational (tra–tra mechanisms). In the following subsections, C1i C1o mechanical advantage is obtained for the each group. C oi Rot–rot mechanisms In this group of mechanisms both the input and output variables are rotational, so following relations Figure 1. Straight lines the six instant centers lie on. can be written.

(2) Pin = Tinωin = Toutωout = Pout

Tout ω in = (3) Introducing Eq. (9) into the Eq. (8) leads to, T ω in out r F ω = F C − C ω (10) Therefore, mechanical advantage is obtained as in in in out 1i io in follows. So mechanical advantage for this group of F r T r ω mechanisms is M.A. = out = ( in )( out ) = ( in )( in ) (4) F r T r ω in out in out out F r M.A. = out = in (11) Figure 1 shows the geometric relationship among Fin C1i − Cio the six instant centers. With reference to Fig. 1, velocity of instant center C , C& , can be written as Tra–rot mechanisms io io In this type of mechanisms, the input variable is translational and the output variable is rotational and

C&io = C1i − Cio ωin = C1o − Cio ωout (5) following relation can be written. which results in Pin = FinVin = Toutωout = rout Foutωout = Pout (12)

ω C − C ( in ) = 1o io (6) ωout C1i − Cio Cik C1k

So C C1i ∞ ok r C − C M.A. = ( in ) 1o io (7) rout C1i − Cio C1o Coi

Rot–tra mechanisms In this group of mechanisms, the input variable is Figure 3. Straight lines the instant centers lie on rotational and the output variable is translational and when C is at infinity. following relation can be written. 1i

As it is shown in Fig. 3 instant center C is P = T ω = r F ω = F V = P (8) 1i in in in in in in out out out located at infinity. Velocity of the input link is equal to the velocity of instant center Cio, so

Vin = C1o − Cio ωout (13) Cik Introducing Eq. (13) into the Eq. (12) leads to

C1k Cok Fin C1o − Cio ωout = rout Foutωout (14) ∞ C1o So mechanical advantage is obtained as; C1i C oi F C − C M.A. = out = 1o io (15) F r in out

Figure 2. Straight lines the instant centers lie on Tra–tra mechanisms when C1o is at infinity. In this group of mechanisms, both the input and output variables are translational and following According to Fig. 2, instant center C1o is located at relation can be written. infinity. Velocity of the output link is equal to the Velocity of instant center C , So io Pin = FinVin = FoutVout = Pout (16)

Vout = C1i − Cio ωin (9)

With reference to Fig. 4, the instant centers C1i, Rot–rot mechanisms C1oand Cio are located at infinity, so different method The analysis of Eq. (7) brings to the conclusion must be used to compute the mechanical advantage. that an inverse kinematic singularity (M.A.=∞) Every point on input link has the same velocity as occurs when (Fig. 5). the instant center Cik , so

C1i = Cio (21) Vin = C1k − Cik ωk (17) On the other side, Eq. (7) shows that a direct Also every point on output link has the same kinematic singularity (M. A.=0) occurs when (Fig. 6). velocity as the instant center Cok , so C1o = Cio (22)

Vout = C1k − Cok ωk (18) Finally, combined singularity occurs when Eqs. (21) and (22) are satisfied simultaneously for the Where ωk is angular velocity of link k. Introducing Eq. (17) and (18) into the Eq. (16) results same configuration of the mechanism. in

Fin C1k − Cik ωk = Fout C1k − Cok ωk (19) C1k Cko So mechanical advantage is C1o

F C − C M.A. = out = 1k ik (20) Fin C1k − Cok Cik, Cio, C1i,

Figure 5. An example of type-(I) singularities of the C1k rot–rot mechanisms in which condition (21) is matched.

Cik Cok ∞ Coi

∞ C1k Coi, Cok, C1o C1i ∞

C1o Cik Figure 4. Straight lines the instant centers lie on

when C1i and C1o are at infinity. C1i

SINGULARITY ANALYSIS OF PLANAR MECHANISMS WITH ONE DEGREE OF Figure 6. An example of type-(II) singularities of the FREEDOM rot–rot mechanisms in which condition (22) is According to the classification presented in [4], matched. there exist three types of singularities: (1) Type (I) singularities in which mechanical Rot–tra mechanisms advantage becomes infinite. Since denominators in Eqs. (7) and (11) are (2) Type (II) singularities in which mechanical identical, geometric condition (21) identifies the advantage theoretically1 become zero. inverse kinematic singularities for this case too. (3) Type (III) singularities in which two previous However, in the rot–tra mechanisms, the singularities occur simultaneously. configuration of mechanism corresponding to Eq. In the following subsections, the above conditions (21) is different from the one of the same equation in will be applied on Eqs. (7), (11), (15) and (20) and the rot–rot mechanisms. For Example, Fig. 5 geometric conditions identifying singularities will be becomes Fig. 7 in the rot–tra mechanisms. given for all groups of mechanisms. On the other side, Eq. (11) brings to the conclusion that direct kinematic singularitis (M.A.=0) occur when Cio locates at infinity. Note that, according to Aronhold–Kennedy theorem, instant centers Cio, C1o 1 Note that long before mechanical advantage and C1i lie on the same line; therefore this type of becomes zero, the mechanism breaks down. singularities occurs when the following geometric condition is matched (See Fig. 8).

∞ Cio Cio = C1o = ∞ (23) Cik

Finally, combined singularity occurs when Eq. Cok, C1k ∞ C1i (21) together with Eq. (23) are satisfied for the same configuration of the mechanism. C1o

C1k Figure 9. An example of type-(I) singularities of the C tra–rot mechanisms in which condition (24) is 1o ∞ Cok matched.

C1k C 1i ∞ Cik Cio, Cik, C1i

Figure 7. An example of type-(I) singularities of the rot–tra mechanisms in which condition (21) is matched. Coi, Cok, C1o

Figure 10. An example of type-(II) singularities of C io ∞ the tra–rot mechanisms in which condition (22) is matched.

C Cik, C1k C1o 1o ∞ Cok ∞

Cik C1i ∞ Coi Cok, C1k ∞ C1i

Figure 8. An example of type-(II) singularities of the rot–tra mechanisms in which condition (23) is Figure 11. An example of type-(I) singularities of the matched. tra–tra mechanisms in which condition (25) is matched. Tra–rot mechanisms The analysis of Eq. (15) brings to the conclusions Tra–tra mechanisms that an inverse kinematic singularity occurs when Cio Considering Eq. (20), inverse kinematic singularity locates at infinity. Again, according to Aronhold– occurs when (Fig. 11). Kennedy theorem, this condition leads to (Fig. 9). C = C (25) C = C = ∞ (24) 1k ok io 1i and direct kinematic singularity occurs when (Fig. Comparing Eq. (15) and (7), one can see that 12). geometric condition (22) identifies the direct kinematic singularities for this case too. However, in C = C (26) the tra–rot mechanisms, the graphic representations 1k ik of Eq. (22) are different from the ones of the same equations in the rot–rot mechanisms. For instance, Finally, combined singularities occur when both Fig. 6 becomes Fig. 10 in the tra–rot mechanisms. conditions (25) and (26) are satisfied for the same Finally, type-(III) singularities occur when Eq. configuration of the mechanism. (22) together with Eq. (24) are satisfied for the same Now singularities of single-dof planar mechanisms configuration of the mechanism. can be found using the above conditions.

C23

3 Coi ∞ Cik, C1k Cok C34 C1o ∞

2 4 C ∞ C12, C24 14

C1i

Figure 12. An example of type-(II) singularities of Figure 14. The intersecting four bar mechanism at a the tra–tra mechanisms in which condition (26) is type (I) singularity. matched. C34 ILLUSTRATIVE EXAMPLES In the following subsections, singularities of two 3 single-dof planar mechanisms are analyzed to show C23 the method presented.

2 4 Singularity analysis of an intersecting four bar C12 mechanism C14, C24 This mechanism is shown in Fig. 13. With reference to Fig. 13, Link 2 is the input link; link 4 is the output link and link 1 is the reference link used to Figure 15. The intersecting four bar mechanism at a evaluate the rate both of the input variable and of the type (II) singularity. output variable. θ21 and θ41 are the input and output variables, respectively, so ‘‘i’’ = 2, ‘‘o’’ = 4 and the C12 C14 C23 C34 2 4 3 mechanism is in the group of rot–rot mechanisms.

C34 Figure 16. The intersecting four bar mechanism at a type (III) singularity. 3 C23 Singularity analysis of a six bar single-dof 4 mechanism Figure 17 shows a six bar single-dof mechanism 2 together with its instant centers. Link 2 is the input link; link 6 is the output link; link 1 is the reference C12 θ14 θ12 C14 C24 link used to evaluate the rate both of the input and output variables. Arbitrary point A is fixed to link 1. θ21 is the input variable; S61 is the output variable; Figure 13. The intersecting four bar mechanism at a Therefore, ‘‘i’’ = 2, ‘‘o’’ = 6, and the mechanism is generic configuration with its instant centers. included in the group of the rot–tra mechanisms. Singularity condition (21) brings to the conclusion According to the condition (21), type (I) that type-(I) singularities occur when C26 coincides singularities occur when Cio coincides with C1i, i.e. with C12. Considering Fig. 17 one can see that when C24 coincides with C12. An example of this type coincidence of these two instant centers occur when of singularities is shown in Fig. 14, in which links 2 link 2 is perpendicular to link 4 which is the known and 3 are collinear. geometric condition identifying the dead-center The condition (22) shows that type (II) positions of S61. Fig. 18 shows the mechanism at a singularities occur when C24 coincides with C14. type-(I) singularity. Figure 15 shows an example of this type of According to condition (23), type (II) singularities singularities for the mechanism under study in which occur when C26 coincides with C16. With reference to links 3 and 4 are collinear. Fig. 17, C26 coincides with C16 when the direction of Type (III) singularities occur when two previous motion of link 6 is parallel to link 4 and θ12 is at its singularities occur simultaneously which is dead-center position, See Fig. 19. equivalent to coincidence of C12, C24 and C14; this Finally type (III) singularities occur when both condition is satisfied when C is not determined and 24 previous singularities occur simultaneously, i.e. C26 identifies a configuration in which the mechanism is must coincide with C12 and C16 simultaneously; this flattened, see Fig. 16. condition is satisfied when the position of C26 is not

∞ C13, C36 C46 A 1 C26 2 3 C C23 C23 12 2 θ21 ∞ C14 3 4 C12 C13 C15 5 4

1 C45 C14 C15 C45 Figure 20. The six bar mechanism at a type (III) 5 singularity. 1

Figure 17. The six bar mechanism at a generic CONCLUSION configuration with its instant centers. A geometric method for singularity analysis of single dof planar mechanisms was presented. First, 6 one dof planar mechanisms were classified into four groups, based on the type of input and output

C46 variables; then mechanical advantage for each group of the mechanisms was obtained using the concept of C12, C26 instant centers and geometric conditions ∞ corresponding to different types of singularities were C36 found for each group. Finally two illustrative C23 2 examples were presented to show the method. 3 The method is simple and comprehensive and can 4 be used to find singular configurations of all types of ∞ C14 C13 single dof planar mechanisms.

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Journal of Mechanical Engineering, Vol. ME 41, No. 1, June 2010 Transaction of the Mech. Eng. Div., The Institution of Engineers, Bangladesh