# Single DOF Spherical Mechanisms Using Instantaneous Poles Biology Essay

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Instantaneous poles have properties that are the spherical counterparts of instant centers; however, they are not completely used to study the kinematic behavior of spherical mechanisms. Instantaneous kinematics of a mechanism becomes undetermined when it is in a singular configuration (singularities); this indeterminacy has undesirable effects on static and motional behavior of the mechanism, so these configurations must be found and avoided. This paper presents a new geometrical method to find singularities of single-dof spherical mechanisms. First, mechanical advantage of single-dof spherical mechanisms is obtained using the concept of instantaneous poles; and then it is used to find geometric conditions corresponding to each type of singularities, occurring in this class of mechanisms.

## 1. INTRODUCTION

For two co-spherically moving shells, there exist two instantaneously coincident points, each belonging to the respective shell, the linear velocities of which are identical. The place of these common points is called Instantaneous Pole, henceforth referred to as Instant Pole, of the two shells [1]. Instant poles have the same properties as instant centers in planar mechanisms.

Instant centers are used to fully describe the instantaneous kinematics of planar mechanisms, e.g. [2-8]. Daniali [3] introduced a fast method to find singular configurations of planar parallel manipulators, based on the properties of instantaneous center of rotations. Di Gregorio [9] presented an algorithm that computes the instant centers in all single-dof planar mechanisms, and an exhaustive analytical and geometrical study [4] about the singularity conditions occurring in single-dof planar mechanisms, which is based on the instant centers. He also found [5] singular configurations of n-dof planar mechanisms by considering them as the union of n one dof mechanisms.

There is a close relation between singularities of a one-dof mechanism and its stationary configurations; in this case, Yan and Wu [7, 8] gave a geometric criterion to identify which instant centers coincide at stationary configurations of planar mechanisms [7] and developed a geometric methodology to generate planar one-dof mechanisms in dead center positions [8].

All of the above mentioned works can b done for spherical mechanisms using the concepts of Instant poles. For instance, deducing from his previous work [9], Di Gregorio presented an exhaustive algorithm to determine the instant poles' positions of single-dof spherical mechanisms [10].

In contrast to the planar mechanisms, singularity analysis of spherical mechanisms is more challenging and most of the presented methods are analytical (see for instance [11-13]). Here, singularities of spherical mechanisms are analyzed using the concept of the instant poles. The presented method is a 3D graphical method which is not as simple as the planar ones.

This paper is organized as follows: next section will present some notations and definitions. In section 3, mechanical advantage of single-dof spherical mechanisms is computed using the concept of instant poles, and then section 4 presents geometric conditions for each type of singularities which occur in this group of mechanisms. Two illustrative examples are presented in section 5 to show the method. Finally, some conclusions for this research activity are presented in section 6.

## 2. NOTATION

Spherical mechanisms can be studied by projecting them through the spherical-motion center onto the reference sphere (Fig. 1). So doing, the instant poles become points of the reference sphere. The links become spherical shells that move on the reference sphere. With reference to a Cartesian system whose origin is at the spherical motion center, intersections between the reference sphere and xy plane, yz plane, and zx plane will be called primary great circle (PGC), infinity great circle (IGC), and declination great circle (DGC), respectively. The IGC cuts the reference sphere into two hemispheres: the one (positive hemisphere) whose points have a positive x coordinate and the other (negative hemisphere) whose points have a negative x coordinate. Two great circles with different slopes have only one intersection in the positive (negative) hemisphere, whereas two great circles with the same slope (i.e. that belong to the same pencil of meridians) do not intersect each other in the positive (negative) hemisphere. An Instantaneous pole Axis that does not lie on the yz plane intersects the reference sphere at two diametrically opposite points: one lying on the positive hemisphere and the other lying on the negative hemisphere. An Instantaneous pole Axis that lies on the yz plane cuts the IGC into two diametrically opposite points: one either with positive y coordinate, or with zero y coordinate and positive z coordinate, and the other either with negative y coordinate, or with zero y coordinate and negative z coordinate. Therefore, the points of the positive (negative) hemisphere plus the points of the IGC either with positive (negative) y coordinate or with zero y coordinate and positive (negative) z coordinate are sufficient to identify all the possible instant pole. This set of points, which is a subset of the RS, will be called positive (negative) shell.

Figure 1. Reference sphere and Cartesian reference system fixed to the frame: IGC=infinity great circle, PGC=primary great circle, DGC=declination great circle.

As it is shown in the next section, mechanical advantage of a single-dof spherical mechanism can be computed using only instant poles of the relative motions among three links: input link (i), output link (o), reference link (1), i.e. reference sphere, used to evaluate the rate of input and output variables. The input (output) variable is a geometric parameter that defines the pose (position and orientation) of link ''i'' (''o'') with respect to link ''1''.

For the three links mentioned above, the different relative motions are three, and they will be denoted ''1i '', ''oi'' and ''1o '' (the second letter indicates the link from which motion of the link, denoted by the first letter, is observed). Instant poles of these three motions will be denoted Pmn, where mn Î {1i, oi, 1o}. Pmn will denote the position vector that locates the instant pole Pmn on the surface of the sphere (it is meant that all the position vectors are defined in a unique reference system fixed to the center of the reference sphere).

In a one-dof spherical mechanism, the positions of all instant poles between any two links of mechanism uniquely depend on its configuration. Such positions can be found by graphical methods, using properties of the instant poles (Aronhold-Kennedy theorem, etc.).

Aronhold-Kennedy theorem in plane kinematics can be adopted for spherical kinematics and stated as follows [1]:

## Three instant poles of the three co-spherically moving links lie on a unique great circle.

According to the theorem, the three instant poles P1i, Pio and P1o must lie on a unique great circle.

Since instant poles' positions are sufficient to fully describe the first-order kinematics of spherical mechanisms, the first-order kinematics of the spherical mechanisms can be studied by using only one (either positive or negative) shell of the reference sphere [10]. Hereafter the positive shell will be used.

The only kinematic pairs that appear in spherical mechanisms are revolute pairs, rolling contacts and slipping contacts. The relative motion between two links joined by any of these pairs is uniquely determined by the position of its instant poles. Note that, for a slipping contact, the correspondent instant pole lies on the intersection of great circles that are normal to both the spherical curves which slip on each other.

In this paper, only the single input-single output mechanisms (SISO mechanisms) are considered; however a one-dof single input-multiple output mechanism (SIMO mechanism) can be considered as n independent SISO mechanisms working in parallel.

Figure 2. Projection of reference sphere and instant poles , and on a plane parallel to the plane of the paper.

## 3. MECHANICAL ADVANTAGE AND INSTANT POLES

If we assume that a mechanism is a conservative system (i.e. energy losses due to friction, heat, etc., are negligible compared to the total energy transmitted by the system), and if we assume that there are no inertia forces, input power (Pin) is equal to output power (Pout). On the other hand, if we consider a single-dof spherical mechanism as an input-output device, then following relation can be written.

Pin = Tinwin = Toutwout = Pout (1)

or

(2)

Where Tin (Tout) denotes the magnitude of input (output) torque and win (wout) is the magnitude of angular velocity of input (output) link. Therefore, mechanical advantage (M.A.) is obtained as follows.

(3)

In which Fin (Fout) is the magnitude of force applied on the input (output) link and rin (rout) is the input (output) torques' arm.

With reference to Fig. 2, velocity of instant pole Pio, , can be written as

(4)

is the angle between vector and angular velocity vector of input link and is the angle between vector and angular velocity vector of output link. Considering Eqs. (3) and (4) simultaneously, leads to the following relation.

(5)

## 4. SINGULARITY ANALYSIS OF SPHERICAL MECHANISMS WITH ONE DEGREE OF FREEDOM

Gosselin and Angeles [14] identified three types of singularities:

Type (I) singularities (inverse kinematic singularities) occur when inverse instantaneous kinematic problem is indeterminate. In one-dof mechanisms, such singularities occur when the output link reaches a dead center, i.e. when an output variable reaches a border of its range; also in type (I) singularities mechanical advantage becomes infinite because in this configurations, at least one component of output torque (force), applied to the output link, is equilibrated by the mechanism structure without applying any input torque (force) in the actuated joints.

Type (II) singularities (direct kinematic singularities) occur when direct instantaneous kinematic problem is unsolvable. In one-dof mechanisms, such configurations occur when the input link reaches a dead center. In type (II) singularities, a (finite or infinitesimal) output torque (force), applied to the output link, need at least one infinite input torque (force) in the actuated joints to be equilibrated which in one-dof mechanisms, corresponds to a theoretically zero mechanical advantage.

Type (III) singularities (combined singularities) occur when both the inverse and direct instantaneous kinematic problems are unsolvable, i.e. when two previous singularities occurs simultaneously; In this type of singularities the input-output instantaneous relationship, used out of such singularities, holds no longer and the mechanism behavior may change. In one-dof mechanisms, these singularities lead to one or more additional uncontrollable dofs.

Considering the above discussion, geometric conditions of singularities occurring in single-dof mechanisms can be obtained through Eq. (5). The analysis of Eq. (5) brings to the conclusion that an inverse kinematic singularity (M.A.=∞) occurs when

(6)

On the other side, Eq. (5) shows that a direct kinematic singularity (M.A.=0) occurs when

(7)

Finally, combined singularities occur when two previous singularities occur simultaneously for the same configuration of the mechanism.

## 5. ILLUSTRATIVE EXAMPLES

In the following subsections, singularities of two single-dof spherical mechanisms are analyzed to show the method presented.

## 5.1. SINGULARITY ANALYSIS OF AN INTERSECTING FOUR BAR SPHERICAL MECHANISM

This mechanism is shown in Fig. 3; link 2 is the input link; link 4 is the output link, q21 and q41 are the input and output variables, respectively; link 1 (sphere) is the reference link used to evaluate the rate both of the input variable and of the output variable; so i = 2 and o = 4. P24 is located at the intersection of two great circles P12 P14 and P23 P34.

According to the condition (6), type (I) singularities occur when Pio coincides with P1i, i.e. when P24 coincides with P12. An example of this type of singularities is shown in Fig. 4, in which links 2 and 3 are located on the same great circle and output variable q41 is at its dead center position.

Figure 3. The intersecting four bar spherical mechanism at a generic configuration with its instant poles.

Figure 4. The intersecting spherical four bar mechanism at a type (I) singularity.

condition (7) shows that type (II) singularities occur when Pio coincides with P1o ,i.e. when P24 coincides with P14, Fig. 5 shows an example of this type of singularities for the mechanism under study in which links 3 and 4 are located on the same great circle and input variable q21 is at dead center position.

Type (III) singularities occur when two previous singularities occur simultaneously which is equivalent to coincidence of P12, P24 and P14; this condition is satisfied when P24 is not determined and identifies a configuration in which the mechanism is flattened, see Fig. 6.

Figure 5. The intersecting spherical four bar mechanism at a type (II) singularity.

Figure 6. The intersecting spherical four bar mechanism at a type (III) singularity.

## 5.2. SINGULARITY ANALYSIS OF A SIX BAR SINGLE-DOF SPHERICAL MECHANISM

Figure 7 shows a six bar single-dof spherical mechanism together with its instant poles. Link 2 is the input link; link 6 is the output link; link 1 (sphere) is the reference link used to evaluate the rate both of the input and output variables; therefore, ''i'' = 2, ''o'' = 6. q21 is the input variable. Arbitrary point A is fixed to link 1. s61 is the output variable which is the distance between point A and a specific point on output link 6. The location of instant pole P26 can be obtained as follows: (i) P14 is located at the intersection of two great circles P15 P45 and P46 P16; (ii) P13 is located at the intersection of two great circles P14 P34 and P12 P23; (iii) P36 is located at the intersection of great circles P13 P16 and P34 P46 and finally (iv) P26 is located at the intersection of two great circles P36 P23 and P16 P12.

Singularity condition (6) brings to the conclusion that type-(I) singularities occur when P26 coincides with P12. Considering Fig. 7, one can see that coincidence of these two instant poles occur when axis of link 2 is spherically perpendicular to axis of link 4 which is the known geometric condition identifying dead-center positions of s61. Fig. 8 shows the mechanism at a type-(I) singularity.

According to condition (7), type (II) singularities occur when P26 coincides with P16. With reference to Fig. 7, P26 coincides with P16 when axis of link 4 and direction of motion of link 6 are located on the same plane; in this type of singularities q12 is at its dead-center position, See Fig. 9.

Finally type (III) singularities occur when both previous singularities occur simultaneously, i.e. P26 must coincide with P12 and P16 simultaneously; this condition is satisfied when the position of P26 is not determined, see Fig. 10.

Figure 7. The spherical six bar mechanism at a generic configuration with its instant poles.

Figure 8. The spherical six bar mechanism at a type (I) singularity.

Figure 9. The spherical six bar mechanism at a type (II) singularity.

Figure 10. The six bar spherical mechanism at a type (III) singularity.

## 6. CONCLUSION

A new geometric method for singularity analysis of single dof spherical mechanisms was presented. First, mechanical advantage of single dof spherical mechanisms was obtained using the concept of instant poles; then geometric conditions corresponding to different types of singularities were found for these mechanisms using the obtained mechanical advantage. Finally two illustrative examples were presented to show the method. The method is simple and comprehensive and can be used to find singular configurations of all types of single dof spherical mechanisms.