Gr¨uebler's criterion

Gr¨uebler's criterion calculates the theoretical number of D.O.F within a mechanism. This is also known as the mechanism's F number.

D.O.F is the number of independent joint variables which must be specified in order to define the position of all links within a mechanism. A body restricted to planar motion has at most three D.O.F. The link is a rigid body. The joint is a contact (or permanent

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connection) between two links. The number of links may be denoted by l. The number of joints may be denoted by j. The type of joint (or connection) defines the relative motion of the two connected links.

There are five categories of contacts in spatial motion, they allow for fj

D.O.F between the connected bodies, where 1≤ f_j ≤5. A f_j = 6 contact would be a non-contact. The theoretical D.O.F F within a mechanism calculated by the Gr¨uebler's criterion is expressed as follows.

where λ is the mobility of the space in which the mechanism operates (λ = 3 for general plane mechanism, λ = 6 for spatial mechanism). The number of independent loops of the mechanism is denoted by L. According to Euler’s formula it is obtained that L = j – l + 1. The above equation is then written into

∑ −

= f L

F _j λ (2.2)

The above equation is useful in mechanism synthesis design when the number of independent loop usually indicates the complexity of the mechanism.

2.2.1 Joint

An artificial joint, joint with no D.O.F ( = 0), is introduced in this chapter for the following derivations. The joints with

are then called the normal joints. The joints connect the links can be revolute (f composed by the lower order joints f_j ≤3. The high order joints may

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not be generally useful in the parallel manipulators for their difficulty in fabrication. In this chapter, we will look only into the joints with . The normal joint attached with an actuator is called the active joint; on the other hand, it is called the passive joint.

≤3 fj

2.2.2 Dyad, Triad, and Quad

The F number within a mechanism calculated by the Gr¨uebler's criterion is provided with the existence of a ground link. The mobility ground link is subtracted from the overall F number. In case that the kinematic structure is floating with respect to the ground, the mobility M of the kinematic structure may be derived from Gr¨uebler's criterion as

The mobility of the limb is derived from eq. (2.3) that +∑

= _ji

i f

M λ _, (2.4)

where ∑ f_j,i denotes the total number joint D.O.F on the ith limb.

The limb with one joint connecting two links is called a dyad. The limb with two joints in series connection to three links is called a triad.

The limb with three joints in series connection to four links is called a quad. For convenience, the total number of D.O.F of the joints is denoted by T that

=∑ f_j

T . (2.5)

The total number of joint D.O.F of the ith limb, according to eq.

(2.4), is derived as

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λ +

= T

M (2.6)

In order to avoid the under-constraint condition of the Gr¨uebler's criterion, the limb should have the total joint D.O.F T no more than λ.

Otherwise, there will be either an uncontrollable rotation or an uncontrollable translation in the limb. The limb with T = λ is called the saturated limb. The ambient space of a saturated limb is the space in which the mechanism operates.

Table 1 lists some of the useful spatial (λ = 6) triads using the aforementioned joints. The term “class” i.e. s^T, in Table 1 is used to denote the class of limbs with total joint D.O.F of the limbs equal to T.

In Table 1, the SS triad, as shown in Fig. 4, is a saturated limb, however, there is an uncontrollable rotation between two spherical joints. The uncontrollable rotation presents as an under-constraint of the Gr¨uebler's criterion.

Table 2 lists some of the useful spatial (λ = 6) quads using the aforementioned joints.

In order to classify limbs, we define the limb embodying an active joint the active limb; it is a passive limb if no actuator is attached. Different versions of actuators are shown in Fig. 5. The actuators being connected to the ground are called ground actuators;

others are called the floating actuators.

2.2.3. Open chain

A limb is defined as the limb with one end grounded. A convenient way to represent an open-chain is the graph representation as shown in Fig.6

In the analysis point of view, the ambient space of the open-chain 9

is given by the Cartesian product of the joint spaces of all the joints that make up the open-chain [22].

2.2.3. Parallel manipulator

A parallel manipulator as shown in Fig. 7 is regarded as a set of limbs connected in parallel to a common rigid body, known as the end-effector. The reduced graph may be expanded into individual graphs representing different limbs.

Let an integer n denote the number of limbs, an integer li denote the number of links of the ith limb respectively, and an integer j_i denote the number of joints of the ith limb respectively. The total number of joints including the artificial joints and links in the parallel manipulators are

j = ∑ ji+ 2n (2.7)

l = ∑l_i +2 (2.8)

Knowing that the artificial joints are with zero joint D.O.F., the total joint D.O.F. is calculated as

=∑ ∑

∑ f_j f_j,i (2.9)

Since all limbs are grounded on one end and connected to the end-effector on the other end, the end-effector becomes a common link of all open-chains made of the limbs. According to previous statement that the ambient space of the open-chain, the ambient space of end-effector (the common link) is then the intersection of ambient spaces of all individual open-chains. For example, the intersection of PUP quad and PS triad open-chain will result maximally an ambient space of one axis of translation with two axes of rotation to the end effector.

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