A Unifying Framework for Classification and

D. Zlatanov R. G. Fenton

Computer Integrated Manufacturing Laboratory, Department of IVlectianical Engineering, University of Toronto, 5 King's College l^oad, Toronto, Ontario, Canada, M5S 1A4

1

A Unifying Framework for Classification and Interpretation of Mechanism Singularities This paper presents a generalized approach to the singularity analysis of mechanisms with arbitrary kinematic chains and an equal number of inputs and outputs. The instantaneous kinematics of a mechanism is described by means of a velocity equation, explicitly including not only the input and output velocities but also the passive-joint velocities. A precise definition of singularity of a general mechanism is provided. On the basis of the six types of singular configurations and the motion space interpretation of kinematic singularity introduced in the paper, a comprehensive singularity classification is proposed.

Introductitm

In certain configurations, usually referred to as "singular," "special," or "critical," the kinematic (and static) properties of mechanisms change dramatically. The study of such configurations is of sig»ificant importance for the synthesis and control of mechanisms (LipJdn and PoM, 1991). The singularity problem has been studied extensively for open-loop kinematic chakis (Warldron et al., 1985; Hunt, 1986; Lai and Yang, 1986; Wang and Waldron, 1987; Lipkin and Pohl, 1991; and Burdick, 1991). For a seri^ manipalator, a configuration is well defined as singular when the end-effector loses degrees of freedom and the Jacobian matrix becomes rankdeficient, i.e., when the input-output velocity map x = f(0) is singular. Other authors have addressed singularity of parallel and hybrid-chain manipulators (Agrawal, 1990; Kumar, 1990; Merlet, 1989; Shi and Fenton, 1992). For parallel manipulators, the usual definition of singularity is dual to the one for serial chains: a configuration is singular when the end-effector acquires additional degrees of freedom and the Jacobian matrix of the inverse kinematics becomes rank-deficient, i.e., the inverse input-output map 0 = fix) is singular. However, this duality is incomplete since parallel manipulators can also have configurations where the end-effector has reduced degrees of freedom and it is natural to consider such configurations as singular as well. Thus, for a mechanism, singularity cannot be solely associated with the degeneration of the derivative of an input-output map. To surmount this obstacle, Gosselin and Angeles (1990) analyzed the singularities of both open and closed chains using the derivative of a more general input-output relationship of the type/(;
q e. [W] ^ r, < n q e {\0} ro < n,i and r, < «, ro < «, =* 9 e {RI) or ^ e {RPM } r, < n, => ^ e { RO} or ^ e {RPM }

Proof. (i) - (vi) (vii)

(viii)

Follow directly from the definitions of the singularity types, If ro < rig, there are non-zero motion vectors projected onto zero by po (i.e., Ker po * 0). If such a vector M in Ker po belongs also to Ker p,, then an RPM-type singularity is present due to (vi). Otherwise, an Rl-type singularity is implied by (iii). The proof is analogous to the proof of (vii). D

the singular configuration must belong to at least one 1-type and one R-type. 6.2 Enumeration of All Possible Combinations. Below, the velocity-space formulation of the singularity problem is applied to find all feasible combinations of the six singularity types for the general case of an arbitrary kinematic chain. First, in the following proposition the rules for the simultaneous occurrence of the singularity types are stated. Proposition 4. (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) Proof. (i)

6 Classification of Singularities 6.1 Singularity-type Combinations. For any configuration, singular or non-singular, r, ^ n ^ n^ and ro ^ n ^ n^. A configuration is non-singular, only if r/ = « = n, and ro = n = n^. The cases in which these equalities do not hold are analyzed below: Case 1. n < n„ This is an IIM-type singularity. It can be noted that in this case r, < w, and ro < n^. Therefore, as implied by Proposition 3, (vii) and (viii), an RPM-type singularity or a singularity belonging to both the RI- and the RO-type must be present as well. Case 2. / - , < « = «, and ro = n = n^ This case is a combination of the II and RO singularity types. Indeed, r, < n implies an Il-type singularity according to Proposition 3, (i). According to Proposition 3, (viii), either an ROtype or an RPM-type singularity is present. But, if the configuration were an RPM-type singularity, according to Proposition 3, (vi), ro would be smaller than «,.

(ii) (iii) (iv) (v) (vi) (vii)

(viii) (ix)

Case 3. n„ and ro < n This case is symmetrical to Case 2 and is a combination of the lO- and Rl-type singularities. The reasoning is the same as above. Case 4. r, < n = n^ and ro < n = n^ This case is a combination of the II- and lO-type singularities together with either an RPM-type singularity or any combination of at least two different R-type singularities. The Il-type and the lO-type singularities are implied by Proposition 3, (i) and (ii), while (vii) and (viii) show that in this case there should be either an RPM-type singularity or a singularity of at least two different R-types. The above discussion of the four cases provides the proofs for the following theorems: Theorem 2. e (RPM)

q is singular =*
q & {II} or ^ e {lO} or ^ e {IIM} Indeed, each individual singular configuration belongs to exactly one of Cases 1 to 4, and for each case, it was shown that Journal of Mechanical Design

^ G {RI} =» 9 e {lO) o r ^ e {IIM} ? G {RO} =* 9 G {II} o r ^ G {IIM}