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Research Papers

# Decoupled and Homokinetic Transmission of Rotational Motion via Constant-Velocity Joints in Closed-Chain Orientational Manipulators

[+] Author and Article Information
Marco Carricato

Department of Mechanical Engineering (DIEM), University of Bologna, Viale Risorgimento 2, 40136 Bologna, BO, Italymarco.carricato@mail.ing.unibo.it

A rigid body is said to have Schoenflies motion if it can freely translate in space and rotate about a constant direction.

Different terminology may also be encountered in the literature, addressing $J=Jdir−1Jinv$ as the direct kinematic Jacobian matrix and $J−1$ as the inverse one (26).

Standard Euler-type angles represent sequential body rotations about the axes of an orthogonal frame. However, the orthogonality condition is not essential and it will not be imposed here, thus the angle between the axes of the pairs $Ri$ being left generic.

If the constant elements of $K$ in Eq. 7 are obtained as ratios of proportional variable functions, direct singularities are not necessarily ruled out (see Sec. 5).

Equation 10 provides a more general result than that deducible from Porat's study. Indeed, Porat (22) examined a CV transmission that may be shown to be equivalent to a special arrangement of that portrayed in Fig. 3, with $a1$ and $a2$ being, respectively, set collinear with $a3̱$ and perpendicular to the plane determined by $a3̱$ and $a3$ (this depends on the particular choice of the Cardan angles adopted to describe the output body orientation). Porat provides an expression of $ω30$ as a function of $q̇3$, $φ̇1$, and $φ̇2$, by which the reader may verify that $q̇3=φ̇3$ if and only if $φ̇1=0$, i.e., if $ω20$ is parallel to $a2$ and thus perpendicular to the plane of the shaft axes. Indeed, this is the only configuration that locates $ω20$ on the bisecting plane, given the particular location of $a1$ and $a2$. Eq. 10 proves that, in a more general case, having $ω20$ orthogonal to $a3̱$ and $a3$ is not a necessary condition for homokinetic transmission (though it is a sufficient one).

Such an actuator is, to any dynamical extent, analogous to a “base-mounted” actuator, since it is equivalent to an inertia revolving about a fixed axis.

In ball-in-track couplings, an (instantaneous) $RSR$ linkage equivalent to each track-sphere-track pairing contact may be obtained by connecting the ball-and-socket joint with two revolute pairs, whose axes pass through the centers of curvature of the track centerlines and are perpendicular to the corresponding osculating planes. If the grooves are rectilinear, the revolute pairs are replaced by prismatic ones and a $PSP$ chain is obtained, which is equivalent to an $RER$ chain.

$vF$ vanishes when $r$ is perpendicular to $Σm̱n$. In this case, the axes of the pairs $R$ are collinear and the inner members of $Gm̱n$ may rotate finitely and rigidly about $r$, thus the position of the joint $E$ being indeterminate. For the sake of brevity, this situation is investigated only in the case of the Koenigs connecting chain (Sec. 61).

The end-effector of the device presented in Ref. 7 is connected to the frame via a spherical subchain $M1R2$, with $R2$ being actuated by a transmission of type $M2ER$. The constraints transmitted by the subchain $ER$ constitute a pencil of parallel forces. Such a pencil belongs to the constraint system generated by $M1R2$ twice for each full revolution of $M2$, thus originating an uncertainty singularity (which is independent of the simultaneous singularity exhibited by the parallelogram chosen by Gogu to implement $M2$).

Other solutions are also possible. Herchenbach (44), for instance, used a guiding disk sliding on the bisecting plane, planarly coupled with the central member of the $UU$ chain and cylindrically coupled with two spheres fixed to the shaft ends.

J. Mechanisms Robotics 1(4), 041008 (Sep 18, 2009) (14 pages) doi:10.1115/1.3211025 History: Received July 09, 2008; Revised May 20, 2009; Published September 18, 2009

## Abstract

This paper presents novel 2DOF and 3DOF closed-chain orientational manipulators, whose end-effector motion is actuated in a decoupled and homokinetic way by frame-located motors via holonomic transmissions based on constant-velocity couplings. The functioning of these couplings is investigated and the conditions applying for homokinetic transmission to be preserved during simultaneous motor drive are revealed and implemented. As a result, decoupled and configuration-independent relations between the motor rates and the time-derivatives of the variables describing the end-effector orientation are achieved. The attainment of analogous relations between the motor speeds and the components of the end-effector angular velocity is conversely proven to be unfeasible. The problem of singularities is furthermore examined, showing that input-output homokinesis is not a sufficient condition for a globally uniform kinetostatic behavior of the mechanism, which may, indeed, possibly reach uncertainty singular configurations. The connecting chains of the most typical constant-velocity couplings are analyzed, in order to obtain analytical expressions for the functions on which such singularities depend. The influence of design parameters is accordingly inspected. The results are valuable for the type and dimension synthesis of closed-chain wrists free from direct kinematic singularities, and characterized by simple kinematics and regular input-output kinetostatic relations.

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## Figures

Figure 1

Closed-chain wrist actuated by base-mounted actuators

Figure 2

CV coupling between intersecting shafts

Figure 3

CV coupling between shafts whose relative location is varied by a serial spherical chain

Figure 4

Decoupled and homokinetic actuation of a 2DOF closed-chain wrist via a CV-joint-based transmission

Figure 5

Remote homokinetic actuation of the third revolute pair of a serial spherical wrist

Figure 6

Decoupled and homokinetic actuation of a 3DOF closed-chain wrist via CV-joint-based transmissions

Figure 7

General connecting chain of a CV coupling for intersecting shafts

Figure 8

Geometry of the RER connecting chain

Figure 9

Uncertainty-configuration loci of the RER connecting-chain for (a) λ=0, (b) λ=0.5, and (c) λ=1

Figure 10

A decoupled and homokinetic 2DOF wrist employing a self-supporting Koenigs coupling

Figure 11

Geometry of the RSR connecting chain

Figure 12

Uncertainty configuration loci of the RSR connecting-chain for ρs=0, ψ=π/2, and λ∊[0,1.5]

Figure 13

A decoupled and homokinetic 3DOF wrist employing Clemens and Hooke connecting chains

Figure 14

A UU connecting chain plus a centering device: (a) constraint system imposed by the UU connecting chain and (b) centering device

Figure 15

Geometry of the UU connecting chain

Figure 16

A decoupled and homokinetic 2DOF wrist employing a UU connecting chain plus a centering device

Figure 17

Examples of homokinetic transmissions for: (a) linear motion between translating members, and (b)–(c) rotational motion between parallel shafts

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