## 1 Introduction

## 2 Mathematical Foundation and Index Definition

## 3 Transmissibility of Typical Branches

### 3.1 Planar RRR and Spatial RSS Branches

_{ j1}and l

_{ j2}are unit link vectors. Link I is the swing bar, and link II is the two-force bar. Forces are indicated with dotted-line arrows, while actuated joint is labeled with the solid-line arrow. The actuated revolute joint is attached to the base, and the JTI value of the joint is 1. For the end-effector joint, the JTI value of joint 3 is also 1. The transmission performance of the branch is completely determined by joint 2. As indicated in the figure, the afferent force of joint 2 is \(\varvec{f}_{{j2{\text{A}}}}\), perpendicular to the link I. The efferent force of joint 2 is \(\varvec{f}_{{j2{\text{E}}}}\), along the link II. Therefore, the BTI value of the jth branch can be determined as

### 3.2 PUS and UPS Branches

## 4 Transmissibility of End Effectors

_{1}and f

_{2}are unit force vectors exerted on the end effector by two branches, along l

_{12}and l

_{22}vectors respectively, which are unit structural vectors of passive links. According to the geometric relation, the transmission index can also be written as \(\varGamma_{\text{ETI}} = {\text{ort}}\left( {\varvec{L}} \right)\) and \(\varvec{L} = \left[ {\begin{array}{*{20}l} {\varvec{l}_{12} } \hfill & {\varvec{l}_{22} } \hfill \\ \end{array} } \right]\). For the 5R parallel manipulator, the ETI value equals the sine value of the angle between f

_{1}and f

_{2}vectors (l

_{12}and l

_{22}vectors). Obviously, when the angle becomes 0° or 180°, limb force vectors are collinear, and the end effector loses the force output capacity along the direction perpendicular to limb force vectors. Accordingly, a singularity occurs, and the ETI value is 0. If the angle is 90°, the transmissibility of the end effector is the best, and the ETI value equals 1.

_{ j }acting on the end effector by the jth limb is along the limb, through rotational center of the spherical joint. As illustrated in Figure 6, the geometric center of the end effector is point o. Thus, direction of the unit radial vector c

_{j}is from point o to the rotational center of the jth spherical joint. Considering the geometric structure, the unit branch vector l

_{ j2}can be used instead of the limb force vector f

_{ j }. The matrix E becomes the unit diagonal matrix, and all diagonal elements are 1. Then, we can deduce the \(\varGamma_{\text{ETI}}\) expression of the Stewart manipulator as

## 5 Examples

Parameter | Value |
---|---|

Horizontal distance of actuated joints D/m | 1.0 |

Length of the actuated pendulum L
_{1}/m | 0.8 |

Length of the passive link L
_{2}/m | 1.6 |

Parameter | Value |
---|---|

Radius of the base R/m | 0.45 |

Radius of the end effector r/m | 0.225 |

Joint distribution angle of the base α/(°) | 10 |

Joint distribution angle of the end effector β/(°) | 25 |