**5.3 Sensitivity**

54 Serial and Parallel Robot Manipulators – Kinematics, Dynamics, Control and Optimization

For evaluation of kinematic transmissibility of a manipulator, Yoshikawa (1984) defined the

The manipulability can geometrically be interpreted as the volume of the ellipsoid obtained by mapping a unit n-dimensional sphere of joint space onto the Cartesian space (Cardou et al., 2010). It also can be interpreted as a measure of manipulator capability for transmitting a certain velocity to its end-effector (Mansouri & Ouali, 2011). To have a better performance for a manipulator, It is more suitable to have isotropic manipulability ellipsoid (Angeles &

Lopez-Cajun, 1992). The isotropy index for manipulability can be defined as:

*iso*

det( ) *<sup>T</sup>*

*Min*

*Max* 

*Min* are maximum and minimum of singular values of jacobian matrix (*J*),

respectively. The isotropy index is limited between 0 and 1. When the isotropy index is equal to 1, it indicates the ability of manipulator to transmit velocity uniformly from actuators to the end-effector along all directions. Inversely, when the isotropy index is equal to zero, the manipulator is at a singular configuration and cannot transmit velocity to the

The accuracy of a mechanism is dependent on the condition number of the jacobian matrix,

where *J* is the jacobian matrix and *J* denotes the norm of it and is defined as follows:

<sup>1</sup> || || ( ) *<sup>T</sup> J tr JJ*

where *n* is the dimension of the square matrix *J* that is 3 for the manipulator under study.

1 || ||.|| || *J J*

The local dexterity can be changed between zero and one. The higher values indicate more accurate motion generated at given instance. When the local dexterity is equal to one, it denotes that the manipulator is isotropic at the given pose. Otherwise, it implies that the

To evaluate the dexterity of a manipulator over the entire workspace (*W* Gosselin & Angeles

1

*J J* . (31)

, (32)

<sup>1</sup> *kJ J* || ||.|| || , (33)

*<sup>n</sup>* , (34)

) as a criterion for measuring the kinematics

. (35)

**5.1 Manipulability** 

manipulator index,

where *Max* and

end-effector.

**5.2 Dexterity** 

which is defined as follows:

accuracy of a manipulator,

Gosselin (1992) defined the local dexterity (

manipulator has reached a singular configuration pose.

(1991) have introduced the global dexterity index (GDI) as:

Evaluating of the kinematic sensitivity is another desirable concept in the performance analysis of a manipulator. The kinematic sensitivity of a manipulator can be interpreted as the effect of actuator displacements on the displacement of the end-effector. Cardou et al. (2010) defined two indices ( *<sup>r</sup>* , *<sup>p</sup>* ) for measuring the rotation and displacement sensitivity of a manipulator. They assumed the maximum-magnitude rotation and the displacement of the end-effector under a unit-norm actuator displacement as indices for calculating the sensitivity of a manipulator. The sensitivity indices can be defined as:

$$
\pi\_r = \begin{array}{c}
\|\,\|\,f\_r\|\,\|\,\|\\
\nu\_r
\end{array} \tag{38}
$$

and

$$\left| \tau\_p = \right| \left| J\_p \right| \left| \right|\_{\text{A}} \tag{39}$$

where *rJ* and *pJ* are rotation and translation jacobian martices (Cardou et al., 2010), respectively, where || ||stands for a *p*-norm of the matrix. Cardou et al. (2010) suggested to use 2-norm and ∞-norm for calculating the sensitivity.
