**5. Performance indices**

With the increasing demand for precise manipulators, search for a new manipulator with better performance has been intensive. Several indices have been proposed to evaluate the performance of a manipulator. Merlet reviewed the merits and weaknesses of these indices (Merlet, 2006). The dexterity indices have been commonly used in determining the dexterous regions of a manipulator workspace. The condition number of the jacobian matrix is known to be used as a measuring criterion of kinematic accuracy of manipulators. Salisbury & Craig (1982) used this criterion for the determination of the optimal dimensions for the fingers of an articulated hand. The condition number of the jacobian matrix has also been applied for designing a spatial manipulator (Angeles & Rojas, 1987).

The most performance indices are pose-dependant. For design, optimization and comparison purpose, Gosselin & Angeles (1991) proposed a global performance index, which is the integration of the local index over the workspace.

The performance indices are usually formed based on the evaluation of the determinant, norms, singular values and eigenvalues of the jacobian matrix. These indices have physical interpretation and useful application for control and optimization just when the elements of the jacobian matrix have the same units (Kucuk & Bingul, 2006). Otherwise, the stability of control systems, which are formed based on these jacobian matrices, will depend on the physical units of parameters chosen (Schwartz et al., 2002). Thus, different indices for rotations and translations should be defined (Cardou et al., 2010).

#### **5.1 Manipulability**

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

$$\mu = \sqrt{\det(\boldsymbol{J}^T \boldsymbol{J})} \,. \tag{31}$$

Exploiting Higher Kinematic Performance –

index over the entire workspace,

(2010) defined two indices ( *<sup>r</sup>*

indices, and singularity analysis.

 , *<sup>p</sup>* 

to use 2-norm and ∞-norm for calculating the sensitivity.

**6.1 Reachable points and workspace comparison** 

g and h are the radii of the fixed and moving platforms, respectively.

mechanism can successfully reach to the locations within this cubic space.

sensitivity of a manipulator. The sensitivity indices can be defined as:

**5.3 Sensitivity** 

and

Using a 4-Legged Redundant PM Rather than Gough-Stewart Platforms 55

*W*

 

*GDI*

*W*

which is the average value of local dexterity over the workspace. This global dexterity index can be used as design factor for the optimal design of manipulators (Bai, 2010; Li et al., 2010; Liu et al., 2010; Unal et al., 2008). Having a uniform dexterity is a desirable goal for almost all robotic systems. Uniformity of dexterity can be defined as another global performance index. It can be defined as the ratio of the minimum and maximum values of the dexterity

*dW*

, (36)

. (37)

) for measuring the rotation and displacement sensitivity

*r r* || || *<sup>J</sup>* , (38)

*p p* || || *<sup>J</sup>* , (39)

*dW*

*Min*

*Max* 

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.

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

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

In order to investigate the kinematic performance of 4-legged mechanism, the response of the mechanisms are compared in several different aspects; reachable points, performance

Consider the 3-legged and the 4-legged mechanisms with g=1 m and h=0.5 m, respectively;

By assuming a cubic with 1m length, 1 m width and 1 m height located 0.25 m above the base platform, we are interested in determining the reachability percentage in which each

*iso*

**6. Comparison between 4-legged mechanism and other mechanisms** 

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:

$$
\mu\_{\rm iso} = \frac{\sigma\_{\rm Min}}{\sigma\_{\rm Max}} \,\,\,\tag{32}
$$

where *Max* and *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 end-effector.

#### **5.2 Dexterity**

The accuracy of a mechanism is dependent on the condition number of the jacobian matrix, which is defined as follows:

$$k = \lfloor \lfloor f \rfloor \rfloor \lfloor . \rfloor \mid \lfloor f^{-1} \rfloor \mid \vert \tag{33}$$

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

$$<\langle J \mid \mid = \sqrt{\frac{1}{n} \underline{M}^T}) \; , \tag{34}$$

where *n* is the dimension of the square matrix *J* that is 3 for the manipulator under study. Gosselin (1992) defined the local dexterity ( ) as a criterion for measuring the kinematics accuracy of a manipulator,

$$\nu = \frac{1}{||\||\||\||\cdot||\||^{-1}||\||}\,. \tag{35}$$

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 manipulator has reached a singular configuration pose.

To evaluate the dexterity of a manipulator over the entire workspace (*W* Gosselin & Angeles (1991) have introduced the global dexterity index (GDI) as:

$$GDI = \frac{\int \nu \cdot d\mathcal{W}}{\int \mathcal{W}},\tag{36}$$

which is the average value of local dexterity over the workspace. This global dexterity index can be used as design factor for the optimal design of manipulators (Bai, 2010; Li et al., 2010; Liu et al., 2010; Unal et al., 2008). Having a uniform dexterity is a desirable goal for almost all robotic systems. Uniformity of dexterity can be defined as another global performance index. It can be defined as the ratio of the minimum and maximum values of the dexterity index over the entire workspace,

$$
\nu\_{\rm iso} = \frac{\nu\_{\rm Min}}{\nu\_{\rm Max}} \,. \tag{37}
$$
