Deterioration Rate

Deterioration rate is an index of measuring the variation in the dynamic performance of a manipulator before and after including the influence of the errors which are resulted from the manufacturing, assembling, and operating processes. Deterioration rate is defined as the ratio of the deviation between the acceleration radius without and with the influence of the errors resulted from the manufacturing, assembling, and operating processes to the acceleration radius without these errors. The acceleration radius without the influence of these errors can be found out by using the processes provided in Section 2.2. However, finding out the acceleration radius with the influence of these errors which is due to the manufacturing, assembling, and operating processes is more complicated than finding the radius without the influence of these errors. How to conduct the acceleration radius with the influence of the errors resulted from the manufacturing, assembling, and operating processes will be explained in the following.

The first step of conducting the acceleration radius with the influence of the errors resulted from the manufacturing, assembling, and operating processes is to include the effect of these errors into the D-H parameters.

The conventional D-H transformation matrix which has four D-H parameters is not sufficient to fully include the influence of these errors because it cannot include the angular error which is about the y axis. To cope with this insufficiency, an improved D-H transformation matrix which composes of five D-H parameters was proposed [56-58] and successfully includes all the effect of the errors resulted from the manufacturing, assembling, and operating processes into the transformation matrix. This kind of D-H transformation matrix which owns five D-H parameters is

called modified D-H transformation matrix hereafter. In the modified D-H transformation matrix, the extra parameter, _i, is the rotation angle of the two consecutive frames about y axis and is shown in Figure 8. The _i modified D-H transformation matrix, ^i1A_i^' can be conducted by post-multiplying the conventional D-H transformation matrix, ^i1A_i , with the rotation homogeneous matrix of _i as shown in (33).

Figure 8: Diagrammatic definitions of modified D-H parameters

1 ' 1

Where A(y ,β ) is the rotation homogeneous matrix of _{i i} _i. When

i is equal to zero, (33) is fully equivalent to the conventional D-H transformation matrix as shown in (12). In fact, the purpose of existence of

i is to include the rotation error about the y axis which is resulted from _i the manufacturing, assembling, and operating processes. Because _i is used to introduce the influence of the angular error about the y axis, _i _i, into the modified D-H transformation matrix, _i itself has no intended function and is always assigned to be zero in practical applications. When the errors resulted from the manufacturing, assembling, and operating processes exist in the modified D-H parameters, the corrective modified D-H transformation matrix which includes the influence of these errors can be expressed as the sum of the original modified D-H transformation matrix and the differential change matrix. The corrective modified D-H transformation matrix can be expressed as (34).

1 1 '

i C i

i i i

A A dA

    (34)

Where ^i1A_i^C is the corrective modified D-H transformation matrix;

1 ' i

Ai

 is the modified D-H transformation matrix with nominal modified D-H parameters; dA is the differential change matrix resulted from the _i influence of the errors caused by the manufacturing, assembling, and operating processes. Because _i is always zero, then ^{i1 '}A_i  ^i1A_i , always much smaller than the corresponding nominal modified D-H parameters, dA_i can be presented as the linear combination of these errors

without significantly losing the representativeness, and it can be expressed modified D-H transformation matrix can be expressed as (37).

1 1 ' 1 1 ' ( 1 ) 1 '

1

After conducting the corrected modified D-H transformation matrix, the total corrected modified D-H transformation matrix can be shown as (38).

Because the kinematic deviation resulted from the error items,  ^i1A_i, is relatively small, the influence of the second and higher order terms can be omitted without any significant influence on the result. In the following discussion, only the first order approximation of (38) will be utilized, and it can be presented as (39).

1 1 1

Where R_n^C is the rotation portion of T_n^C; P_n^C is the position portion

When the corrected modified D-H transformation matrix and the total corrected modified D-H transformation matrix have been conducted, what should be done next is conducting the Jacobian matrix and acceleration ellipsoid which includes the influence of the errors which are resulted from the manufacturing, assembling, and operating processes. Because the process of conducting the Jacobian matrix and acceleration ellipsoid with the influence of the errors is similar with the one explained in Section 2.2, the following just shows and gives brief interpretations of some important concluded equations. The same as (14), _n and w can be expressed as _n (40) and (41) respectively.

1 described in the reference frame and equivalent to the 3rd column vector of

1 C

Ri_ , ^i1p_n^C is the corrective position vector from end-effecter to the origin of the i frame and is also described in the reference frame, 1 ^_i is the angular velocity value of the ith revolute joint, and d_i is the linear velocity value of the ith prismatic joint.

From the conduction shown in (39), z_i^C_1 and ^i1p_n^C can be presented as (42) and (43) respectively.

1 1

1 1 1 1 1 the reference frame which is equivalent to the 3rd column vector of R_i1;

1 i

pn

 is the position vector from end-effecter to the origin of the i 1 frame and also is described in the reference frame. The subscript “P” means the translation part of the bracketed transformation matrix and the subscript

“Z” means the direction of z axis of the bracketed rotation matrix equivalent to the 3rd column vector.

From (40) and (41), the corrective Jacobian matrix can be expressed as (44). for a prismatic joint.

Substitute (42) and (43) into (44) and eliminate the second order term,

C

J can be presented as (45). i C

i i i

J J dJ (45)

Where J is the ith column of the nominal Jacobian matrix without _i the influence of the errors resulted from the manufacturing, assembling, and operating processes, and dJ is the differential change Jacobian matrix _i resulted from the influence of these errors.

In (46) and (47), they show J and the first order _i dJ of a revolute _i joint respectively.

1 joint, respectively.

1

position error resulted from the influence of the configuration errors which are from the end-effecter to the i link. 1

When the D-H transformation matrix and Jacobian matrix with and without the effect of the errors caused by the manufacturing, assembling, and operating processes are known, the acceleration radius with and without the effect of these errors can be conducted through the method proposed in Section 2.2. When the acceleration radiuses with and without the effect of these errors are known, deterioration rate can be defined as (50) [53].

Where DR_P is the deterioration rate of a manipulator with a certain

configuration at a specific posture; r_e and r_i are the acceleration radius of a manipulator with a certain configuration at a specific posture with and without the influence of the errors resulted from the manufacturing, assembling, and operating processes respectively.

For a certain workspace or region, the deterioration rate over this workspace or region can be defined as (51), and it can be taken as a representative index of derating margin of the manipulation over this workspace or region [53].

w( )

W

w

dr dw

DR dw

 

 (51)

Where DR is the deterioration rate over a prescribed workspace or _W region, _w( )dr dw is the integral of the deterioration rate which is over this workspace or region , _wdw presents the workspace or region in discussion, dr is the differential function of deterioration rate, and dw presents the differential area of the workspace or region.

With the help of deterioration rate, designers of manipulators can estimate the degradation on the dynamic performance of a manipulator which is caused by configuration errors and handle the specification of the dynamic performance of a manipulator more correctly.

III. Evaluate the Variation in Dynamic Performance of a Manipulator