Xuaogang Thanks you. It seems that you looked to the problem as a control problem. For the estimation problem the system can be excited in open loop and the position of end effector is not primary objective.To make the question brief, I asked about the effect of n-1 (n the number of links) first link on the last link dynamic behavior but yo can see that the problem is more general. I mean that according to the formulation presented in the paper the n-m first links have no effect on the dynamic behavior of m last links dynamic behavior.

Based on my works I think that I found the answer of my question. As I pointed in my question it makes no sense that the last link of a serial manipulator be isolated from other links dynamics. The paper looks this problem in a different way. without loosing the generality we can assume that gravity is negligible (like space-robots)

For the last link, if someone draws the free diagram, the external forces and torques are just related to the joint (W[n]). Then for last link as a rigid body with inertia parameters vector of Theta=[m  rc(3*1) Ixx Ixy Ixz Iyy Iyz Izz]^T one can write:

W[n] = PHI[n]*Theta[n]                                                        (1)

where PHI[n] is the regressor matrix (Eq. 8 in the paper)

For the link before the last link there are two external force and torques:

1- force and torques resulted from Joint n-1(W[n-1])

2- force and torques resulted from Joint n.(W[n] in the opposite direction)

Considering that W[n] is defined in n th body coordinate system it should be transformed to n-1 th body coordinate system. So we will have:

W[n-1] - T[n]*W[n] = PHI[n-1]*Theta[n-1]                                  (2)

W[n-1] = T[n]*W[n] + PHI[n-1]*Theta[n-1]                                 (3)

W[n-1] = T[n]*PHI[n]*Theta[n] + PHI[n-1]*Theta[n-1]

             =[PHI[n-1]  T[n]*PHI[n]][Theta[n]   Theta[n-1]]^T          (4)

and it can be continued until the base in the same manner.

So joint force and torques should be measured which is not applicable necessarily. If there were measurable there were no need to replace W[n] with PHI[n]*Theta[n] in the Eq. (2) but if it is not measurable using PHI[n]*Theta[n] is necessary in the proposed estimation process.It should be noted that In the proposed estimation process the joint forces and moments would be projected to the orientation which is measurable.

In this regard the difference between the first link and the last link is:

- The last link joint force and moments should be measured 

- The first link joint force and moment is calculated from other links dynamics

It can be reversed easily and in this regard there is no difference between first and last links.

I hope that the text be clear and my answer be convincing. If not it can be discussed more.

Thanks