Huang XH, et al. Sci China Tech Sci July (2016) Vol.59 No.71
•Article •July 2016 Vol.59 No.7: 1–?
doi: 10.1007/s11431-016-6069-3
Huang XH, et al. Sci China Tech Sci July (2016) Vol.59 No.71
Appendix A: Necessary results for the planar FFSR model
Some necessary results for the planar FFSR are presented in this part, based on the modeling in Ref. [8].
First, the result of components of the Jacobian matrix between and , , , , is given as
(A.1)
where
(A.2)
is a cross product operator for the planar vectors and the related arithmetics are
(A.3)
where, are arbitrary 2-dimensional planar vectors, and is an angular vector perpendicular to the plane, which is denoted by the third element of the vector for short in our planar modeling.
The basic 2-dimensional link vectors are
(A.4)
where , and we can get the derivatives of , with respect to the state variable :
(A.5)
From eqs. (A.2), (A.4) and (A.5), we can get that all the derivatives of , with respect to the base attitude are zeros:
(A.6)
Eventually, by the definitions of , , , their derivatives with respect to the base attitude are all zeros, too:
(A.7)
Eq. (A.7) is needed for the main body of the paper. And next, we work for another result needed for the paper:
(A.8)
where is solving the residual of divided by .To prove eq. (A.8), we first compute the partial derivatives of any :
(A.9)
which implies that is a repetend of the derivatives (), with a period of four. Using this result, we can find that the situation is the same for the derivatives of , , i.e., for any variable , it yields
(A.10)
and is a repetend.
Calculating the derivatives of , , of order with respect to , with eqs. (A.9) and (A.10), finally we have
(A.11)
That’s to say
(A.12)
Appendix B: Design of a simple self-correcting motion cycle
A simple and intuitive self-correcting motion cycle is designed in the joint space here, for the purpose of comparison with the optimal self-correcting motion proposed in Sec. 5.
Define a parallelogram cyclewith its side vectors denoted by , , , and , as shown in Figure 10. The constraints are chosen as follows:
(1) A square cycle is assumed with a same side-length as that of the optimal self-correcting motion cycle in Sec. 4.2, which implies
(B.1)
(2) Since the side vectors are independent of their start points, set the current point as a relative zero for simplicity. The two end points of the first side are chosen as and, respectively, where is another unknown parameter. This implies
(B.2)
(3) The diamond cycle plane is perpendicular to the plane and thus
(B.3)
Equations (B.1) through (B.3) yield seven equations with seven unknowns, which can be solved for the desired correcting motion cycle. As a result, the four side vectors of the simple cycle are obtained:
(B.4)
Figure10 A simple self-correcting motion in joint space.
Huang XH, et al. Sci China Tech Sci July (2016) Vol.59 No.71