Tham khảo tài liệu 'advances in robot kinematics - jadran lenarcic and bernard roth (eds) part 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Inverse Kinematics of a Fragment of Protein Backbone 205 between Cl and C2 except at T2 0 and T2 n where it coincides with Cl and C2 respectively Fig. 3 . For any fixed T1 G S1 the set RT1LRT2Lz is the circle obtained by rotating LRT2 Lz by T1 around the z axis. Thus for every point s in the region between C1 and C2 RT1LRT2 Lz contains s for two distinct values of T1. We conclude that n-1 has two values Tk Tk k 1 2. In C1 s z and n-1 s t1 0 T1 G S1 . For any s G C2 n-1 s has a single value of the form T1 n . Elsewhere n-1 s is empty. Corresponding to each value t1 t2 of n-1 s there is a unique value of T3 given by Eq. 5 hence a single value of p-1 R . Thus as we initialize an orientation R G SO 3 not in the critical sets C1 and C2 P-1 R is the disjoint union of two 3-D tori written Mk k 1 2. 4. Inverse Position Map Restriction to Mk. We now study p-1 X where X gR3 and Pk k G 1 2 is the position map p with its domain restricted to Mk. Since 02j-1 02j j 1 2 3 are constant on Mk and equal to T each point on Mk is uniquely defined by the values of 01 03 and 05. Eq. 1 yields Pk S1 3 -R3. 01 03 05 Vo k R1 RTkLR3 RTkLRTkLR5 v2 where v0 k RTkL RTkLRTk L v1 is a constant vector and R1 v2 RTkLR3v2 and R kLRTkLR5v2 are constant circles of radius Ể1 contained in three different planes. Computing P-1 X amounts to solving the equation X Pk -02 03 05 R-2V2 LR3V2 LRT2kLR5V2 6 where X Rt X v0 k and R-2 is the rotation of 02 around z. Critical set. Here we directly determine the critical positions X where the number of solutions of pk changes. We rewrite Eq. 6 as X r w q t u 7 where we rename the variables as t 02 u 03 w 05 and Y Tk. X r w is a unit circle centered at X and q t u spans a quartic surface Q in R3. Q is the Minkowski sum of two circles so it is bounded and connected. Eq. 7 can be solved by computing the intersections between X r w and the coss-section curve of Q by the plane containing X r w . We computer w xcw ysw. X sịc7 c2a sYsa saca 1 cY T and y sasY cY casY T form