GEOM MECH ROBOTS 1
GEOM MECH ROBOTS 1 EML 6281
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Date Created: 09/18/15
Chapter 12 Solution TE 1 Prove that successive rotations by the angles a Eradians anal a respectively about the x TE y anal z axes are equivalent to a single rotation of Eradians about the y axis The rotations about the X y and z axes can be represented respectively by the quaternions ql qz and q3 as I I cos s1n 1 ql 2 2 cos E sin n 39 qz 4 4 J a 1 Pk cos sin 13 2 2 The operator which will transform a point by the three successive rotations may be written as q3qul O 114 1 q3391 The product qqu is evaluated as I 7 q2q1cos cos9cos sin icos9 sing jsin sing k 4 2 4 2 2 4 4 2 The product ngqu is evaluated as I 1 cos9 cosE cos sing sin sin9 q3q2ql 2 4 2 4 2 2 I I 75 I 1 TE TE cos cos s1n 1s1n cos Js1n s1n k 2 4 2 4 2 4 2 cos cos sing k 4 2 2 1 k 0 0 sin9 39 2 I I 1 TE TE TE cos s1n s1n cos s1n s1n 4 2 4 2 4 2 Regrouping this expression gives 2l 29 cos sin 131qu 4 cos 2 4 s1n 2 TE 1 cos s1n cos s1n s1n cos 4 2 2 4 2 2 z n 2 TE 1 st cos 3 cos s1n 3 k sinE sin9 cos9 cosE sin cos 4 2 2 4 2 2 Simplifying this expression gives z 7c q q q COS cos s1n 5111 3 2 1 4 4 2 TE TE 1 s1n COS COS S111 2 2l 4 4 j Sinsin29 coszgj 2 9 2 2 k sin9 cos9 cosE sing 2 2 4 4 Substituting sin E cos g and sinzg 0052 1 gives cos E sin 5 39 13 12 11 4 4 J Since the product ngqu equals qz the total rotation caused by ngqu is equivalent to the rotation caused by q alone 2 The quaterm39ons q and q2 are given as q1 cos 300 Sm 530 3i 4k sin 60quot cos60 5 2k 12 29 Solve the following equation for the quaternion q3 QIQ3 Q2QI Premultiplying both sides of the equation by q1 yields q3 111 qz q1 Since q1 is a unit quaternion its inverse will equal its conjugate Thus the product ql39lqz is given by 1 sin 30 sin 60quot cos 30O 314k cos 600 5 2k q1 q2 5 Jz g J q1 q 03044 01716 i 06000j 07198 k The product q3 q1 q q1 may be evaluated as q3 05 04800 i 02349j 06816 k 3 q is a unit quaternion and p is a quaternion with no scalar component Under what conditions willp q1pq1391 Since q1 is a unit quaternion it models a rigid body rotation Since p has no scalar component it models the coordinates of a point Postmultiplying both sides by q1 gives P 11 11 P This multiplication is commutative only if 1 q1 has no vector component ie the sine of half the angle of rotation equals zero This means that the angle of rotation is equal to 0 or some scalar multiple of 271 2 p has no vector component ie p represents the origin 3 The vector part of q1 is proportional to the vector part of p This implies that p is located on the aXis of rotation 4 A box is movedfrom position 1 to position 2 and then to position 3 as shown in Figure 129 Determine the axis and angle of rotation which would move the box directly from position 1 to position 3 The rst rotation is performed by the operator q1 q1 where q1 cos 45 sin 45 k The second rotation is performed by the operator q2 q1 where q cos 45 sin 45 j The body could have been moved directly from position 1 to position 3 by using the quatemion operator q3 q339 where q3 qqu q3 is evaluated as q3 cos 45 sin 45 j cos 45 sin 45 k q3 cos2 45 cos 45 sin 45 k cos 45 sin 45 j sin2 45 i q305 05 ij k The vector part of q3 may be normalized so that q3 may be written as L i L J5 5 E 39 The scalar part of q3 equals the cosine of half the angle of rotation The sine of half the angle of rotation equals 0866 Thus the net rotation is equivalent to a rotation of 120 about an axis parallel to ijk q3 05 0866 5 A box has been rotated 40 degrees about an axis parallel to 2ijk It was then rotated 60 degrees about an axis parallel to i3j 2k measured with respect to the fixed coordinate system You wish to return the box to its original orientation with one rotation Determine the angle and axis of rotation measured with respect to the fixed coordinate system which will accomplish this The rst rotation can be modeled by the quatemion operator q1 q1 where q1 is given by 39 39 q1 cos 20 sin 20 The second rotation can be modeled by the quatemion operator q2 q2391 where q is given by q2 cos 30 sin 30 The net rotation is modeled by the quatemion q3q2q1 which is evaluated as q3 07578 04607 i 04043j 02235 k Normalizing the vector part of q3 gives q3 07578 06524 07061 i 06197j 03426 k The quatemion q3 represents a rotation about the vector 07061 i 06197 j 03426 k The cosine of half the angle of rotation is 07578 and the sine of half the angle of rotation is 06524 The angle of rotation is thus 81460 The box can be returned to its original position by rotating 8l46O about the axis 07061 i 06197j 03426 k 2 6 Line segmentAB is rotated 75 degrees Ending posmon about an axis parallel to i1lc which passes through point C as shown in Figure 1210 C 022 The coordinates of points A B and C are as Kiwis of roldm follows A 520 B 130 y C 022 X A s 2 Stu 39hg D Use quaternions to determine the coordinates 39 mm of the endpoints of line segmentAB after the rotation is accomplished A new coordinate system named the D coordinate system is introduced whose axes are parallel to the XYZ of the xed system but whose origin is located at point C The coordinates of points A and B in the D coordinate system are DPA 51 2k DPB i j 2k Points A and B can be rotated in the D coordinate system by the operator q1 q1 where ql is given by 39 39 q1cos375O Sln3750 J3 q10793403515ijk The coordinates of points A and B as measured in the D coordinate system may be written as the quatemions pA and p3 where pA5l2k pBlj2k The coordinates of these points after the rotation as measured in the D coordinate system may be determined from q1pAq1391 and q1pBq1391 as q1pAq1391 31506 i 31626j 30119 k q1pBq1391 19319 i 14142j 05176 k The coordinates of the rotated points A and B in the original coordinate system are obtained via translation respectively as 31506i 11626j 50119k and 193l9i 05858j 14824k 7 a Assume the quaternioris p and q are given as follows 17 1 1 2 3 q 3 2 1 05 What is the product pq What is qp What is p39Jq pq 45 315 l35 qp 45 7 115 35 p391 q 01 00667 00333 09667 b Under what circumstances would the product pq equal qp List all cases pq will equal qp if The vector part of p is zero The vector part of q is zero The vector part of p is proportional to the vector part of q c What is the result of the quaterniori product i j k ijk 1
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