HW4_sol - Texas A & M University Department of...

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Unformatted text preview: Texas A & M University Department of Mechanical Engg. MEEN-408 Intro. to Robotics Solution HW4 Instructor: Dr. Darbha Swaroop 1. Solution to Problem 1 For the given robot,the axes for the links are as shown in fig.1. Also, we have DH parameter table as folows. Figure 1: Coordinate frames Link a d 1 2 = 90- 2 l 1- 2 Table 1: DH parameters where is the angle z 1 makes with x . Based on DH parameters, the link transformation matrices are as follows T 1 = T z ( 1 ) T z ( d 1 ) T x ( a 1 ) T x ( 1 ) T 1 = c s s- c 1 1 T 1 2 = T z ( 2 ) T z ( d 2 ) T x ( a 2 ) T x ( 2 ) 1 T 1 2 = c- s s c- 1 l 1 1 T 2 = T 1 T 1 2 T 2 = cc- s sc l 1 s sc c- ss- l 1 c s c 1 (a) Angular Velocity Jacobian-Link 1 For link 1, the angular velocity is as follows L 1 = k where k is the unit vector along z . Therefore, angular velocity of link 1 in terms of principle coordi- nates , is as follows L 1 = J 1 L 1 = 0 0 0 0 1 0 (b) Angular Velocity Jacobian-Link 2 For the second link, the angular velocity is given by L 2 = k + k 1 Hence, from the transformation matrix T 1 ,k 1 can be written in the base frame as k 10 = s- c Therefore, angular velocity jacobian for link 2 is L 2 = J 2 L 2 = s- c 1 (c) Velocity Jacobian for Link-1 The velocity of the center of mass (CM) for link 1 is given by V cm 1 = L 1 r cm 1 In case of link is r cm 1 = l 1 2 k 1 . Hence, we can write coordinates of k 1 in base coordinate frame as 2 r cm 1 = l 1 2 s i- l 1 2 c j Therefore, assuming = 1 rad/s , we get velocity jacobian as V cm 1 = k ( l 1 2 s ) i- ( l 1 2 c ) j ) V cm 1 = l 1 2 c i + l 1 2 s i V cm 1 = l 1 2 c l 1 2 s (d) Velocity Jacobian for Link-2 The CM of link-2 has velocity due to rotation about axis fixed to link-1 and velocity due to rotation of link-1 about fixed frame. In order to determine first colummn of the velocity jacobian, we assume = 0 , = 1 rad/s . V cm 2 = k r cm 2 r cm 2 = r k 2 x cm 2 y cm 2 z cm 2 1 = T 2 r 1 = r sc + l 1 c- r ss- l 1 s r c 1 Therefore velocity of CM of link-2 due to rotation of link-1 is V cm 2 = k ( rsc + l 1 s ) i- ( rss + l 1 c ) j + rc k V cm 2 = ( r ss + l 1 c ) i + ( r sc + l 1 s ) j V x cm 2 V y cm 2 V z cm 2 = r ss + l 1 c r sc + l 1 s Second column of velocity jacobian: In order to determine first colummn of the velocity jacobian, we assume...
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This note was uploaded on 09/10/2009 for the course MEEN MEEN612 taught by Professor Swaroopdarbha during the Spring '09 term at Texas A&M.

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HW4_sol - Texas A & M University Department of...

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