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by: Miss Damien Crooks
Miss Damien Crooks
GPA 3.52


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Class Notes
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This 30 page Class Notes was uploaded by Miss Damien Crooks on Monday October 19, 2015. The Class Notes belongs to ECSE 4490 at Rensselaer Polytechnic Institute taught by Staff in Fall. Since its upload, it has received 42 views. For similar materials see /class/224784/ecse-4490-rensselaer-polytechnic-institute in ELECTRICAL AND COMPUTER ENGINEERING at Rensselaer Polytechnic Institute.



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Date Created: 10/19/15
24 The Manipulator J acobian Part 11 Topics Be able to derive an expression for the joint torques and forces induced by the tooltip contact forces and moments calculate the arm sti hess and compliance matrix from the manipulator J acobian locate the joint space singularities for a speci c robot External Forces and Moments EMNM 7 fun and ntoal are produced upon contact A force and moment are generated at the tool tip due to physical contact Problem Find a relationship between the external forces and moments generated at the tool tip and the internal joint torques for revolute joints and forces for prismatic joints that they induce Use this relationship to calculate the arm stiffhess and compliance matrices Induced Joint Torques and Forces Derive an expression for the induced joint torques and forces Let F 00 Let rk torque about axis k dp d and recall du The work done by an in nitesimal tool displacement is Example Find the joint torques in a four axis SCARA robot induced by F 0 alSla2S12 alClazC12 0 0 0 1 aZC12 0 0 0 1 O 1 0 0 0 0 0 O 0 0 1 251 2 J T01 Stiffness sti hesskeeps the link from de ecting small de ections will be corrected by a restoring torque LetI diagy1 yz yn Compliance Find the tool de ection when F is applied at the end of the arm Gq is the endof ann compliance matrix It measures the sensitivity of the arm to de ections at various points q For n 2 6 Jq will have full rank as long as q is not a joint space singularity Gquotq is the endof arm Sti ness matrix F m1 is very large near a singularity Joint Space Singularities du Jq dq for n lt 6 or for certain q rankJq lt 6 In this case there exist du that are not realizable by any dq For example consider the SCARA dp1 dp2 dp3 dcp1 dcp2 quot61151 11251 2 a1C1 2 C1 2 02C1 2 0 H00 61251 2 0 O dq1 dq2 dq3 OOOOO dq4 Alternatively rankJq nullityJq n The null space of Jq is 0 Jqdq Similarly I JTqFtool Example Examine the SCARA robot for joint space singularities Am q2on2ni Solution 1 Z J Tq Ftool alSlale2 alC1azCk2 0 0 0 1 J T01 12S12 aZC3912 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 Locate the singularities Check the torques for q2 0 one of the singularities Check the torques when Ftool 2 C1 S1 0 nltool nztool 0T Example Find the tool con guration J acobian matrix and the manipulator J acobian for a three axis planar robot Solution C S123 0 a1C1 aZCIZ 123 Two S123 C123 0 aISI aZSIZ base 0 O 1 d3 0 0 0 l The tool con gurqtion vector is 01C a2C12 als 1 aZS 12 I Shouldnr awkq ijQ aq 144 For the manlpulator J acoblan Jq 34 C1 S1 0 alc1 C1 S1 0 alCl C2 52 o 212CZ T101 31 C1 0 alSl T2 S1 C1 0 alSl S2 C2 0 azs2 0 0 0 1 0 o 0 1 o 0 0 1 o o o 0 1 o 0 0 1 0 0 0 1 39 1sn39 r9n 1512 0 a wazq a1sz 0 Arum 0 D 0 D l 1 l 18 Dynamic Model of a SCARA Robot Topics Be able to nd the LagrangeEuler dynamic model of an nlink manipulator that has revolute and prismatic joints 0 describe the physical signi cance of each term in the model Ref Schilling pp 212 220 a 24104 Summary of the LagrangeEuler Equations Recall the Lagrangian L014 Tqq39Uq 12q39TDq6jchq where Dq manipulator inertia tensor mk mass oflink k Dkq inertia tensor wrr L0 and Hg E m 5q k1 andquot H1 Tok q Ack k A Also k J k q link Jacobian B It can be shown that d a a L L 1 dt 8 q q aq q q and the first term is equal to i the second term in Fi is 8Lq 61 I aqi n 3 De ne hiq Z gk mj AkJiUI k1 j1 which is the gravity loading vector and ago i Dijon 12 1 Dag aqk which is a velocity coupling matrix LagrangeEuler Equations BilMg Ck qkqjhiltqgt Halq note ti Fi 1914 or in matrix form for all n links I Dq i W C61 6 1161 16139 To obtain the dynamic model we need to find inertia tensor of link k wrt base coordinates center of mass of link k wrt base coordinates and the link J acobian Example Find the dynamic model for a three axis SCARA robot oo 0 3 Join d a a 3 Hum m V m 2 a Tquot 1 d n q 17 o quotI a l 3 2 3 s 72 i q 3 a i Y I n v 77 1 ll Solution q 617 629 d3T Start with Link 1 Ac1 cm of link 1 in L1 coordinates 1 1 C1 0 1S1 To 0 0 1 d1 0 0 0 1 D1inertia tensor of link 1 in base coordinates Find the link 1 J acobian J1q Find Dq A1T m1 A1 B1T D1 B1 1 0 0 1 0 O 1 0 0 Dqltm1a 4gto o o m1aE12o o 0 m1a1230 0 0 0 0 0 0 0 0 0 O 0 Repeat for link 2 Ac2 D2 T02 T01 T12 39Cl 51 0 alCl C2 52 0 a2C2 51 C1 0 0151 32 C2 0 a2 0 0 1 d1 0 0 1 0 0 0 0 1 0 0 0 1 IC1 2 51 2 0 a2C1 2 a1C1 51a C12 0 a2S1 2 a1S1 0 0 1 d1 0 0 0 1 NOW use this to find 5 2 Find D2 inertia tensor of link 2 wrt L0 Compute the link 2 J acobian J 2 Add the inertia from link 2 to the manipulator inertia tensor 0 O Dqm1a123 0 0 0 2Tm2A 2B 2TDZBZ 000 1 2 m a m a 1 1 2 2 2 mzar1 m2a1a2C2 Dq m2a22 mzalazC 2 3 2 0 Repeat for link 3 center of mass in L3 is A03 for a thin rod D3 7 mza2 mzalazC2 O quot 3 2 mzaz 0 3 0 0 C1 2 51 2 0 a1C1a2C1 2 1 0 0 0 SH Cl4 0 ss4 01 00 0 0 1 d1 0 0 1 d3 0 0 0 1 O 0 0 1 C1 2 51 2 0 aICI a2C1 2 51 2 C12 0 a151ale2 0 0 1 dl q3 0 0 0 1 Link 3 center of mass in base coordinates is 53 The inertia tensor in base coordinates is D3 Compute the link 3 Jacobian J3q a1S1a2S1 2 251 2 a2C1 25391C1 a2C1 2 0 0 J3q 0 0 0 OO Add the effect of link 3 to the manipulator inertia tensor DQquotD61 A 3T M314 3 3 3TD33 3 m2 m3afm22m3a1a2C2m3a D q 1 m3azz m3a1a2CZ 0 m2 2 m2 3 m3cz2 2 m3a1a2C2 0 m 2 m3a22 0 0 m3 Compute the velocity coupling matrix for joint 1 a ij D1j 1 1 a D aqk 2 aql W I gt I a 6 C D D 11 aq 11 q 2 aql 11 q 1 0 123 6 1 a C1 D D 21 aqz 11q 2 aql 21Q a 1 a C1 D D 22 aqz 12 q 2 aql 22 1 1 C23 2 Calculate the gravitational loading at joint 1 g 2 0 g0T Calculate the required torque at joint 1 Calculate the velocity coupling matrix for joint 2 a a C2 D k aqk 21W 1 a D 2 aqz 1901 2 C13 2 C21 2 C22 2 C23 2 C3 1 C32 2 C33 2 a 1 D aql 22q 2 a 1 D aql 234 2 a 1 D aqz 21q 2 a D 6613 21 q a 1 D 8613 2261 2 012 61 D1361 D21 q D 6 3161 D aqz 32q Calculate the required torque at joint 2 Calculate the velocity coupling matrix for joint 3 Ans C3 0 Compute the gravitational loading at joint 3 Compute the required force at joint 3


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