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# ROBOTICS I CSCI 4480

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This 27 page Class Notes was uploaded by Santos Fadel on Monday October 19, 2015. The Class Notes belongs to CSCI 4480 at Rensselaer Polytechnic Institute taught by John Wen in Fall. Since its upload, it has received 29 views. For similar materials see /class/224842/csci-4480-rensselaer-polytechnic-institute in ComputerScienence at Rensselaer Polytechnic Institute.

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Date Created: 10/19/15

Robotics amp Automation Lecture 26 Wheeled Robot and Nonholonomic Systems John T Wen November 24 2008 J TWRA26 RPI E CSECS CI 4480 Robotics I Integrability of Constraints Given a set of constraints on a serial robot M61 0 M61 6 Rm we can differentiate it and write it as Now we ask the converse question if we are given Gqq 0 does there eXist M61 such that ith row of laGq 63quot If the answer is yes then the ith constraint is integrable otherwise it s non integrable An integrable constraints is called a holonomic constraint A non integrable constraint is called a nonholonomic constraint The number of nonholonomic constraints with in G is called the degree of nonholonomy of G 24 2008Copyrighwd by John T Wen Page 1 J TWRA26 RPI E CSECS CI 4480 Robotics I Kinematics of 3 Wheel Unicycle I Consider a wheel that cannot fall a fat unicycle Constraints 0 Planar constraint 1 Z r o No tilt constraint 2 O a a d o No slip constraint rc0 gtlt z E O velocity constraint We shall see that the rst two constraints written in terms of velocity are integrable holonomic constraints and the third is non integrable non holonomic constraint Px In the inertial frame 9 py mo Ry then pz Planar pTz pz r No tilt zTRy O No slip rcAozp O 24 2008C0pyrighted by John T Wen Page 2 RPI E CSECS C1 4480 Robotics I Unicycle Robot One parametrization of R is R 62 eye so no tilt constraint is satis ed J TW RA26 Differentiate R we obtain 0 Z 62 So the no slip constraint becomes we now use Qty for 1 py X rc 4 e y rs Complete differential kinematics ignoring wheel rotational angle X rc O 0 e y rs I I O l 24 2008C0pyrighwd by John T Wen Page 3 J TWRA26 RPI E CSECS CI 4480 Robotics I Unicycle A Simple Derivation velocity u L E x Wheel cannot translate sideway 17 O in base frame sin 6X cos 6y O This means X cos 6 y sine 24 2008C0pyrighwd by John T Wen Page 4 J TWRA26 RPI E CSECS CI 4480 Robotics I Differential Drive Many mobile robots are differential drive Differential kinematics X c O i 4 O lteiergt y s 2 r or 6g 1 0 Through a simple transformation this becomes the same model as unicycle 24 2008C0pyrighwd by John T Wen Page 5 J TWRA26 RPI E CSECS CI 4480 Robotics I Rearwheel Drive Car No sideway motion for rear wheels 17 r O in base frame Xsine ycos 6 O No sideway motion for front wheels 17 c x E 39 f O in base frame Xsin6 1 ycos6 1 cos MG O Differential kinematics X cos I O y i sinq O u G taTm 0 w I O 1 24 2008C0pyrighwd by John T Wen Page 6 J TWRA26 RPI E CSECS CI 4480 Robotics I Frontwheel Drive Car Tricycle No sideway motion for front wheels 17 f O in base frame Xsin6 1 y cos6 1 O No sideway motion for rear wheels 17 c x E r O in base frame Xsine ycos 6 is 0 Differential kinematics X cos6 l O y sin6 l O u 3 y 0 w i 0 1 Reduce to unicycle X cos6 l O u y s1n6l O yujrw 6 l O 1 24 2008C0pyrighted by John T Wen Page 7 J TWRA26 RPI E CSECS CI 4480 Robotics I Special Cases o Simple car I pmax lt Maximum turning radius t anim o Reeds Shepp Car speed of car restricted to three values 14 6 10 1 o Dubins Car car can only go forward 14 6 01 24 2008C0pyrlghwd by John T Wen Page 8 J TWRA26 RPI E CSECS CI 4480 Robotics I Kinematic Control of a Unicycle I Differential kinematics ignoring Wheel angle X rc O 0 e y rs I I O 1 We can now treat 6 1 as control input and the goal is to drive xy 1 to arbitrary desired position and orientation Note that there are only two inputs to control three states and there is no drift term like Ax in linear systems So linearized system is never controllable Very tough problem 24 2008Copyrighted by John T Wen Page 9 J TWRA26 RPI E CSECS CI 4480 Robotics I Approaches to Address Nonholonomic Constraints I 0 Remove nonholonomic constraint e g omnidirectional mobile robot 0 Unicycle Control x y rst then steer in place 0 Apply control theoretic tools e g optimal control nonlinear control etc Linearized driftless control systems are never controllable so nonlinear techniques must be applied Major approaches Open loop control It qfcyclicpath configuration W ace A 39 39 39 A Txog1 u A 4 xo 0 o Contmuatlon method x0 T 24 2008Copyrighwd by John T Wen Page J TWRA26 RPI E CSECS CI 4480 Robotics I Closed loop control There does not exist a time invariant continuous stabilizing feed back control Brocket 83 o Time varying feedback gain o Discontinuous feedback gain o Dynamic feedback extended J acobian See survey papers on the course webpage 24 2008C0pyrighwd by John T Wen Page 1 1 Robotics amp Automation Lecture 24 J acobian of Parallel Mechanisms John T Wen November 13 2008 J TWRA24 RPI E CSECS CI 4480 Robotics I Differential Kinematics 4bar39 linkage Example Let q q1q2q3q4T Differentiate the 3 constraints 1 orientation 2 position to get ch O Differentiate task variable q to get XT JTq Decompose q to active and passive joint variables 61a Qth 612761344 Write the velocity constraint and task velocity equation as Joaqa Jcpqp O XT JT a JTp61p If Jcp is invertible square for of constraints of passive joints we can solve for 611 as Q JC1JCaqa Then the overall differential kinematics from active joint velocity to task velocity is M 1T JTchcha qa 1T 13 2008C0pyrighwd by John T Wen Page 1 J TWRA24 RPI E CSECS CI 4480 Robotics I Singularity Parallel mechanism exhibits a different type of singularity than serial mechanism When Jcp loses rank there are internal joint motion no task motion even when all the active joints are locked Dual perspective certain task spatial force cannot be resisted by the active joints This type of singularity is called unstable singularity or actuator singularity Parallel mechanism can also contain unmanipulable singularities as in serial chain e g boundary singularity In this case Jcp is invertible but 7T loses rank 13 2008C0pyrighwd by John T Wen Page 2 Robotics amp Automation Lecture 10 Inverse Kinematics John T Wen September 25 2008 J TWRAI 0 RPI E CSECS CI 4480 Robotics I Motivation Elbow Manipulator I 91792793 Consider a 3DOF elbow manipulator Since it s 3DOF we only consider the po sition kinematics Suppose that p04 is given we want to nd all the corresponding 2008C0pyrighled by John T Wen Page 1 J TWRAI 0 RPI E CSECS CI 4480 Robotics I 3D Elbow Inverse Kinematics First note that p02 is a xed vector in the base frame so 1924 I904 I902 130113120923 R23P34 Take the norm of both sides we get 1904 I902I I923 R23p347 which can be used to solve 63 by Subproblem 3 up to 2 solutions elbow up and elbow down Once 63 is found we can go back to 1924 I904 I902 130113120923 R23I934 and solve for 6162 with Subproblem 2 up to 2 solutions shoulder right shoulder left 2008C0pyrighted by John T Wen Page 2 J TWRAI 0 RPI E CSECS CI 4480 Robotics I Subproblem 0 we know ZI 51 cosG So we can nd 6 from cos 1 51 1 Hill This is not numerically attractive since cos 6 has near zero slope for small 6 e tt tan lt gt H1i 2 pq If is in the same direction as 13 x Z then 6 is positive otherwise it s negative Alternative we can use Recall that we also saw subproblem 0 in DH parameterization 2008C0pyrighled by John T Wen Page 3 J TWRAI 0 RPI E CSECS CI 4480 Robotics I Subproblem 1 Spinning I or 51 about generates a cone Let the projection of I and Z to the top of the cone corresponding to the tip of 13 as 131 and 511 a 1 k aka ME Then 6 is the angle of rotation from 131 to 671 about E This is just subproblem 0 2008C0pyrighled by John T Wen Page 4 J TWRAI 0 RPI E CSECS CI 4480 Robotics I Subproblem 2 If 1 and 2 are collinear we just have subproblem 1 So assume that they are not collinear Now consider two cones spinning 13 about 2 and spinning Z about 1 There may be 0 1 or 2 intersections between the cones which are the solutions Let E be the vector of intersection Then 3 exp 91k1gtlt1 6x19lt92k2gtlt13 Represent E as Z ch 1l3jc2 7 1gtlt2 Since exp 6kgtlt b use EulerRodriguez formula we have 13 063g1 2 1 23 0c 123213 2008C0pyrighled by John T Wen Page 5 J TWRAI 0 RPI E CSECS CI 4480 Robotics I Subproblem 2 C0nt This can be written as 1 1 2 X i 1 1 2 1 3 2 3917 We can now solve for 0c 3 since 1 and 2 are not collinear 1 1 E X i 12 1 3 1 E 22 E 2008C0pyrighled by John T Wen Page 6 J TWRAI 0 RPI E CSECS CI 4480 Robotics I Subproblem 2 Cont It remains to solve 7 Note that g g g g 2 320 2l3220 l3k1quot932 lt1sz ll llz Since CL 3 have been found 7 can be solved 1 2 Z If y s are imaginary then there is no solution two cones do not intersection If y 0 then there is 1 solution two cones intersection at the tangent of the cones If 7 have two positive solutions then there are 2 solutions two cones intersecting at two Y I H HZ ocZ BZ ZOLBE 1 ng points Once Z is found 6162 are found by solving Subproblem 1 twice 2008C0pyrighled by John T Wen Page 7 J TWRAI 0 RPI E CSECS CI 4480 Robotics I Subproblem 3 The solutions correspond to the intersections between the cone generated by spin ning 13 about and the sphere centered around the tip of c with radius 8 First project 13 to the top of the cone 131 13 iii Then project 67 to the plane parallel to the top of the cone 5111 3c Let the distance of the projection of Z exp6kgtlt 13 to the top of the cone be 81 Then 6 82 k 17 512 2008C0pyrighled by John T Wen Page 8 J TWRAI 0 RPI E CSECS CI 4480 Robotics I Subproblem 3 Cont Now we can just focus on the action at the top of the cone 71 rotates 6 to the rst intersection then over an angle say I to be aligned with 671 then over I again to the second intersection By the law of cosine a 2 a 2 a a H191 611 2I91611C05 5 Solving cos I we get 171 H2 6712 51 2 171 511 II If the magnitude of the right hand side is greater than 1 there is no solution If the cosq magnitude is 1 there is one solution If the magnitude is less than 1 there are two solutions 2008C0pyrighled by John T Wen Page 9 J TWRAI 0 RPI E CSECS CI 4480 Robotics I MATLAB code i Subproblem solutions subproblem0m subproblem1m subproblem2m subproblem3m Examples of Inverse Kinematics o PUMA 560 see puma560indin m o SCARA 2008Copyrighled by John T Wen Page

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