Kinematic Design of Redundant Robotic Manipulators that are - PowerPoint PPT Presentation

Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions � Kinematic Design of Redundant Robotic Manipulators that are Optimally Fault Tolerant Presented by: Khaled M. Ben-Gharbia Septmber 8 th , 2014 1/45

Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions � Why Fault Tolerance? Applications hazardous waste cleanup space/underwater exploration anywhere failures are likely or intervention is costly Common failure mode locked actuators 2/45

Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions � Simple Redundant Robot Planar 3DOF ˙  θ 1  � ˙ � x 1 ˙ = J ( θ ) θ 2   x 2 ˙ ˙ θ 3 3/45

Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions � Jacobian ( J ) Matrix 3R planar manipulator Forward Kinematics: x = f ( θ ) x 1 = a 1 cos( θ 1 ) + a 2 cos( θ 1 + θ 2 ) + a 3 cos( θ 1 + θ 2 + θ 3 ) x 2 = a 1 sin( θ 1 ) + a 2 sin( θ 1 + θ 2 ) + a 3 sin( θ 1 + θ 2 + θ 3 ) = ⇒ taking the derivative w.r.t. time x = J ( θ ) ˙ = ⇒ ˙ θ θ 1 + θ 2 + θ 3 Geometrically x 2 θ 1 + θ 2 J 3 × 3 = [ j 1 j 2 j 3 ] = [ z 1 × p 1 z 2 × p 2 z 3 × p 3 ] θ 1 x 1 where z i is the rotation axis, and here it is [0 0 1] T Spatial manipulator � v i � j i = ω i 4/45

Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions � Jacobian ( J ) Matrix 3R planar manipulator Forward Kinematics: x = f ( θ ) j 2 x 1 = a 1 cos( θ 1 ) + a 2 cos( θ 1 + θ 2 ) + a 3 cos( θ 1 + θ 2 + θ 3 ) x 2 = a 1 sin( θ 1 ) + a 2 sin( θ 1 + θ 2 ) + a 3 sin( θ 1 + θ 2 + θ 3 ) = ⇒ taking the derivative w.r.t. time p 2 x = J ( θ ) ˙ = ⇒ ˙ θ θ 1 + θ 2 + θ 3 Geometrically x 2 θ 1 + θ 2 J 3 × 3 = [ j 1 j 2 j 3 ] = [ z 1 × p 1 z 2 × p 2 z 3 × p 3 ] θ 1 x 1 where z i is the rotation axis, and here it is [0 0 1] T Spatial manipulator � v i � j i = ω i 4/45

Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions � Jacobian ( J ) Matrix 3R planar manipulator Forward Kinematics: x = f ( θ ) j 2 x 1 = a 1 cos( θ 1 ) + a 2 cos( θ 1 + θ 2 ) + a 3 cos( θ 1 + θ 2 + θ 3 ) x 2 = a 1 sin( θ 1 ) + a 2 sin( θ 1 + θ 2 ) + a 3 sin( θ 1 + θ 2 + θ 3 ) = ⇒ taking the derivative w.r.t. time p 2 x = J ( θ ) ˙ = ⇒ ˙ θ θ 1 + θ 2 + θ 3 Geometrically x 2 θ 1 + θ 2 J 3 × 3 = [ j 1 j 2 j 3 ] = [ z 1 × p 1 z 2 × p 2 z 3 × p 3 ] θ 1 x 1 where z i is the rotation axis, and here it is [0 0 1] T Spatial manipulator � v i � j i = ω i 4/45

Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions � Kinematic Dexterity Measures: Function of Singular Values of the Jacobian � σ 1 � � cos φ � 0 0 − sin φ V T V = I D = U = 0 σ 2 0 sin φ cos φ 5/45

Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions � Approaching Kinematic Singularities Singularity: σ 2 = 0 Value of σ m is a common dexterity measure 6/45

Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions � Optimal Fault Tolerant Configurations for Locked Joints Failures Fault-free Jacobian: J m × n = [ j 1 j 2 . . . j n ] Failure at joint f : f J m × n − 1 = [ j 1 j 2 . . . j f − 1 j f +1 . . . j n ] f J = � m f σ i f ˆ f ˆ v T u i i =1 i f σ m Worst-case remaining dexterity: K = min n f =1 � n − m Isotropic & Optimal J : K opt = σ for all f n 7/45

Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions � Isotropic and Optimally Fault Tolerant Jacobian equal σ i ’s equal f σ m for all f equal � j i � for all i Example: 2 × 3 isotropic and optimally fault tolerant J :  � � �  2 1 1  − 3 6 6 J = [ j 1 j 2 j 3 ] =  � � 1 1 0 − 2 2 � T � � � � 1 1 1 Null space equally distributed: ˆ n J = 3 3 3 8/45

Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions � Example: 2 × 3 Optimally Fault Tolerant Jacobian j 3 120° j 1 120° 120° j 2 � f σ 2 = 1 Remaining dexterity: K opt = 3 , for all f 9/45

Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions � Example: 2 × 3 Optimally Fault Tolerant Jacobian j 3 p 3 p 2 120° j 1 120° 120° j 2 p 1 � f σ 2 = 1 Remaining dexterity: K opt = 3 , for all f 9/45

Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions � Example: 2 × 3 Optimally Fault Tolerant Jacobian j 3 p 3 p 2 120° j 1 120° 120° p 1 j 2 � f σ 2 = 1 Remaining dexterity: K opt = 3 , for all f 9/45

Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions � Different Optimally Fault Tolerant Jacobians Sign change and permuting columns do not affect the isotropy and the optimally fault tolerant properties of J , e.g. But each new J may belong to a different manipulator 10/45

Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions � Goal of This Research Show that multiple different manipulators possess the same desired local properties described by a Jacobian (designed to be optimally fault-tolerant) Study optimally fault tolerant Jacobians for different task space dimensions Illustrate the difference between these manipulators in terms of their global fault tolerant properties 11/45

Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions � 3DOF Planar Manipulators The total possible Jacobians are 48 (2 3 × 3!) y y � j ✁ j ✂ ] � ✄ j ✁ j ✂ ] [ j [ j 2 2 2 � j ✂ j ✁ ] 2 � ✄ j ✂ j ✁ ] ( 0 , ) [ j ( 0 , ) [ j 3 3 3 3 All of these Jacobians 2 2 2 2 3 3 -3 -3 correspond to only 4 different -2 -2 1 1 x x -1 -1 2 2 2 ( − 0 , ) ( − 0 , ) 3 3 2 3 3 2 3 manipulators -2 -3 -2 2 -3 3 � j ✁ j ✂ ] � -j ✁ j ✂ ] -[ j -[ j ✁ ] � j ✂ j ✁ ] � -j ✂ j -[ j -[ j Link lengths: (a) (b) y y � j ✁ ✄ j ✂ ] � -j ✁ ✄ j ✂ ] [ j [ j 2 2 2 � j ✂ -j ✁ ] 2 � -j ✂ -j ✁ ] ( 0 , ) [ j ( 0 , ) [ j 3 3 3 3 2 2 Robot a 1 a 2 a 3 2 2 3 3 -3 -3 1 L s L s L s -2 -2 1 1 x x -1 -1 2 2 2 L s L l L s ( − 0 , ) ( − 0 , ) 3 3 2 3 3 2 3 L l L s L s -2 2 -3 -2 2 -3 3 3 � j ✁ ✄ j ✂ ] � -j ✁ ✄ j ✂ ] -[ j -[ j 4 L l L l L s � j ✂ -j ✁ ] � -j ✂ -j ✁ ] -[ j -[ j (c) (d) √ � L s = 2 / 3 and L l = 2 12/45

Introduction Kinematics Fault Tolerance Goal 3R Planar 4R Planar To Spatial 4R Spatial 7R Spatial Conclusions � 3DOF Planar Manipulators The total possible Jacobians are 48 (2 3 × 3!) y y 2 2 � j ✁ ] � j ✁ j [ j ✂ ] [ j ✂ -j 2 -3 2 -3 2 � j ✁ ] 2 � -j ✁ j 3 [ j ✂ j 3 [ j ✂ ] ( − , ) ( − , ) 2 4 2 4 1 1 2 2 All of these Jacobians -3 -3 2 2 3 3 2 2 -1 x -1 x correspond to only 4 different 2 2 1 1 -2 -2 3 3 3 3 manipulators -2 -2 -1 2 -1 2 2 2 3 3 ( , − ) ( , − ) 2 4 � j ✁ ] 2 4 � j ✁ j -[ j ✂ ] -[ j ✂ -j 3 3 � j ✁ ] � -j ✁ j -[ j ✂ j -[ j ✂ ] (a) (b) Link lengths: y y 2 � -j ✁ ] 2 � -j ✁ ] [ j [ j ✂ -j ✂ j 2 2 -3 � j ✁ -j -3 � -j ✁ -j 2 2 3 3 [ j ✂ ] [ j ✂ ] ( − , ) ( − , ) 2 4 2 4 1 1 2 2 2 2 Robot a 1 a 2 a 3 -3 3 -3 3 2 2 3 3 2 2 1 L s L s L s -1 x -1 x 1 1 -2 -2 2 L s L l L s 3 3 -2 -1 -2 -1 2 2 2 3 2 3 ( , ) ( , − ) − 3 L l L s L s � -j ✁ ] 2 4 � -j ✁ ] 2 4 -[ j -[ j ✂ j ✂ -j 3 3 � -j ✁ -j � j ✁ -j -[ j ✂ ] -[ j ✂ ] 4 L l L l L s (c) (d) √ � L s = 2 / 3 and L l = 2 12/45