Introduction to Robotics Tutorial II Alfred Bruckstein Yaniv Altshuler

Denavit-Hartenberg Reminder Specialized description of articulated figures Each joint has only one degree of freedom rotate around its z-axis translate along its z-axis

Denavit-Hartenberg Link length ai The perpendicular distance between the axes of jointi and jointi+1

Denavit-Hartenberg Link twist αi The angle between the axes of jointi and jointi+1 Angle around xi-axis

Denavit-Hartenberg Link offset di The distance between the origins of the coordinate frames attached to jointi and jointi+1 Measured along the axis of jointi

Denavit-Hartenberg Link rotation (joint angle) φi The angle between the link lenghts αi-1 and αi Angle around zi-axis

Denavit-Hartenberg Compute the link vector ai and the link length Attach coordinate frames to the joint axes Compute the link twist αi Compute the link offset di Compute the joint angle φi Compute the transformation (i-1)Ti which transforms entities from linki to linki-1

This transformation is done in several steps : Denavit-Hartenberg This transformation is done in several steps : Rotate the link twist angle αi-1 around the axis xi Translate the link length ai-1 along the axis xi Translate the link offset di along the axis zi Rotate the joint angle φi around the axis zi

Denavit-Hartenberg

Denavit-Hartenberg

Denavit-Hartenberg

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Multiplying the matrices : Denavit-Hartenberg Multiplying the matrices :

DH Example 3 revolute joints Shown in home position R L1 L2 joint 1 Link 2 Link 3 Link 1 joint 2 joint 3 L1 L2

DH Example Shown with joints in non-zero positions Z0 x3 z3 2 3 x2 1 x0 Z1 Observe that frame i moves with link i

1 = 90o (rotate by 90o around x0 to align Z0 and Z1) DH Example 1 = 90o (rotate by 90o around x0 to align Z0 and Z1) R Z0 L1 L2 1 x1 x2 x3 1 x0 3 2 Z1 Z3 Z2 Link Var  d  a 1 1 90o R 2 2 L1 3 3 L2

DH Example Link Var  d  a 1 1 90o R 2 2 L1 3 3 L2

DH Example

DH Example z0 x3 z3 2 3 x2 x1 z2 1 x0 z1 x1 axis expressed wrt {0} y1 axis expressed wrt {0} z1 axis expressed wrt {0} Origin of {1} w.r.t. {0}

DH Example z0 x3 z3 2 3 x2 x1 z2 1 x0 z1 x2 axis expressed wrt {1} y2 axis expressed wrt {1} z2 axis expressed wrt {1} Origin of {2} w.r.t. {1}

DH Example z0 x3 z3 2 3 x2 x1 z2 1 x0 z1 x3 axis expressed wrt {2} y3 axis expressed wrt {2} z3 axis expressed wrt {2} Origin of {3} w.r.t. {2}

DH Example where

Example – the Stanford Arm

Example – the Stanford Arm Y1 Z1 X2 Z2 X3 Z3 X4 X5 X6 Z4 Z5 Z6 X7 Z7

Example – the Stanford Arm Y1 Z1 X2 Z2 X3 Z3 X4 X5 X6 Z4 Z5 Z6 X7 Z7 i ai di i i 1 d1 90° 1 2 3 4 5 6

Example – the Stanford Arm Y1 Z1 X2 Z2 X3 Z3 X4 X5 X6 Z4 Z5 Z6 X7 Z7 i ai di i i 1 d1 90° 1 2 d2 2 3 4 5 6

Example – the Stanford Arm Y1 Z1 X2 Z2 X3 Z3 X4 X5 X6 Z4 Z5 Z6 X7 Z7 i ai di i i 1 d1 90° 1 2 d2 2 3 d3 (var) 4 5 6

Example – the Stanford Arm Y1 Z1 X2 Z2 X3 Z3 X4 X5 X6 Z4 Z5 Z6 X7 Z7 i ai di i i 1 d1 90° 1 2 d2 2 3 d3 (var) 4 d4 4 5 6

Example – the Stanford Arm Y1 Z1 X2 Z2 X3 Z3 X4 X5 X6 Z4 Z5 Z6 X7 Z7 i ai di i i 1 d1 90° 1 2 d2 2 3 d3 (var) 4 d4 4 5 d5 0° 5 6

Example – the Stanford Arm Y1 Z1 X2 Z2 X3 Z3 X4 X5 X6 Z4 Z5 Z6 X7 Z7 i ai di i i 1 d1 90° 1 2 d2 2 3 d3 (var) 4 d4 4 5 d5 0° 5 6 d6 6