Ferrier_kinematics3

8/4/2019 Ferrier_kinematics3

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Kinematics Frame Assignment using

Denavit-Hartenberg Convention

Professor Nicola Ferrier

ME Room 2246, [email protected]


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Coordinate Transformations

Goal

Base

Supply

End-effector

Table


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Coordinate Transformations

Robot forward

kinematic model


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Motion is composition of elementarymotions for each link

Base

End-effector

Manipulator Forward Kinematics


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Relative Pose between 2 links

i-1

i


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Relative Pose between 2 links

Frames can be chosen arbitrarily

Denavit-Hartenberg convention is usedto assign frames described in 3.2.2of Spong, Hutchinson, Vidyasagar Text

Iterative process (start at base, assign

frames for each link from base to end-effector)


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DH Frame assignment

Frame {i} moves with link i when joint i is actuated

Zi axis is along joint axis i+1

Zi is axis of actuation for joint i+1

Link i

Zi-1

Zi

Link i+1Link i-1


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DH convention: Assign Z axes

Use actuation as a guide

Prismatic joint slides along zi

Revolute joint rotates around zi

Establish base frame {0}:

Nearly arbitrary

Start at base and assign frames 1,,N

Pick x-axis and origin

y-axis chosen to form a right hand system


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Robot Base

Often base is given orsome fixed point on the

work-table is used. z0 is along joint axis 1

Original:

any point on z0 for origin

Modified DH: {0} is defined to be

completely co-incidentwith the referencesystem {1}, when the

variable joint parameter,d1 or q1 , is zero.


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DH convention: Assign X axes

Start at base and assign frames 1,,N

Pick x-axis and origin

y-axis chosen to form a right hand system

Consider 3 cases for zi-1 and zi:

Not-coplanar

Parallel Intersect


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DH convention: x axis

zi-1 and zi are not-coplanar

Common normal to axes is the link axis

Intersection with zi is origin

Xizi-1

zi

Usually, xipoints fromframe i-1 to i


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zi and zi-1 are parallel

Infinitely many common normals

Pick one to be the link axis

Choose normal that passes through origin of

frame {i-1} pointing toward zi Origin is intersection of x

iwith z

i

Xi

zi-1 zi


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If joint axes zi-1

and ziintersect, xi is normal to

the plane containing theaxes

link i

Xi

xi= (z

i-1 z

i)

zi-1

zi


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DH convention: Origin non-coplanar Z

Origin of frame {i} is placed at intersection of jointaxis and link axis

xi

zi


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DH convention: y axis

Yi is chosen to make a right hand frame

xi

Zixi pointsfrom framei-1 to i

Yi


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DH convention: Origin parallel Z


Origin is intersection of xi with zi

xi

zizi-1


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DH convention: x axis - parallel Z


Origin is intersection of xi with zi

Yi is chosen to make a right hand frame

xi

ziyizi-1


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DH convention: origin

link i

xi

zi-1

zi

If joint axes intersect, theorigin of frame {i} isusually placed at

intersection of the jointaxes


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DH convention: y axis

link i

xi

zi-1

zi

Yi is chosen to make aright hand frame

yi


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End-Effector Frame Frame to which

the gripper is

attached Sometimes {n} isused

denoted by {e} (or{n+1} in many

texts) Often simple

translation along Xnaxis

Z4

Ze

Xe


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End-Effector Frame Frame to which

the gripper isattached denoted by {e} (or

{n+1} in manytexts)

Often simpletranslation along Xn

axis Often:

Origin betweengrippers

Z points outward

(approach) Y points along

pinch direction(sliding)

X points normal

Z4

ze

xe

ye

k


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Link Parameters

ai+1

Zi-1Zi

Zi+1

Zi

ai+1

ai

ai

Link i

i


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Joint Parameters

qi+1

di

di+1

qi

qi

O i i l DH


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-1

Original DH

Frame is placed at

distal end of link

xi screw motionzi-1 screw motion

DH F d P t


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DH Frames and Parameters

R b t R l t J i t DH


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Robot Revolute Joint DH

P i ti J i t DH


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Prismatic Joint DH

Li k T f ti


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Link Transformations

Described by 4 parameters:

ai : twist

ai : link length

di : joint offset

qi : joint angle

Joint variable is di or qi Build Table with values for each link:

Link Var q d a a

1 q1 q1 090o

L12 d2 0 d2 0 0

Li k T f ti


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Link Transformations

Described by 4 parameters:

ai : twist

ai : link length

di : joint offset

qi : joint angle

Joint variable is di or qi Link Transformation is

xiscrew motionzi-1 screw motion

A mat ices


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A-matrices

Ai= contains only one variable

or

Equation 3.10 in Spong,

Hutchinson, Vidyasagar

Original DH


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-1

Original DH

Frame is placed at

distal end of link

zi-1 screw motion xi screw motion

Modified DH


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Zi Zi+2

Zi+1

Modified DH

Frame is placed atproximal end of

link

xi

zi yi

zi screw motionxi-1 screw motion

Modified DH text figure


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Modified DH text figure

DH Example: academic manipulator


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3 revolute joints

Shown in home position

Link 1 Link 3

Link 2

joint 1

joint 2 joint 3

R

L1 L2



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Z1

Z0

Z2

q1

q2q3

Z0 and Z1 are not co-planar

Z1 and Z2 are parallel



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Z1

Z0

Z2

q1

q2q3

x0

x1 x2 x3

Z3

Z0 and Z1 are not co-planar:

x0 is the common normal



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Z1

Z0

Z2

q1

q2q3

x0

x1 x2 x3

Z3



Z1 and Z2 are parallel :x1 is selected as the common

normal that lies along the

center of the link



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Z1

Z0

Z2

q1

q2q3

x0

x1 x2 x3

Z3



Z2 and Z3 are parallel :x2 is selected as the common

normal that lies along the

center of the link



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Shown with joints in non-zero

positions

Z1

Z0

Z2q1

q2 q3

x0

x1

x2

z3

x3

Observe that frame i moves withlink i



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Z1

Z0

Z2

Link lengths given

a1 = 90o(rotate by 90o around x0 to

align Z0 and Z1)

x0

x1 x2 x3

Z3

R

L1 L2

a1



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Z1

Z0

Z2

q1

q2q3

Build table

x0

x1 x2 x3

Z3

Link Var q d a a1 q1 q1 0

90o R

2 q2 q2 0 0 L1

3 q3

q3

0 0 L2

R

L1 L2

a1



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Link Var q d a a

1 q1 q1 0 90o

R

2 q2 q2 0 0 L1

3 q3 q3 0 0 L2



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z1

z0

z2q1

q2

q3

x0

x1

x2

z3

x3

Origin of {1}w.r.t. {0}

x1 axis expressedwrt {0}

y1 axis expressedwrt {0}

z1 axis expressedwrt {0}



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z1

z0

z2q1

q2

q3

x0

x1

x2

z3

x3







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z1

z0

z2q1

q2

q3

x0

x1

x2

z3

x3







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where



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DH Example: academic manipulator alternate end-effector frame


Z1

Z0

Z2

q1

q2q3

Z0 and Z1 are not co-planar

Z1 and Z2 are parallel

Pick this z3



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Z1

Z0

Z2

q1

q2q3

x0

x1

Z3a1

x2

Would need

to rotateabout y2

here!

y2



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Z1

Z0

q1

q2q3

x0

x1

x2

Z3a1

x2

Solution: Add

offset to rotationabout z2(q3+90

o )



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Z1

Z0

Z2

q1

q2q3

x0

x1

x2 x3

Z3

L2

a1

x2

Now can

rotate aboutx to align z2

and z3



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Link Var q d a a

1 q1 q1 090o R

2 q2 q2 0 0 L1

3 q3 q3 +90o 0 90o

0e - L2



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Z1

Z0

Z2

q1

q2q3

x0

x1

x2

x3

Z3

Link Var q d a a1 q1 q1 0

90o R

2 q2 q2 0 0 L1

3 q3 q3 +90o 0 90o 0

R

L1 L2

a1

x2



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Z1

Z0

Z2

q1

q2q3

x0

x1

x2

x3

Z3

R

L1 L2

a1

x2

Z3


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