Introduction Robotics Lecture3

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Introduction Robotics

Introduction Robotics, lecture 3 of 7

dr Dragan Kosti

WTB Dynamics and Control

September-October 2009


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Recapitulation

Forward kinematics

Outline


Inverse kinematics problem


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Robot manipulators

Kinematic chain is series

of links and joints.SCARA

eometr


Types of joints:

rotary (revolute, )

prismatic (translational, d).

schematic representations of robot joints


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Common geometries of robot manipulators



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Forward kinematics problem

Determine position and orientation of the end-effector as

function of displacements in robot joints.



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DH convention for homogenous transformations (1/2)

An arbitrary homogeneous transformation is based on 6

independent variables: 3 for rotation + 3 for translation.

DH convention reduces 6 to 4, by specific choice of


.

In DH convention, each homogeneous transformation has the form:


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DH convention for homogenous transformations (2/2)

Position and orientation of coordinate frame i with respect to

frame i-1 is specified by homogenous transformation matrix:

q0

qi+1

xnz

0

zn


ai

qi

x0

xi-1

xi

zi

zi-1

y0 yn

di

i0 n

where


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Physical meaning of DH parameters Link length ai is distance fromzi-1 to

zi measured alongxi.

Link twist i is angle betweenzi-1andzi measured in plane normal

qi

q0qi+1

zi

xn

y0

yn

z0

zn

i0 n


i

- .

Link offset di is distance from origin

of frame i-1 to the intersectionxiwithz

i-1, measured alongz

i-1.

Joint angle i is angle fromxi-1 toximeasured in plane normal tozi-1

(right-hand rule).

qi

0

xi-1

xi

zi-1di


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DH convention to assign coordinate frames1. Assignzi to be the axis of actuation for joint i+1 (unless otherwise statedzn

coincides withzn-1).

2. Choosex0 andy0 so that the base frame is right-handed.3. Iterative procedure for choosing oixiyizi depending on oi-1xi-1yi-1zi-1 (i=1, 2, , n-1):

a)zi1 andzi are not coplanar; there is an unique shortest line segment fromzi1 to


i, i

line intersectszi is the origin oi; chooseyi to form a right-handed frame,

b)zi1 is parallel tozi; there are infinitely many common normals; choosexi as

the normal passes through oi1; choose oi as the point at which this normal

intersectszi; chooseyi to form a right-handed frame,

c)zi1 intersectszi; axisxi is chosen normal to the plane formed byzi andzi1;

its positive direction is arbitrary; the most natural choice of oi is the

intersection ofzi andzi1, however, any point along thezi suffices;

chooseyi

to form a right-handed frame.


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Forward kinematics (1/2)

Homogenous transformation matrix relating the frame oixiyizi to

oi-1xi-1yi-1zi-1:


Ai specifies position and orientation of oixiyizi w.r.t. oi-1xi-1yi-1zi-1.

Homogenous transformation matrix Tji expresses position and

orientation of ojxjyjzj with respect to oixiyizi:

jjiiij AAAAT 121 ++= K


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Forward kinematics of a serial manipulator with njoints can be

represented by homogenous transformation matrixHn0 which

defines position and orientation of the end-effectors (tip)

Forward kinematics (2/2)


rame on nynzn re a ve o e ase coor na e rame o0x0y0z0:

=

==

1

)()()(

),()()()(

13

00

0

11

00

0

qqq

qq

nn

n

nnnn

xRH

qAqATH K

[ ];00013 =0

[ ]Tnqq L1=q


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Case-study: RRR robot manipulator

-q3

x2x3

1

y2

y3

d2 a3

d3

z3

Introduction Robotics, lecture 3 of 7 x0

q1

q2x1

y0

z0

1

d1

a2

z1

waist

shoulder

e ow

1- twist angle

ai - link lenghts

di - link offsets

qi - displacements


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DH parameters of RRR robot manipulator



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Forward kinematics of RRR robot manipulator (1/2)

Coordinate frame o3x3y3z3 is related with the base frame o0x0y0z0 viahomogenous transformation matrix:

== 32103 (q)(q)A(q)AA(q)T


=

131

0303

0

(q)x(q)R

where

Tqqq ][ 321=q Tzyx ][)(03 =qx

]000[31 =0


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Forward kinematics of RRR robot manipulator (2/2)

,

Position of end-effector:[ ] 132223231 )sin(cos)cos(cos qddqaqqaqx ++++=

[ ] 132223231 )cos(cos)cos(sin qddqaqqaqy +++=


, 122323 sin)sin( dqaqqaz +++=

Orientation of end-effector:

++

++

++

=

0)cos()sin(

cos)sin(sin)cos(sinsin)sin(cos)cos(cos

3232

1321321

132132103

qqqq

qqqqqqqqqqqqqq

R


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Inverse Kinematics


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Inverse kinematics problem

Inverse kinematics (IK): determine displacements in robot joints that

correspond to given position and orientation of the end-effector.



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Illustration: IK for planar RR manipulator (1/2)

Elbow down IK solution



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Illustration: IK for planar RR manipulator (2/2)

Elbow up IK solution



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The general IK problem (1/2) Given a homogenous transformation matrixHSE(3)

find multi le solution s ,, to e uation


Here,Hrepresents the desired position and orientation of the tip

coordinate frame onxnynzn relative to coordinate frame o0x0y0z0of the base; T0n is product of homogenous transformation

matrices relating successive coordinate frames:


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The general IK problem (2/2)

Since the bottom rows of both T0n andHare equal to [0 0 0 1],

equation


gives rise to 4 trivial equations and 12 equations in n unknownsq1,,qn:

Here, Tij andHij are nontrivial elements of T0

n andH.


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Illustration: Stanford manipulator



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FK of Stanford manipulator



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Example of IK solution for Stanford manipulator Rotational equations (correspond toR06):


Positional equations (correspond to o06): One solution:


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Nature of IK solutions FK problem has always unique solution whereas IK problem may

or may not have a solution; if IK solution exists, it may or may not

be unique; solving IK equations, in general, is much too difficult.

It is preferable to find IK solutions in closed-form:


faster computation (e.g. at sampling time of 1 [ms]),

if multiple IK solutions exist, then closed-form allows us to developrules for choosing a particular solution among several.

Existence of IK solutions depends on mathematical as well asengineering considerations.

We assume that the given position and orientation is such that atleast one IK solution exists.


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Kinematic decoupling (1/3)

General IK problem is difficult BUT for manipulators having 6 jointswith the last 3 joint axes intersecting at one point, it is possible to

decouple the general IK problem into two simpler problems:


inverse position kinematics and inverse orientation kinematics. IK problem: for givenR and o solve 9 rotational and 3 positional

equations:


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Spherical wrist as paradigm.


Let oc

be the intersection of the last 3 joint axes; asz3,z4, andz5intersect at oc, the origins o4 and o5 will always be at oc;

the motion of joints 4, 5 and 6 will not change the position of oc;

only motions of joints 1, 2 and 3 can influence position of oc.


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q1, q2, q3

q4, q5, q6


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Articulated manipulator: inverse position problem


Inverse tangent functionAtan2(xc,yc) is defined for all (xc,yc)(0,0)and equals the unique angle 1 such that:

,cos22

1

cc

c

yx

x

+

= .sin22

1

cc

c

yx

y

+

=


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Articulated manipulator: left arm configuration

r


*

*


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Articulated manipulator: right arm configuration

Introduction Robotics, lecture 3 of 7 +

*

*

r


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*

r

Articulated manipulator: IK solution for 3


Law of cosines:

+ elbow down; - elbow up


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Articulated manipulator: IK solution for 2


1

2 2=1- 2

1=Atan2(r,s)

2=Atan2(a2+a3cos3,a3sin3)


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Four IK solutions 1-

3for articulated manipulator

PUMA robot asan example of

the articulated


geometry.


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Articulated manipulator: inverse orientation problem

Introduction Robotics, lecture 3 of 7 Equation to solve:


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Articulated manipulator: IK solutions for 4

and 5

Equations given by the third column in :


If not both right-hand sides of the first two equations are zero:

If positive square root is chosen in solution for 5:


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Articulated manipulator: IK solutions for 6

The first two equations given by the last row in :

-s5c6 = s1r11 - c1r21

s5s6 = s1r12 - c1r22


Analogous approach if negative square root is chosen insolution for 5.

From these equations it follows:

Introduction Robotics Lecture3

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