introduction to robotics lecture ii
DESCRIPTION
Introduction to Robotics Lecture II. Alfred Bruckstein Yaniv Altshuler. Denavit-Hartenberg. 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. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/1.jpg)
Introduction to Robotics
Lecture II
Alfred BrucksteinYaniv Altshuler
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Denavit-Hartenberg
•Specialized description of articulated figures
•Each joint has only one degree of freedom
•rotate around its z-axis
•translate along its z-axis
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Denavit-Hartenberg
•One degree of freedom : very compact notation
•Only four parameters to describe a relation between two links :
•link length
•link twist
•link offset
•link rotation
![Page 4: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/4.jpg)
Denavit-Hartenberg• Link length ai
• The perpendicular distance between the axes of jointi and jointi+1
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• Link twist αi
• The angle between the axes of jointi and jointi+1
• Angle around xi-axis
Denavit-Hartenberg
![Page 6: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/6.jpg)
• 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
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• Link rotation (joint angle) φi
• The angle between the link lenghts αi-1 and αi
• Angle around zi-axis
Denavit-Hartenberg
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Denavit-Hartenberg
1.Compute the link vector ai and the link length
2.Attach coordinate frames to the joint axes
3.Compute the link twist αi
• How to compute the parameters to describe an articulated figure :
![Page 9: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/9.jpg)
Denavit-Hartenberg
4.Compute the link offset di
5.Compute the joint angle φi
6.Compute the transformation (i-1)Ti which transforms entities from linki to linki-1
![Page 10: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/10.jpg)
Denavit-Hartenberg
1.Compute the link vector ai and the link length
2.Attach coordinate frames to the joint axes
3.Compute the link twist αi
4.Compute the link offset di
5.Compute the joint angle φi
6.Compute the transformation (i-1)Ti which transforms entities from linki to linki-1
• Let’s do it step by step
![Page 11: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/11.jpg)
Denavit-HartenbergThe link length ai is the shortest
distance between the joint axes jointi and jointi+1.
Let the joint axes be given by the expression :
Where pi is a point on axis of jointi and ui is one of its direction vectors (analogous for jointi+1).
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Denavit-Hartenberg
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Denavit-Hartenberg
•There are three methods to compute the link vector ai and the link length
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Denavit-Hartenberg
The shortest distance ai is the length of the vector connecting the two axes, and perpendicular to both of them.
Which can be expressed :
Method 1 : The Pseudo-naive approach
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Denavit-HartenbergLet’s find the points oi and oai where
this distance exists.
![Page 16: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/16.jpg)
Denavit-HartenbergWe can go some distance s from pi along
axisi, and then the distance ai along the unit vector
and finally some distance t along axisi+1 to arrive at point pi+1.
![Page 17: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/17.jpg)
Denavit-Hartenberg
Multiplying respectively by ui and ui+1, we obtain the two following equations:
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Denavit-Hartenberg
Solution :
![Page 19: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/19.jpg)
Denavit-Hartenberg
Finally, using and we obtain :
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Denavit-Hartenberg
We can go some distance s from point pi along the axisi, and then go some distance k along ui x ui+1. Finally go some distance t along the axisi+1 to arrive at point pi+1.
The vector ui x ui+1 gives the perpendicular vector to both axes. Let’s find out where it is located on the joint axes.
Method 2 : The Geometric approach
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Denavit-Hartenberg
We obtain the equation :
There are three unknowns.
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Denavit-Hartenberg
Let’s first eliminate the unknown k from the equation :
by multiplying by ui :
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Denavit-Hartenberg
by multiplying by ui+1 :
Let’s first eliminate the unknown k from the equation :
![Page 24: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/24.jpg)
Denavit-Hartenberg
by multiplying by ui x ui+1 :
Now we shall eliminate the s and t from the equation :
![Page 25: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/25.jpg)
Denavit-Hartenberg
We have obtained a system of three equations in the unknowns s, t, k :
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Denavit-HartenbergFrom
,it can be seen that the shortest distance between jointi and jointi+1 is given by the vector :
Where
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Denavit-Hartenberg
From
and ,
we can compute s and t :
![Page 28: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/28.jpg)
Denavit-Hartenberg
Finally, using and we obtain :
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Denavit-Hartenberg
The distance between two arbitrary points located on the joint axes jointi and jointi+1 is :
Method 3 : The Analytic approach
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Denavit-Hartenberg
The link length of linki, ai, is the minimum distance between the joint axes :
![Page 31: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/31.jpg)
Denavit-Hartenberg
A necessary condition is :
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Denavit-Hartenberg
Which is equivalent to their numerators being equal to 0 :
![Page 33: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/33.jpg)
Denavit-Hartenberg
Rewriting this system yields :
![Page 34: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/34.jpg)
Denavit-HartenbergWhose solution are :
![Page 35: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/35.jpg)
Denavit-Hartenberg
Finally, using and we obtain :
![Page 36: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/36.jpg)
Denavit-Hartenberg
oi and oai are the closest points on the axes of jointi and jointi+1.
We deduce that the link vector ai and the link length ai :
![Page 37: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/37.jpg)
Denavit-Hartenberg
The link vector ai :
![Page 38: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/38.jpg)
Denavit-Hartenberg
Calculating the scalar products and, both equal to 0, proves that the vector ai is perpendicular to both axes of jointi and jointi+1
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Denavit-Hartenberg
•Three methods
•How do we actually compute ai and ||ai||2 ?
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Denavit-Hartenberg
The link vector ai is perpendicular to both of the axes of jointi and jointi+1. The unit vector :
is parallel to the link vector ai.
![Page 41: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/41.jpg)
Denavit-HartenbergGiven two points pi and pi+1 on the axes
of jointi and jointi+1, the link length can be computed as :
And the link vector :
![Page 42: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/42.jpg)
Denavit-HartenbergSpecial cases :
•The joint axes intersect
•The shortest distance ai is equal to zero
•The link vector is the null vector
![Page 43: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/43.jpg)
Denavit-Hartenberg•The joint axes are parallel
•There is no unique shortest distance
oi can be chosen arbitrarily, so we should chose values that offset the most of Denavit-Hartenberg parameters
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Denavit-Hartenberg
•The first joint
•There is no link preceding it
•We use a base link : link0
•Its link frame should coincide with the link frame of link1
•Most of the Denavit-Hartenberg parameters will be equal to zero
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Denavit-Hartenberg
•The last joint
•There is no link succeding it
•We use arbitrary values so that most of Denavit-Hartenberg parameters are equal to zero
![Page 46: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/46.jpg)
Denavit-Hartenberg
1.Compute the link vector ai and the link lenght
2.Attach coordinate frames to the joint axes
3.Compute the link twist αi
4.Compute the link offset di
5.Compute the joint angle φi
6.Compute the transformation (i-1)Ti which transforms entities from linki to linki-1
![Page 47: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/47.jpg)
Denavit-Hartenberg
•Identify the joint axes
•Identify the common perpendiculars of successive joint axes
•Attach coordinate frames to each joint axes
![Page 48: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/48.jpg)
Denavit-Hartenberg
Identifying the joint axes
![Page 49: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/49.jpg)
Denavit-Hartenberg
Remember, is the point where the shortest distance to jointi+1 exists
Identifying the common perpendiculars
![Page 50: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/50.jpg)
Denavit-Hartenberg
the origin
Attaching the frames
![Page 51: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/51.jpg)
Denavit-Hartenberg
1.Compute the link vector ai and the link lenght
2.Attach coordinate frames to the joint axes
3.Compute the link twist αi
4.Compute the link offset di
5.Compute the joint angle φi
6.Compute the transformation (i-1)Ti which transforms entities from linki to linki-1
![Page 52: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/52.jpg)
Denavit-Hartenberg
Remember
![Page 53: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/53.jpg)
Denavit-Hartenberg
Let’s define the function
![Page 54: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/54.jpg)
Denavit-Hartenberg
We then have :
![Page 55: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/55.jpg)
Denavit-Hartenberg
1.Compute the link vector ai and the link lenght
2.Attach coordinate frames to the joint axes
3.Compute the link twist αi
4.Compute the link offset di
5.Compute the joint angle φi
6.Compute the transformation (i-1)Ti which transforms entities from linki to linki-1
![Page 56: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/56.jpg)
Denavit-Hartenberg
![Page 57: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/57.jpg)
Denavit-Hartenberg
1.Compute the link vector ai and the link lenght
2.Attach coordinate frames to the joint axes
3.Compute the link twist αi
4.Compute the link offset di
5.Compute the joint angle φi
6.Compute the transformation (i-1)Ti which transforms entities from linki to linki-1
![Page 58: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/58.jpg)
Denavit-Hartenberg
![Page 59: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/59.jpg)
Denavit-Hartenberg
1.Compute the link vector ai and the link lenght
2.Attach coordinate frames to the joint axes
3.Compute the link twist αi
4.Compute the link offset di
5.Compute the joint angle φi
6.Compute the transformation (i-1)Ti which transforms entities from linki to linki-1
![Page 60: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/60.jpg)
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
iiii xxzzi
i RotTransTransRotT 1
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Denavit-Hartenberg
1000
0cossin0
0sincos0
0001
ii
iixi
Rot
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Denavit-Hartenberg
1000
0100
0010
001 i
x
a
Transi
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Denavit-Hartenberg
1000
100
0010
0001
iz d
Transi
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Denavit-Hartenberg
1000
0100
00cossin
00sincos
ii
ii
ziRot
![Page 65: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/65.jpg)
Denavit-Hartenberg
Multiplying the matrices :
iiii xxzzi
i RotTransTransRotT 1
1000
cossin0
sinsincoscoscossin
cossinsincossincos
1
iii
iiiiiii
iiiiiii
ii
d
a
a
T
![Page 66: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/66.jpg)
Denavit-HartenbergBy specifying the transformation
for all joints we can specify a transformation form linkN to the base link0 :
Let’s denote the joint parameters of jointi.
We obtain :
![Page 67: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/67.jpg)
End effector
•End effector - the last coordinate system of figure
•Located in joint N.
•But usually, we want to specify it in base coordinates.
![Page 68: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/68.jpg)
End effectorA transformation from the link N to the
base :
![Page 69: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/69.jpg)
End effector•We can also express it as
•three rotations (around each of the coordinate axes)
• followed by a translation
•How can we establish a relation with the other expression ?
![Page 70: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/70.jpg)
End effector•Location
•The origin of a coordinate frame relative to some base coordinate frame is specified by the translation :
![Page 71: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/71.jpg)
End effector•Orientation
•Any 3D orientation relative to some base coordinate frame can be specified by :
three rotations, one around each of the coordinate axes.
![Page 72: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/72.jpg)
End effector•Orientation
•These rotations are named roll, pitch, yaw
•We do them in this order : around x, y, z.
![Page 73: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/73.jpg)
End effector•Orientation
•The yaw transformation can be expressed :
![Page 74: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/74.jpg)
End effector•Orientation
•The pitch transformation can be expressed :
![Page 75: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/75.jpg)
End effector•Orientation
•The roll transformation can be expressed :
![Page 76: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/76.jpg)
End effector•Orientation
•The roll, pitch and yaw transformation is then expressed :
![Page 77: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/77.jpg)
End effector•Finally, the transformation from a
coordinate frame to the base frame is expressed :
![Page 78: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/78.jpg)
End effectorRemember that can be expressed as a 4x4 matrix with elements mij - functions of joint parameters θ1,...,θN.
Let’s omit them for readability :
![Page 79: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/79.jpg)
End effectorAs the two transformations and do the same thing :
![Page 80: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/80.jpg)
End effector
We obtain directly the translation vector :
![Page 81: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/81.jpg)
End effector
We can obtain the yaw angle :
Because :
arctan is π-periodic. Let’s use our function arctan2 to get the right angle.
![Page 82: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/82.jpg)
End effector
Knowing the yaw angle, we can obtain the pitch angle :
Because :
Again, let’s use our function arctan2 :
![Page 83: Introduction to Robotics Lecture II](https://reader035.vdocuments.site/reader035/viewer/2022062304/568135a6550346895d9d1607/html5/thumbnails/83.jpg)
End effector
We can obtain the roll angle :
Because :
Again, let’s use our function arctan2 :
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End effectorLet’s define the state vector
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End effectorAs previously shown,
The state vector is composed of elements of this matrix. It’s also a function of joint parameters :