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PR 502 Robot Dynamics & Control 11/9/2006

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PR 502: Robot Dynamics & Control

Robot Kinematics:Articulated Robots

Asanga RatnaweeraDepartment of Mechanical Engieering

9 November 2006 Asanga Ratnaweera, Department of Mechanical Engineering

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Articulated Robots


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Denavit-Hartenberg (DH) Representation

Developed by Denavit and Hartenberg in 1955 for kinematic modeling of lower pairsHas become a standard way of representing robots and modeling their motions.However, direct modeling techniques learned before are faster and straight forwardQuite useful for Articulated robot modeling


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Links, Joints and Their ParametersMechanical manipulator consists of sequence of rigid bodies (links) connected either revolute or prismatic joint.Each joint-link pair constitutes one degree of freedom (dof).Hence, for n dof system has n number of links.Usually the first link (link 0) is attached to a supporting base and last link is attached with the tool.


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DH Coordinate frames

Establishing base coordinate system:A right handed Cartesian coordinate system XYZ or xo, yo, zo is assigned to the base of the manipulator with the zo axis lying along the axis of motion for the 1st link (joint 1) and pointing towards the shoulder of the robot.


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DH Coordinate framesEstablishing the joint axis:

All joints without exception are represented by a Z axis.If the joint is revolute the Z axis is the axis of rotation.If the joint is prismatic, Z axis is along the direction of the linear motion.


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DH Coordinate framesEstablishing the joint axis:

Defining X axisAssign the X axis along the common normal between two Z axes.If two z axes are parallel then assign X axis along the common normal to the previous jointIf two Z axes are intersecting each other, assign the x axis along a line perpendicular to the plane formed by the two Z axes


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Common normal between two z axes of joint 1 and 2


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Z1X1

Y1

X2Z2

Y2

Z3

X3

Y3

1 2 3


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Z(i - 1)

X(i -1)

Y(i -1)

α( i - 1)

a(i - 1 )

Z i Y i

X i a i d i

i

Z(i - 1)

X(i -1)

Y(i -1)

α( i - 1)

a(i - 1 )

Z i Y i

X i a i d i

i

Denavit-HartenbergParametersConsider any link i


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DH ParameterFour parameters (a,α,d,θ) are associated with each link of a manipulator.

a : the perpendicular distance between the adjacent joint axes (Z axes).

ex: ai-1 is the perpendicular distance between Z(i) and Z(i-1) axes.


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DH Parameters

ai-1

αi-1

αi-1


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DH Parameterα : Amount of rotation around the

common perpendicular so that the joint axes are parallel.

Ex: αi -1 is how much you have to rotate Z(i-1)about X(i-1) axis so that the Z(i-1) is pointing in the same direction as the Z(i) axis.Positive rotation follows the right hand rule.


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ai-1

αi-1

αi-1


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DH Parameterdi : The displacement along the Zi-1 axis needed to align the ai-1 common perpendicular to the ai common perpendicular. In other words, displacement along the Zi to align the Xi-1and Xi axes.Note: a, α are called link parameters and d, θ are called joint parameters


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Denavit-HartenbergParameters

Z(i - 1)

X(i -1)

Y(i -1)

α( i - 1)

a(i - 1 )

Z i Y i

X i a i d i

i

Z(i - 1)

X(i -1)

Y(i -1)

α( i - 1)

a(i - 1 )

Z i Y i

X i a i d i

i


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Denavit-HartenbergTransformation


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Denavit-HartenbergTransformation


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The Denavit-HartenbergMatrix

Just like the Homogeneous Matrix, the Denavit-Hartenberg Matrix is a transformation matrix from one coordinate frame to the next. Using a series of D-H Matrix multiplications and the D-H Parameter table, the final result is a transformation matrix from some frame to your initial frame.

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡−−

−

−−−−

−−−−

−

1000cosαcosαsinαcosθsinαsinθsinαsinαcosαcosθcosαsinθ

0sinθcosθ

i1)(i1)(i1)(ii1)(ii

i1)(i1)(i1)(ii1)(ii

1)(iii

dd

a


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Example: Robot with three revolute joints

Tool


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Z0

X0

Y0

Z1

X2

Y1

Z2

X1

Y2


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Z0

X0

Y0

Z1

X2

Y1

Z2

X1

Y2

d2

a0 a1


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Z0

X0

Y0

Z1

X2

Y1

Z2

X1

Y2

d2

a0 a1

i α(i-1) a(i-1) di θi

0 0 0 0 θ0

1 0 a0 0 θ1

2 -90 a1 d2 θ2

Denavit-Hartenberg Link Parameter Table


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Ex: robot with 3 Revolute Joints Z2Y2

Z0

X0

Y0

Z1

X2

Y1

X1

d2

a0 a1

Notice that the table has two uses:

1) To describe the robot with its variables and parameters.

2) To describe some state of the robot by having a numerical values for the variables.

Denavit-Hartenberg Link Parameter Table


0 0 0 0 θ0

1 0 a0 0 θ1

2 -90 a1 d2 θ2


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Z0

X0

Y0

Z1

X2

Y1

Z2

X1

Y2

d2

a0 a1


0 0 0 0 θ0

1 0 a0 0 θ1

2 -90 a1 d2 θ2

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

1VVV

TV2

2

2

000

Z

Y

X

ZYX T)T)(T)((T 12

010=

Note: T is the D-H matrix with (i-1) = 0 and i = 1.


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⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡ −

=

1000010000cosθsinθ00sinθcosθ

T 00

00

0


0 0 0 0 θ0

1 0 a0 0 θ1

2 -90 a1 d2 θ2

This is just a rotation around the Z0 axis

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−

−

=

100000cosθsinθ

d100a0sinθcosθ

T22

2

122

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This is a translation by a0 followed by a rotation around the Z1 axis

This is a translation by a1 and then d2followed by a rotation around the X2 andZ2 axis

T)T)(T)((T 12

010=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡ −

=

1000000000cosθsinθa0sinθcosθ

T 11

011

01


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More examples

Y0ˆ

Y1ˆ

Y3ˆ

Y2ˆ

X0ˆ

X1ˆ

X2ˆ

X3ˆ

θ 30L203

θ 2 0L102

θ 10001

θ i d ia i - 1α i - 1i


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Ex: PUMA 650 Robot

θ 6 00-90°6

θ 50090°5

θ 4d4a3-90°4

θ 3d3a203

θ 2 00-90°2

θ 10001



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Ex: PUMA Robot

θ 6 00-90°6

θ 50090°5

θ 4d4a3-90°4

θ 3d3a203

θ 2 00-90°2

θ 10001


TTTTTTTT 65

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100=

Px = c1 [ a2c2 + a3c23 – d4s23] – d3s1

Py = s1 [ a2c2 + a3c23 – d4s23] + d3c1

Pz = – a3s23 – a2s2 – d4c23

Where,C – cos, S - sin

Ci=cosθi ; Cij = cos(θi + θj)


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Px = c1 [ l2c2 + l3c23]Py = s1 [ l2c2 + l3c23 ]Pz = – l2s2 – l3s23

θ 50090°5

θ 40L304

θ 30L203

θ 2 00-90°2

θ 10001



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Ex: Stanford Robot


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Ex: Stanford Robot


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Example


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Ex: Stanford Robot

X1 Y1

Z1

X2

Z2

X3

Z3

X4

X5

X6

Z4

Z5

Z6

X7

Z7


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DH Notation

Example - Stanford Arm (cont'd)DH Parameters of Stanford Arm

θ60b606

θ50°b505

θ490°004

90°90°b3 03

θ290°b202

θ190°b101

θiαibiaii

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