robot kinematics: articulated...

PR 502 Robot Dynamics & Control 2/28/2007

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PR 502: Robot Dynamics & Control

Robot Kinematics:Articulated Robots

Asanga RatnaweeraDepartment of Mechanical Engieering

28 February 2007 Asanga Ratnaweera, Department of Mechanical Engineering

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Articulated Robots


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Articulated RobotsHome position

When all the joints are in zero position, the manipulator is said to be in home position.


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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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DH Coordinate framesEx: Assign coordinate frames


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Z1X1

Y1

X2Z2

Y2

Z3

X3

Y3

1 2 3


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Denavit-Hartenberg Parameters


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Coordinate frames for DH ParameterFour parameters (a,α,d,θ) are associated with each link of a manipulator.Zi axis is aligned with axis i, its direction being arbitrary. The choice of direction defines the positive sense of joint variable θi.The Xi axis is perpendicular to axis Zi-1 and axis Ziand point away from axis Zi-1. Therefore, Xi is along the common normal CD.The origin of the ith coordinate frame, is located at the intersection of axis of joint (i+1), that is axis i and the common normal between axes (i-1) and i.Yi completes the right and orthogonal coordinate frame i.


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DH Parameterα : Amount of rotation around the common

perpendicular (X axis) so that the joint axes are parallel.

Ex: αi is how much you have to rotate Z(i-1)about Xi 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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DH Coordinate framesParameter α


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DH Parametera : the perpendicular distance between the adjacent joint axes (Z axes). i.e. distance between Z(i-1) and Z(i) along Xi axis

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


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DH ParametersParameter a


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DH Parameters: α and aIf Z – axes (extended if required) are

Collinear lines: α = 0 and a = 0 Parallel lines: α = 0 and a ≠ 0 Intersecting lines : α ≠ 0 and a = 0 Skew lines: α ≠ 0 and a ≠ 0


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DH Parametersdi : The displacement along the Zi-1 axis needed to intersect the ai common perpendicular and Zi axis at the origin of the i-1th frame.

In other words, displacement along the Zi-1 to intersect the Xi and Zi-1 at its origin.


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DH Parametersθi : rotation of X(i-1) about Zi-1 axis needed to align the ai common perpendicular to the ai-1

common perpendicular. (Usually the joint rotation from the home position)In other words, rotation about Zi-1 to align the Xi-1 and Xi axes.Positive rotation follows the right hand rule.Note: a, α are called link parameters and d, θ are called joint parameters


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Denavit-Hartenberg Parameters

Xi is along the common normal of Zi and Z(i-1)hence ai measured along Xi

di measured along Zi-1

θi (rotation of X(i-1) ) about Zi-1

αi (rotation of Z(i-1) ) about Xi

All angle measurements should be based on the right hand rule (counterclockwise positive)Note: the frame i-1 is assigned to joint i and so forth


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Denavit-Hartenberg Transformation

Therefore, the transformation is in the reverse order:

)()()()(1ixixiziz

ii TaTdTTT αθ=−

Thus, the DH transformation matrix:

−

−

1000cossin0

sinsincoscoscossincossinsincossincos

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iiiiiii

iiiiiii

dααθaαθαθθθaαθαθθ


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Manipulator transformation matrix

The position and orientation of the tool frame relative to the base frame can be found by considering n consecutive link transformation matrices relating frames fixed to adjacent links.

nn

n TTTTT 12

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00 ............. −==

i-1Ti – for i=1,2,….,n is Homogeneous link transformation matrix between frame (i-1) and i


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Manipulator transformation matrix

The kinematic model provides the functional relationship between the tool frame (end effector) position and orientation and displacement of each link (qi). )q(fT i=

=

1000rrrrrrrrrrrr

1000paonpaonpaon

34332331

24232221

14131211

zzzz

yyyy

xxxx

For the known joint displacements qi (i=1,2,..,n) the end effector orientation (n,o,a) and position pcan be computed from the above equation.


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Example: 1

Home position of the robot


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Example: 1DH parameter table

DH transformation Matrices C1 – cosθ1, S1 - sinθ1


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Example: 1Therefore, overall transformation matrix

Comparing with the general form of the homogeneous matrix


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Example: 1Therefore, the coefficients

If θ1 = 120 and d2 =200


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Example: 2RPP Manipulator


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Example: 3


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Ex: PUMA Robot

robot kinematics: articulated...

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