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
PR 502 Robot Dynamics & Control 2/28/2007
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28 February 2007 Asanga Ratnaweera, Department of Mechanical Engineering
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Articulated RobotsHome position
When all the joints are in zero position, the manipulator is said to be in home position.
28 February 2007 Asanga Ratnaweera, Department of Mechanical Engineering
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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.
28 February 2007 Asanga Ratnaweera, Department of Mechanical Engineering
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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.
PR 502 Robot Dynamics & Control 2/28/2007
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28 February 2007 Asanga Ratnaweera, Department of Mechanical Engineering
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DH Coordinate frames
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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.
PR 502 Robot Dynamics & Control 2/28/2007
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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
28 February 2007 Asanga Ratnaweera, Department of Mechanical Engineering
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DH Coordinate frames
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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DH Coordinate frames
Z1X1
Y1
X2Z2
Y2
Z3
X3
Y3
1 2 3
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28 February 2007 Asanga Ratnaweera, Department of Mechanical Engineering
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DH Coordinate frames
28 February 2007 Asanga Ratnaweera, Department of Mechanical Engineering
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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.
PR 502 Robot Dynamics & Control 2/28/2007
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28 February 2007 Asanga Ratnaweera, Department of Mechanical Engineering
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DH Coordinate framesParameter α
28 February 2007 Asanga Ratnaweera, Department of Mechanical Engineering
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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.
PR 502 Robot Dynamics & Control 2/28/2007
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28 February 2007 Asanga Ratnaweera, Department of Mechanical Engineering
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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
iii
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
11
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