Forward Kinematics Cedric Fischer and Michael Mattmann Institute of - - PowerPoint PPT Presentation

1 Institute of Robotics and Intelligent Systems Department of Mechanical and Process Engineering (DMAVT) ETH Zurich

Forward Kinematics Cedric Fischer and Michael Mattmann

Forward/Inverse Kinematics

§ Kinematics: To describe the motion of the manipulator without consideration of the forces and torques causing the motion : A Geometric Description.

θ1, θ2, θ3 → R0

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R0

3 → θ1, θ2, θ3

Forward Kinematics To determine the position and

• rientation of the end effector with

the given values for the joint variables. Inverse Kinematics To determine the joint variables with the given the end effectors position and orientation.

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Rigid body motion

q : a point attached to the rigid body

gab(t) = Rab(t) pab(t) 1

• the rigid body motion of the frame B attached to the body,

y, relative to a fixed or inertial frame A

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Forward Kinematics with Screw Theory: POE

§ Forward Kinematics defines a transformation between the joint space and the task space § Joint Space: § Defined by the independent angles theta § Configuration of robot joints § Task Space: § Defined by position and orientation of end-effector § Cartesian space

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Forward Kinematics with Screw Theory: POE

§ General forward kinematics map § Written using the product of exponentials formula: § Product of exponentials uses only two frames!

• Base frame S and tool frame T

gst(θ1, θ2, ..., θn) = e

ˆ ξ1θ1e ˆ ξ2θ2...e ˆ ξnθngst(0)

gst(θ) = e

ˆ ξ1θ1e ˆ ξ2θ2...e ˆ ξnθngst(0)

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Forward Kinematics with Screw Theory: Example

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§ Start from the general formula: § Find and Screw parameter and calculate exponentials: § Compute forward kinematics:

gst(θ) = e

ˆ ξ1θ1e ˆ ξ2θ2...e ˆ ξnθngst(0)

The Denavit-Hartenberg Convention

Ai = Rotz,θiTransz,diTransx,aiRotx,αi =     cθi −sθi sθi cθi 1 1         1 1 1 di 1         1 ai 1 1 1         1 cαi −sαi sαi cαi 1     =     cθi −sθicαi sθisαi aicθi sθi cθicαi −cθisαi aisθi sαi cαi di 1     § In general, we would need 6 independent parameters to define the transformation between two neighboring coordinate frames § The D-H convention reduces the problem to 4 parameters by a clever choice

• f the origin and orientation for the coordinate frames

§ Cancellations occur!

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The Denavit-Hartenberg Convention

§ Assume two features! DH1: The axis Xi is perpendicular to Zi-1 DH2: The axis Xi intersects the axis Zi-1

θi : joint angle di : link offset ai : link length αi : link twist

• angle from xi-1 to xi measured in a plane normal to zi-1
• distance from oi-1 to intersection of xi and zi-1 measured along zi-1
• distance between zi-1 and zi measured along xi
• angle between zi-1 and zi measured in a plane normal to xi

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The Denavit-Hartenberg Convention (Example)

d, the robot is better conditioned. θ1 θ2

§ Forward Kinematics with Denavit-Hartenberg convention

DH1: The axis Xi is perpendicular to Zi-1 DH2: The axis Xi intersects the axis Zi-1 xb yb zb z1 z2 x1 y1 x2 y2 xt yt zt y3 x3 z3 Link a α d θ

1 90 5 90 2 5 θ1 3 3

• 90

θ2 t

• 90

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• Figure, chart, video…

Assignment 4

a) (by inspection) b) Screw parameters h, l, M c) d) e)

d, the robot is better conditioned. θ1 θ2

frame 0 frame 0

g0t(0) gb0(0) gbt(θ) = gb0g0t(θ) = gb0e

ˆ ξ1θ1e ˆ ξ2θ2g0t(0)

ˆ ξi ξi

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