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Jack H.Integration and automation of manufacturing systems.2001.pdf
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page 391

Link/Joint Coordinates - specify where joints, endpoints or centers are located.

Tool coordinates - determine where the tool is and what orientation it is in.

World Coordinates - relates various robots to other robots and devices.

Coordinate transformation - Can map from one set of coordinates to another. Most common method is matrix based. One special case of this is the Denavit-Hartenrberg transformation.

13.1.2 Kinematics

 

 

 

 

0.2m

 

 

 

1m

 

 

 

theta2

TCP

 

 

(xT, yT)

1m

(x1,y1)

 

 

 

 

 

Note: When defining angles is is more

 

 

 

 

 

 

 

 

 

 

convenient to indicate the positive direc-

 

theta1

 

tion and origin (as shown with the solid

 

 

 

 

(xb, yb)

 

line for ‘theta 2’). With more classical

 

 

 

sketching we would use the dashed line,

 

 

 

 

 

 

 

 

Forward kinematics involves finding the endpoint of the robot (xT, yT) given the joint coordinates (theta1, theta2)

There a number of simple methods for finding these transformations,

page 392

-basic geometry

-transformation matrices

-Denavit-Hartenberg transformations

13.1.2.1 - Geometry Methods for Forward Kinematics

For simple manipulators (especially planar ones) this method is often very fast and efficient.

The method uses basic trigonometry, and geometry relationships.

To find the location of the robot above, we can see by inspection,

page 393

xT = xb + l1 cos θ 1 + ( l2 + 0.2) cos 1 + θ 2) yT = yb + l1 sin θ 1 + ( l2 + 0.2) sin 1 + θ 2)

often set to zero

The general form of the operation is as below,

( θ 1, θ 2, … ) → ( xT, yT, zT, θ Tx, θ Ty, θ Tz)

ASIDE: later we will see that the opposite operation maps from tool coordinates, and is called the inverse kinematics.

( θ 1, θ 2, … ) ← ( xT, yT, zT, θ Tx, θ Ty, θ Tz)

Also note that the orientation of the tool is included, as well as position, therefore for the example,

θTx = 0

θTy = 0

θ Tz = θ 1 + θ 2

• The problem with geometrical methods are that they become difficult to manage when more

complex robots are considered. This problem is overcome with systematic methods.

13.1.2.2 - Geometry Methods for Inverse Kinematics

• To find the location of the robot above, we can see by inspection,