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41.1.2 Kinematics

	0.2m
1m
theta2	TCP
	(xT, yT)

1m (x1,y1)

theta1 (xb, yb)

Note: When defining angles is is more convenient to indicate the positive direction and origin (as shown with the solid line for ‘theta 2’). With more classical sketching we would use the dashed line, but this leads to confusion with variable robotic geometries.

•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,

-basic geometry

-transformation matrices

-Denavit-Hartenberg transformations

41.1.2.1 - 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,

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

41.1.2.2 - Geometry Methods for Inverse Kinematics

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