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Chapter 02

Basics of Animation and Forward Kinematics

Page 9

Forward Kinematics and Inverse Kinematics


Let [dx, dy] be the point [destination x, destination y] where we want to move the robot's gripper. It is assumed that the gripper is NOT already at [dx, dy]. In order to move the tip of the gripper [lx(3), ly(3)] to [dx, dy], the link(s) must be rotated from their current positions. It may be required to rotate both the links to certain angle to reach the point [dx, dy]. The counter-clockwise angle between the horizontal axis (X) and the Link-1 is termed as alpha (α). The counter-clockwise angle between the "link-1-extended" and link-2 is termed as beta (β).
alpha beta


If Link-1 is rotated α degrees and Link-2 is rotated β degrees the gripper tip [lx(3) , ly(3)] will reach [dx, dy]. Before attempting to calculate α and β it should be determined that [dx, dy] is in robot's reach. For example if the point [dx, dy] is too far from the robot's base the gripper can't reach it!

Inverse Kinematics
Often it is required to calculate α and β, that is: How much degrees should
Link-1 be rotated and How much degrees the Link-2 is to be rotated in order to reach (dx, dy) ?

α and β are unknown?

The following are known:
L1, L2 the link lengths are known,
(lx[1], ly[1] ) is the base of the robot, its always (0,0) and is known
(lx[2], ly[2] ) is the joint of the L1 and L2, it is known if the current arm position is known
(lx[2], ly[2] ) is the joint of the gripper tip, it is known if the current arm position is known
Since the initial arm position (when the robot was reset is known) or the home position (where robot arm rests, when it has no work) is known, one can keep track on current arm position by updating the arm position after every move.
The above is a problem of Inverse Kinematics (IK).
In short, determining the angles from co-ordinates is an inverse kinematics problem.

Forward Kinematics
Determining co-ordinates from angles is a Forward Kinematics problem.
Example of FK problem:
If lx, ly, L1 and L2 are known (that is: if the current arm position [lx(1), ly(1)], [lx(2), ly(2)], [lx(3), ly(3)], length of Link-1 and length of Link-2) are known then, what will be the new position of the gripper tip [lx(3), ly(3)], if Link-1 is rotated 10 degrees and Link-2 is rotated 73.08 degree?

For further reading, please refer Robot kinematics on Wikipedia

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