What is forward kinematics of a robot and how do you find the end-effector position of a 2-link planar arm?

In one line

Forward kinematics finds the end-effector's position and orientation from known joint variables; for a 2R planar arm it is simple trigonometry of the two links.

Explained simply

If you know how much you bent your shoulder and elbow, you can work out where your hand is. That is forward kinematics. The reverse, deciding how to bend your joints to reach a cup, is inverse kinematics, which is harder.

Technically, forward kinematics maps joint space (\( \theta_1, \theta_2, \ldots \)) to task space (x, y, z, orientation). For a serial robot it always has one unique answer.

How it works

1. Fix the base frame at joint 1, x horizontal.
2. \( \theta_1 \): angle of link 1 from the x-axis. \( \theta_2 \): angle of link 2 relative to link 1.
3. Elbow position: \( (L_1\cos\theta_1, L_1\sin\theta_1) \).
4. Link 2 is at absolute angle \( \theta_1 + \theta_2 \); add its projection.
5. Orientation of the end effector: \( \phi = \theta_1 + \theta_2 \).
6. Inverse kinematics would solve these equations for \( \theta_1, \theta_2 \); it can have two solutions (elbow up / elbow down) or none.

Key formula

\( x = L_1\cos\theta_1 + L_2\cos(\theta_1 + \theta_2) \), \( y = L_1\sin\theta_1 + L_2\sin(\theta_1 + \theta_2) \), \( \phi = \theta_1 + \theta_2 \). L in m, angles in degrees or radians.

Quick example

\( L_1 = 1 \) m, \( L_2 = 0.5 \) m, \( \theta_1 = 30^\circ \), \( \theta_2 = 60^\circ \). \( x = 0.866 + 0.5\cos90^\circ = 0.866 \) m; \( y = 0.5 + 0.5\sin90^\circ = 1.0 \) m; the gripper points straight up (\( \phi = 90^\circ \)).

Common mistakes

• Using \( \theta_2 \) as an absolute angle from the x-axis instead of relative to link 1.
• Swapping the definitions of forward and inverse kinematics.
• Calculator in radian mode while using degrees.

How MU asks it

Theory 3 marks: 'Differentiate between forward and inverse kinematics'. Numerical 5 to 6 marks: 'For the 2-DOF planar arm with given lengths and joint angles, find the end-effector position', often expected using homogeneous transformation or D-H, with this geometric result as a check.