Question 1

Why do two solutions (elbow-up and elbow-down) exist for a 2R robot arm?

Accepted Answer

For a 2R arm with target (x,y), the cosine of θ₂ is uniquely determined as c₂=(x²+y²-L₁²-L₂²)/(2L₁L₂), but the sine s₂=±√(1-c₂²) has two sign choices. The positive sign gives the elbow-down configuration, and the negative sign gives the elbow-up configuration — two distinct arm postures that reach the same target. The choice depends on joint limits and obstacle avoidance requirements.

Question 2

What is a singularity in a robot arm?

Accepted Answer

A singularity occurs when the Jacobian matrix determinant approaches zero, meaning a small displacement of the end-effector in some direction requires infinite joint velocities. For a 2R arm, singularities appear at full extension (θ₂=0 or π). Near singularities, numerical control becomes unstable. Damped least squares (DLS) methods add a damping term λ² to stabilize the pseudo-inverse near singularities.

Question 3

What engineering applications use inverse kinematics?

Accepted Answer

Key applications include: (1) Industrial robot path planning for welding, painting, and assembly operations. (2) Surgical robot precise positioning. (3) Character animation and pose generation in CG. (4) Space robot (e.g., JEMRMS) teleoperation. (5) Rehabilitation device joint angle calculation. In CAE, IK is also used to set initial configurations for multi-body structural and contact analyses.

Question 4

How does the iterative Jacobian pseudo-inverse IK method work?

Accepted Answer

The algorithm computes the error vector Δx between the target and current end-effector position, then uses the Jacobian pseudo-inverse J⁺ to compute the joint angle update Δθ=J⁺Δx. Repeating this minimizes the error iteratively. This tool runs 10 iterations updating the Jacobian each step. A learning rate α=0.5 balances convergence speed and stability. Near singularities, damped least squares (DLS) with damping coefficient λ² is recommended.

Robot Arm Inverse Kinematics Calculator (2/3-DOF)

Analytical IK Equations (2R)