ยท Hankyu Kim ยท Robotics ยท 2 min read
Analytical Inverse Kinematics
This post demonstrates how to compute the inverse kinematics of a 2-link planar robot arm using the Law of Cosines, providing a step-by-step guide for analytical solutions.
Introduction
In robotics, the inverse kinematics (IK) problem involves determining the joint parameters needed to place the end-effector of a robot arm at a desired position and orientation. For a planar arm with two links, this problem can be solved analytically using the Law of Cosines, offering a clear method to compute joint angles.
Problem Setup
Consider a planar robot arm with two links. Let the first link have length L1 and the second link have length L2. Denote the base joint angle as theta1 and the elbow joint angle as theta2. The end-effector is located at a target position (x, y). The goal is to compute the values of theta1 and theta2 that position the end-effector at the target.
Step 1: Compute the Elbow Angle
Using the Law of Cosines, the elbow angle can be determined as follows:
Solving for theta2:
This provides the elbow angle required to reach the desired end-effector position.
Step 2: Compute the Shoulder Angle
Once theta2 is known, the shoulder angle can be calculated using the geometric relationship:
This equation accounts for the relative positions of the links and the end-effector.
Example Calculation
Suppose the first link is 5 units long and the second link is 3 units long. The target end-effector position is (6, 4).
Step 1: Compute theta2
Step 2: Compute theta1
Therefore, the joint angles that position the end-effector at (6, 4) are approximately:
- Shoulder: 14.22 degrees
- Elbow: 53.13 degrees
Conclusion
Analytical solutions using the Law of Cosines provide an efficient method for solving the inverse kinematics of a planar 2-link arm. This approach is useful when precise control of the end-effector is required. Understanding this technique is a foundational skill in robotics and is essential for designing and controlling robotic manipulators.
