Robot Kinematics

Overview

Robot kinematics deals with the mathematical description of robot motion without considering the forces that cause it. It establishes the relationship between joint variables and the position/orientation of the robot end-effector.

Forward Kinematics

Forward kinematics computes the position and orientation of the end-effector given the joint angles.

Problem Statement

Given: Joint angles \(\theta_1, \theta_2, ..., \theta_n\)

Find: End-effector position \([x, y, z]\) and orientation \([roll, pitch, yaw]\)

Solution Methods

1. DH Parameters - Systematic approach using Denavit-Hartenberg convention
2. Product of Exponentials (PoE) - Screw theory based formulation

Inverse Kinematics

Inverse kinematics computes the joint angles required to achieve a desired end-effector pose.

Challenges

• Multiple solutions may exist
• Singularities where solutions become undefined
• Workspace boundaries

Solution Methods

1. Analytical Solutions - Closed-form equations (when available)
2. Numerical Solutions - Iterative methods like Newton-Raphson
3. Geometric Solutions - Trigonometric approaches

Differential Kinematics

Jacobian Matrix

The Jacobian matrix \(J(q)\) relates joint velocities to end-effector velocities:

\[\dot{x} = J(q) \dot{q}\]

The Jacobian is essential for: - Inverse velocity computation - Singularity detection - Force resolution

Examples

2-DOF Planar Robot

For a 2-link planar robot:

\[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)\]

Practice Problems

1. Derive forward kinematics for a 3-DOF robot arm
2. Implement inverse kinematics for a SCARA robot
3. Compute Jacobian for a 6-DOF industrial manipulator

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