Robot Arm Kinematics

• Understand forward kinematics: given joint angles, find end-effector position.
• Understand inverse kinematics: given target position, find joint angles.
• Calculate reach and workspace of a simple 2-link robot arm.
• See how trigonometry applies to robotics.

What is a Robot Arm?

A robot arm consists of rigid links connected by joints. Each joint can rotate by an angle (θ).

• End-effector: the tip of the arm (gripper, tool)
• Degrees of Freedom (DoF): number of independent joint angles
• Workspace: all positions the end-effector can reach

2-Link Planar Arm (2D)

Two links with lengths L₁ and L₂, joint angles θ₁ and θ₂.

Forward Kinematics (angles → position):

• x = L₁ cos(θ₁) + L₂ cos(θ₁ + θ₂)
• y = L₁ sin(θ₁) + L₂ sin(θ₁ + θ₂)

Example: L₁ = 3, L₂ = 2, θ₁ = 30°, θ₂ = 45°:

• x = 3 cos(30°) + 2 cos(75°) ≈ 2.598 + 0.518 ≈ 3.116
• y = 3 sin(30°) + 2 sin(75°) ≈ 1.5 + 1.932 ≈ 3.432

Inverse Kinematics (IK)

Given target (x, y), find θ₁ and θ₂:

1. Distance to target: d = √(x² + y²)
2. Elbow angle: cos(θ₂) = (d² − L₁² − L₂²) / (2 L₁ L₂)
3. Shoulder angle: θ₁ = atan2(y, x) − atan2(L₂ sin(θ₂), L₁ + L₂ cos(θ₂))

Reachability check: |L₁ − L₂| ≤ d ≤ L₁ + L₂

Types of Robot Joints

Applications of Robot Arms

Key Physics Connections

• Torque at joint = Force × distance (lever arm)
• Longer arm → greater reach but more torque needed at shoulder joint
• Gear ratio in motors trades speed for torque

• What is forward kinematics? What does it calculate?
• A 2-link arm has L₁ = 4, L₂ = 3. What is maximum reach? Minimum reach?
• Why can't a robot arm always reach a given target point?
• What is the difference between a revolute and a prismatic joint?
• Name two real-world uses of robot arms.

Open the Practice tab for graded questions on Robot Arm Kinematics.