Robot Kinematics: Forward and Inverse

• Robot kinematics is presented.

Kinematics in Robotics

• Kinematics studies robot motion without considering forces or torques.
• It focuses on:
  • Position: the location of robot parts
  • Velocity: how fast the robot moves
  • Acceleration: the rate of change of velocity

Types of Kinematics

• Forward Kinematics (FK) calculates the end-effector's position and orientation (pose) using joint values (angles or linear positions).
  • Input: Joint angles
  • Output: End-effector position (unknown pose)
  • Knowing a robotic arm's angles allows determining its hand position
• Inverse Kinematics (IK) determines required joint values to move the end-effector to a specific position.
  • Input: Desired end-effector position
  • Output: Joint angles
  • It calculates the angles needed for each joint to reach a certain point

Applications of Kinematics

• Industrial robots: Used for precise positioning in factories
• Autonomous robots: Helps navigation and efficient movement
• Humanoid robots: Enables smooth mimicking of human movements
• Path planning: Ensures robots follow safe, optimized paths

Robot Dynamics

• Robot Dynamics is the study of forces and torques affecting a robot during motion.
• Centers on how external forces like gravity, friction, and torques influence movement
• Ensures robots move precisely, safely, and efficiently
• It uses math equations, like differential equations and Lagrangian mechanics, that describe the robots movement
• Newton's laws of motion (for simpler robots) or Lagrange's equations (for complex systems) are key principles

Kinematics vs. Dynamics

• Kinematics focuses on motion (position, velocity, acceleration), while dynamics focuses on motion + forces & torques.
• Kinematics concerned with "how" a robot moves, dynamics with "why"
• Example: Kinematics calculates hand position, dynamics calculates forces for lifting an object

Example: 2D Robotic Arm Kinematics

• Link 1 Length (L₁) = 5 cm
• Link 2 Length (L₂) = 3 cm
• Joint Angle 1 (θ₁) = 30° (measured from the horizontal axis)
• Joint Angle 2 (θ₂) = 45° (relative to Link 1)
• Using forward kinematics (FK):
  • Convert angles to radians: θ₁ = 30° = 0.524 rad, θ₂ = 45° = 0.785 rad
  • Apply the forward kinematics equations to find the end-effector position (X, Y)
    • X = L₁ cos(θ₁) + L₂ cos(θ₁ + θ₂)
    • Y = L₁ sin(θ₁) + L₂ sin(θ₁ + θ₂)
  • Compute values:
    • X ≈ 5.84 cm
    • Y ≈ 4.1 cm
  • The end-effector (robot hand) reaches the point (5.84, 4.1) in the 2D space

Conclusion: Forward Kinematics helps determine the final position of a robot's hand with joint angles

• Useful in robotic motion planning to predict the end-effector's exact position
• Inverse Kinematics (IK) finds joint angles for a given position

Kinematics for a Wheeled Robot

• A differential drive robot: two wheels that rotate independently
• Movement based on the speeds of the left and right wheels
• Goal: Determine wheel position (X, Y) and orientation (θ) over time.

Example: Wheeled Robot

• Wheel Radius (R) = 0.1 m
• Distance Between Wheels (D) = 0.5 m
• Left Wheel Speed (Vₗ) = 0.3 m/s
• Right Wheel Speed (Vᵣ) = 0.5 m/s
• Initial Position: (X₀, Y₀) = (0, 0)
• Initial Orientation: θ₀ = 0° (facing forward on X-axis)
• Time Step: Δt = 2 seconds

Step-by-step Solution Using Kinematics Equations

• Compute the Linear and Angular Velocity.
  • V = (Vᵣ + Vₗ) / 2 = 0.4 m/s
  • ω = (Vᵣ - Vₗ) / D = 0.4 rad/s
• Compute the New Orientation After Δt.
  • θ = θ₀ + ω * Δt = 0 + (0.4 * 2) = 0.8 rad ≈ 45.8°
• Compute the New Position (X, Y).
  • Final Result After 2 Seconds
    • New Position: (X, Y) ≈ (0.72, 0.30) meters
  • New Orientation: θ ≈ 45.8° (0.8 rad)

Key Insights

• The robot moves in a curved path because the right wheel moves faster than the left wheel
• Final position is (X ≈ 0.72 m, Y ≈ 0.30 m), and the orientation is θ ≈ 45.8°.
• This method is used in autonomous robots, self-driving cars, and mobile robots for motion

Using the Rotation Matrix to Compute a Robot’s New Position:

• When a robot rotates in a 2D plane,the rotation matrix can compute its new coordinates

Example:

• Initial robot position: (X₀, Y₀) = (2, 3)
• Rotation angle: θ = 45°
• The Rotation Matrix (2D Plane Rotation):
• R(θ): cos(θ) - sin(θ)
• Equation to find the new position is: [X'] [cos(θ) - sin(θ)] [X0] [Y'] =[sin(θ) cos(θ)] [Y0]

Solution Steps

• Convert the Angle to Radians: θ = 45° = 0.785 rad
• Compute the Trigonometric Values: sin(0.785) ≈ 0.707, cos(0.785) ≈ 0.707
• Apply the Rotation Matrix:
• New Position after rotating 45°: (X', Y') ≈ (-0.71, 3.54)

Rotation Matrices in 3D

• Rotation Around the X-Axis (Pitch)
• This rotation affects the Y and Z coordinates, while the X coordinate remains unchanged.
• Rotation Around the Y-Axis (Yaw)
• This rotation affects the X and Z coordinates, while the Y coordinate remains unchanged.
• Rotation Around the Z-Axis (Roll)
• This rotation affects the X and Y coordinates, while the Z coordinate remains unchanged.

Applying a 3D Rotation Example

• A robot's arm is at an initial position (X₀, Y₀, Z₀) = (1, 2, 3). Rotate it by 30° around the X- axis, 45° around the Y-axis, and 60° around the Z-axis.
• Convert angles from degrees to radians: for example, θ₁ = 30° = 0.523 rad ####Steps to do this include
• Compute the rotation matrices Ra, Ry, and Rz using trigonometric values
• Compute the final position using the combined transformation
• Rotation Matrices in 3D are widely used in robotics, 3D graphics, drone navigation, and computer vision.