Inverse Kinematics: Math vs AI Approaches

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Learning Objectives

By the end of this lecture, you should be able to:

1. Understand the inverse kinematics problem in robotics
2. Apply trigonometric methods to solve 2-link arm positioning
3. Appreciate how neural networks can learn motion patterns
4. Compare the advantages and limitations of both approaches
5. Recognize when to use mathematical vs AI solutions

The Inverse Kinematics Problem

In robotics, we often need to position the end of a robotic arm (end-effector) at a specific location. This requires determining the joint angles that will achieve the desired position.

Forward Kinematics: Joint angles → End-effector position (easy) Inverse Kinematics: End-effector position → Joint angles (challenging)

Mathematical Approach: Geometric Solution

The traditional approach uses trigonometry and geometric relationships to calculate exact joint angles.

Two-Link Arm Analysis

For a 2-link robotic arm with lengths L₁ and L₂:

r = √(x² + y²)

cos(θ₂) = (r² - L₁² - L₂²) / (2 × L₁ × L₂)
θ₂ = ±arccos(cos(θ₂))

α = atan2(y, x)
β = atan2(L₂ × sin(θ₂), L₁ + L₂ × cos(θ₂))
θ₁ = α - β

import numpy as np

def inverse_kinematics_2link(x, y, L1, L2):
    # Check if target is reachable
    r = np.sqrt(x*x + y*y)
    if r > L1 + L2 or r < abs(L1 - L2):
        return None, None  # Unreachable

    # Calculate elbow angle (two solutions)
    cos_theta2 = (r*r - L1*L1 - L2*L2) / (2*L1*L2)
    theta2_1 = np.arccos(cos_theta2)   # Elbow up
    theta2_2 = -np.arccos(cos_theta2)  # Elbow down

    # Calculate shoulder angles
    alpha = np.arctan2(y, x)
    beta1 = np.arctan2(L2*np.sin(theta2_1), L1 + L2*np.cos(theta2_1))
    beta2 = np.arctan2(L2*np.sin(theta2_2), L1 + L2*np.cos(theta2_2))

    theta1_1 = alpha - beta1
    theta1_2 = alpha - beta2

    return (theta1_1, theta2_1), (theta1_2, theta2_2)

Interactive Mathematical Demo

Try this: Manually set different target positions and observe how the mathematical solution instantly calculates the exact joint angles needed. Notice the two possible configurations (elbow up/down) for most positions.

AI Approach: Neural Network Learning

Instead of deriving equations, we can train a neural network to learn the relationship between positions and joint angles from data.

Neural Network Architecture

import tensorflow as tf

model = tf.keras.Sequential([
    tf.keras.layers.Dense(32, activation='relu', input_shape=(2,)),
    tf.keras.layers.Dense(32, activation='relu'),
    tf.keras.layers.Dense(16, activation='relu'),
    tf.keras.layers.Dense(2, activation='linear')  # θ₁, θ₂
])

def generate_training_data(n_samples=10000):
    theta1 = np.random.uniform(-np.pi, np.pi, n_samples)
    theta2 = np.random.uniform(-np.pi, np.pi, n_samples)

    # Forward kinematics
    x = L1 * np.cos(theta1) + L2 * np.cos(theta1 + theta2)
    y = L1 * np.sin(theta1) + L2 * np.sin(theta1 + theta2)

    positions = np.column_stack([x, y])
    angles = np.column_stack([theta1, theta2])

    return positions, angles

model.compile(
    optimizer='adam',
    loss='mse',
    metrics=['mae']
)

history = model.fit(
    train_positions, train_angles,
    validation_data=(val_positions, val_angles),
    epochs=100,
    batch_size=32,
    callbacks=[
        tf.keras.callbacks.EarlyStopping(patience=10),
        tf.keras.callbacks.ReduceLROnPlateau(patience=5)
    ]
)

Interactive AI Demo

Try this: Manually set target positions and see how the neural network predicts joint angles. Notice how it learns smooth motion patterns and handles the workspace boundaries. The prediction may not be as precise as the mathematical solution but shows learned behavior.

Comparing Both Approaches

Performance Comparison

Real-World Applications

When to Use Mathematical Approaches

• Precision manufacturing: CNC machines, 3D printers
• Simple robots: 2-3 DOF arms where exact solutions exist
• Real-time control: High-frequency control loops
• Safety-critical systems: Medical robots, automotive systems

When to Use AI Approaches

• Complex robots: 6+ DOF industrial arms
• Obstacle avoidance: Path planning with environmental constraints
• Human-robot interaction: Learning from demonstration
• Non-standard geometries: Soft robots, cable-driven systems

Practical Example: Pick and Place Robot

Assignment: Hybrid Approach

Key Takeaways

• Mathematical IK provides exact, fast solutions for simple geometries
• AI approaches excel with complex robots and learned behaviors
• Hybrid systems can combine the best of both approaches
• Problem complexity determines which method is most appropriate
• Real-world robotics often requires both approaches for different situations

The future of robotics lies in intelligently combining precise mathematical foundations with adaptive AI capabilities.

Next Steps

With these fundamentals, you can explore:

• Multi-DOF robots: 6+ joint industrial manipulators
• Redundancy resolution: Choosing optimal solutions when multiple exist
• Real-time planning: Motion planning with dynamic obstacles
• Learning from demonstration: Teaching robots through human examples