Lab 2: Linear Inverted Pendulum Model | Yixin Zhu

Initial Codebase for Gradescope Submission

Background

This assignment focuses on a fundamental concept in bipedal robotics: the Linear Inverted Pendulum Model (LIPM). You will implement and visualize a simplified 2D LIPM system that captures the essential dynamics of walking robots and is widely used in humanoid robotics research.

The Linear Inverted Pendulum Model (LIPM) simplifies bipedal walking dynamics by modeling the robot’s center of mass (CoM) as a point mass moving under gravity while constrained to a constant height. This abstraction enables real-time motion planning and control for walking robots by reducing the complex full-body dynamics to tractable equations.

In this assignment, you will implement key components of LIPM dynamics and foot placement strategies.

Learning Objectives

By completing this assignment, you will:

Understand the dynamics of the Linear Inverted Pendulum Model
Implement CoM trajectory computation for single and multiple steps
Learn capture point theory for stable foot placement

Environment Setup

Prerequisites

Python 3.8 or higher
Linux environment (recommended)

Installation

Unzip and navigate to the lab directory

Create and activate a conda virtual environment:

conda create -n core_lab2 python=3.8
conda activate core_lab2

Install Required Packages
```
pip3 install -r requirements.txt
```

Verify Installation:

python -c "import numpy; import matplotlib; print('Environment setup successful')"

Assignment Tasks

Task 1: Single Step LIPM Dynamics (50 points)

Objective: Implement the core dynamics of LIPM by computing the center of mass trajectory during a single step.

Description: You will complete the updateXtVt() function in LIPM.py. This function computes the current position and velocity of the center of mass based on the LIPM dynamics equations.

Testing Your Implementation

Run the following command to visualize a single step:

python test.py --task single_step

Expected Output

Smooth trajectory of the CoM during one step
Reference output video: ./expected_output/single_step.mp4

Task 2: Multi-Step Walking with Capture Point (50 points)

Objective: Implement orbital energy computation and capture point calculation for stable multi-step walking.

Description: Complete two functions that enable stable walking over multiple steps:

getOrbitalEnergy(): Computes the orbital energy of the LIPM system, a conserved quantity useful for stability analysis
updateCapturePoint(): Calculates the capture point, which determines the optimal foot placement location for maintaining balance

These functions work together to enable leg switching and continuous walking by determining where to place the next step iteratively.

Key Requirements

Correctly compute orbital energy based on current state
Implement capture point dynamics

Testing Your Implementation

Run the following command to visualize multi-step walking:

python test.py --task multi_step

Expected Output

Continuous walking motion over multiple steps
Reference output video: ./expected_output/multi_step.mp4

Implementation Guidelines

Code Structure

Only modify code within ## TODO sections in LIPM.py
Do not change function signatures or class structures

Debugging Tips

Verify energy conservation in your orbital energy calculation
Use breakpoint() to check intermediate results
Compare your output with the provided reference videos

Hand-in Requirements

Submission Format

Submit only one file: LIPM.py
Platform: Gradescope autograder

File Requirements

Your LIPM.py must contain all implemented functions
Do not modify function signatures or imports

Academic Integrity

This is an individual assignment. While you may discuss general concepts with classmates, all submitted code must be your own work. Plagiarism will result in course failure.

Allowed

Discussing theory and concepts
Sharing debugging strategies (without code)
Using course materials and recommended references

Not Allowed

Sharing or copying code solutions
Using external implementations without permission
Collaborating on actual implementation

Good luck with your assignment!