Homework Assignments

Table of contents

1. Homework 1: Model Selection
2. Homework 2: Kinematics Simulation
3. Homework 3: Nonlinear Control Methods
   1. Requirements
   2. For Sliding Mode Control
   3. For Adaptive Control
4. Homework 4: CBF Pedestrian Avoidance
   1. Requirements
5. Homework 5: Potential Fields
   1. Requirements

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Homework 1: Model Selection

Objective: Pick a robot/vehicle model that you will use throughout the course.

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Homework 2: Kinematics Simulation

Objective: Simulate the kinematics of your chosen model.

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Homework 3: Nonlinear Control Methods

Requirements

1. Plot phase portraits of your system’s error dynamics: $e$ (error) vs. $\dot{e}$ (error rate)

2. Choose ONE approach:
   • Robust Sliding Mode Controller OR
   • Adaptive Controller

   Derive the Feedback Linearizable Model for your specific vehicle model.

3. Simulate both the controller and your vehicle model. Test with uncertainties and disturbances.

   Include simulation of:

   \[V_r = \frac{z^T p_b}{|z^T p_b|} \rho(x)\]

For Sliding Mode Control

Ensure the sliding condition:

\[s \dot{s} < -\eta|s|, \quad \eta > 0\]

The control should guarantee reachability and sliding.

For Adaptive Control

Implement parameter update law:

\[\hat{p} = \delta x^T S I_y^{-1}\] \[\hat{P}(t) = \hat{p}(0) + \delta \int_0^t x^T S_b I_y^{-1} dt\]

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Homework 4: CBF Pedestrian Avoidance

Objective: Apply Control Barrier Functions to ensure safe robot navigation around a pedestrian.

Requirements

1. Scenario: Robot must reach goal while avoiding pedestrian obstacle

2. Define geometric constraint:

   \[h(x) = \|x_{\text{robot}} - x_{\text{pedestrian}}\| - d_{\text{safe}} \geq 0\]

3. Apply CBF approach: Use Control Barrier Function to guarantee safety

4. Write 5 equations:
   • System dynamics
   • Barrier function
   • CBF condition
   • Control bounds
   • Distance formula
5. Show solution: Demonstrate robot reaches goal while maintaining safe distance from pedestrian

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Homework 5: Potential Fields

Objective: Implement potential fields showing two scenarios.

Requirements

Demonstrate:

1. Success case: Robot successfully navigates from A to B
2. Failure case: Robot gets trapped in local minimum

Show both scenarios with visualizations.