EECS C106B Safe Control

Project 4 Introduction

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Tasks

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2

3

4

Feedback Linearization

Deadlock CBF-QP

Safety-critical controller for multiple agents

Tracking controller with dynamic extension

Vision CBF-QP

Vision-based safety critical controller

Hardware Implementation

Deploy a braking CBF-QP on turtlebots

Feedback Linearization

• How can we transform a nonlinear system into a linear system using feedback control?
• Once we’ve made the system linear with feedback, how can we design tracking controllers?

Dynamic Extension

• For the turtlebot, we will have to use dynamic extension for feedback linearization
• Augment the state and input vector of the system to linearize
• Implementing in code: use a numerical integral

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Linear Tracking Control

• How can we design an effective linear tracking controller for the system?
• We’d like to track a desired trajectory qd(t) (only care about tracking x(t), y(t) - phi(t) may be anything
• We have access to the first and second time derivatives of qd(t)
• Hint: how can we choose an input z to get stable second order error dynamics, where e is the tracking error?

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Response Plots

Control Barrier Functions

• Imagine we have a system of many turtlebots
• May encode the safety of each turtlebot with respect to the others using a control barrier function h(q)
• We’ll consider a control barrier function between the turtlebot we wish to control (ego turtlebot) and an obstacle turtlebot
• What should h(q) be?

Deadlock CBF-QP

• Let’s use our control barrier function to enable a system of multiple turtlebots to safely track trajectories
• Challenge: if we apply a CBF-QP directly to a feedback linearizing input, turtlebots will get stuck in face-off scenarios called deadlocks
• How can we resolve these deadlocks?

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Deadlock CBF-QP

• We’ll apply a CBF-QP in the innermost linear layer of feedback linearization - then pass the safe z to the outer layers
• To encourage the turtlebot to steer around obstacles, we can weight the steering term in the input differently with a matrix Q
• Apply a barrier function constraint for each turtlebot, with derivatives taken along the trajectories of the linear system

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Deadlock CBF-QP Constraint

• Why do we need a second derivative in the constraint?
• We take the derivatives of the barrier function h(q) along the trajectories of the linear system

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• The input z will not appear until we take the second derivative of our barrier function along the trajectories of the linear system
• Challenge: how can we select the weights in the barrier constraint?
• Try solving the ODE for h(t) - for which k1, k2 is h > 0?

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Deadlock CBF-QP Summary

• Let’s summarize the structure of the CBF-QP:

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Deadlock CBF-QP Demo

Vision-Based CBF-QP

• The turtlebots interact with their environment using LIDAR sensors, which return a pointcloud in the turtlebot frame
• How can we come up with a single barrier function h(q) that encodes the safety of the system based on the pointcloud?
• Closest point, clustering, and many more methods!
• Fine if you get some deadlocks

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Braking CBF-QP

• We’ll implement a simplified CBF-QP on hardware
• We won’t incorporate the full relative degree 2 constraint for simplicity - we’ll just require that our bot brakes for obstacles

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• Now, we’ll apply this CBF-QP directly over the feedback linearizing input, k(q) to get the input usafe
• Now, the derivative of h should be taken along the nonlinear turtlebot dynamics, rather than the linear system

Braking CBF-QP

• The barrier function h(q) should be exactly the same as your vision-based barrier function from simulation
• Implement this controller on hardware
• Stand in the way of the turtlebot as it moves - the braking CBF should slow the turtlebot and prevent it from crashing
• If you move towards the turtlebot, it should evade you

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Braking CBF-QP Demo

Using the TurtleBots

• Remember to switch them OFF to charge
• If a TurtleBot is not sufficiently charged for the next group you may lose points
• Carry them by the base
• Watch where they’re going!
• Be ready to press Ctrl + C
• Refer to the Robot Usage Guide for setup