Overview

Robotic arm real-time centralized planner, with predicted future (fixed horizon) collision avoidance.

Tesseract Setup Execution

Sawyer 20 Steps	ABB 20 Steps

2 Robot Predictive Move	2 Robot Predictive Move

Software Architecture

Algorithm

For any linear system $x_{k+1}=Ax_k+Bu_k$, the future state at $k+N$ could be written as

$$x_{k+N}=A^Nx_k+\begin{bmatrix} A^{N-1}B & A^{N-2}B & ... & B \end{bmatrix} \begin{bmatrix}u_k \\ u_{k+1} \\ ... \\ u_{k+N-1}\end{bmatrix} .$$

N-step lookahead QP

To track a goal at future N-step with a quadratic cost: $|x_d-x_{k+N}|$, it could be solved with quadratic programming

$$min_{\mathbf{u}_{all}} ||x_d-A^Nx_k-J_N\mathbf{u}_{all}||.$$

If the desired goal is moving or the goal is to track a trajectory, similarly

$$\begin{bmatrix}x_{k+1} \\ x_{k+2} \\ ... \\ x_{k+N}\end{bmatrix}=\begin{bmatrix}A \\ A^2 \\ ... \\ A^N\end{bmatrix}x_k+\begin{bmatrix} B & 0 & ... & 0 \\ AB & B & ... & 0 \\ A^{N-1}B & A^{N-2}B & ... & B \end{bmatrix} \begin{bmatrix}u_k \\ u_{k+1} \\ ... \\ u_{k+N-1}\end{bmatrix},$$ $$min_{\mathbf{u}_{all}} ||\mathbf{x_{all}}-A_{stack}x_k-J_{all}\mathbf{u}_{all}||.$$

N-step lookahead Trajectory Optimization

Rather than calculating the new trajectory every loop, it's possible to optimize the trajectory by $\delta\mathbf{x_{all}}$ by adjusting $\delta\mathbf{u_{all}}$

$$min_{\delta\mathbf{u}_{all}} ||x_{k+N}-x_{pred}-J_N\delta\mathbf{u}_{all}||,$$

or

$$min_{\delta\mathbf{u}_{all}} ||\mathbf{x_{all}}-\mathbf{x_{pred}}-J_{all}\delta\mathbf{u}_{all}||.$$

This is more helpful for collision avoidance than standard multi-step QP because collision constraint is a quadratic constraint, this way it's possible to steer the trajectory to the opposite collision direction. It doesn't matter if it's a first-order system, where the control input is velocity that is already $\delta x$.

Collision Constraint

Usually, a control barrier function is used to drive the trajectory away from the collision. Tesseract provides collision distance and vector between two objects, so at $i_{th}$ step this constraint could be written as

$$J_i\delta \mathbf{u_{all}} \vec{d} < h(d),$$

where $\vec{d}$ is the normalized collision vector from a point on the robot to a point in the environment (sign flipped if penetrating), and $h(d)$ is the barrier function depending on the collision distance (negative if penetrating).

Instruction

System

• Ubuntu (20.04) with ROS noetic (wsl2 on win11 OK)

• Python3.8

• Simulation robot service (https://drive.google.com/file/d/1lNFGjh11DI32MYN8jgVuNADaLYteue4R/view?usp=sharing & dotnet3 runtime)

• Swig 4.0.2 or later

• Robot Raconteur C# Native (sudo apt-get install librobotraconteur-net-native)

Python Packages

• catkin_tools: sudo apt-get install python3-catkin-tools
• QP: pip install qpsolvers
• General Robotice Toolbox: pip install general-robotics-toolbox
• Tesseract: pip install tesseract-robotics tesseract-robotics-viewer (may need to upgrade pip first)

ROS Packages (to be built in catkin_ws):

• robotraconteur (noetic branch): https://github.com/robotraconteur/robotraconteur
• RobotRaconteur_Gazebo_Server_Plugin: https://github.com/johnwason/RobotRaconteur_Gazebo_Server_Plugin
• robotraconteur_companion: https://github.com/robotraconteur/robotraconteur_companion

workspace build command (using catkin tools):

rosdep install --from-paths . --ignore-src --rosdistro noetic -y
catkin config --cmake-args -DROBOTRACONTEUR_ROS=1
catkin build

Path & Source

source /opt/ros/noetic/setup.bash
source ~/catkin_ws/devel/setup.bash

export GAZEBO_MODEL_PATH=~/Predictive_planner/models
export GAZEBO_PLUGIN_PATH=~/catkin_ws/devel/lib

Running instructions

• ./start_all to start the simulation environment
• python planner.py to bring up planner
• python client_sawyer.py or python client_abb.py