Safety-Index-Synthesis-via-Sum-of-Squares-Programming

By Weiye Zhao, Tairan He, Tianhao Wei, Simin Liu and Changliu Liu (the code of ``Safety Index Synthesis via Sum of Squares Programming'', American Control Conference, 2023)

Introduction

Safety Index Synthesis Using Sum of Squares Programming is currently implemented under two experimental settings, including (i) varying degrees-of-freedom planar manipulator collision avoidance with limited acceleration, and (ii) unicycle collision avoidance with limited acceleration and angular velocity

Requirements: software

0. MATLAB 2016a or later.

Requirements: hardware

CPU, Windows 7 or later, MAC OS.

Demo

0. Run experiments/unicycle_SI_optimization.m to synthesis an optimal safety index for unicycle collision avoidance. the acceleration limits are $[-1,1]$, and the angular velocity limits are $[-0.1,0.1]$. velocity limits are $[0,1]$, and heading angle range is $[0, \pi/2]$.
1. Run experiments/arm_ndof_SI_optimization.m to synthesis an optimal safety index for n degrees-of-freedom planar manipulator collision avoidance. (The safety index is searched within a limited range, and feasible safety index will be collected and optimal safety index can then be chosen) The acceleration limits for all joints are $[-1,1]$. The state space limits are specified in the code.

Note

0. For Planar Manipulator Experiments, it is recommend to first try experiments/arm_1dof_SI_optimization.m, which is a simplest version where only 1 link planar robot is considered. The explicit SOSP derivative is elaborated in the paper section VII B, the only difference is that code in this repo considers a stricter condtion non-empty set of safe control for all possible states, meaning only $f1,f2,f3,f4$ are included in (18).
1. For Unicycle Experiment, it is recommend to read this paper, which analytically derived the safety index synthesis rule for unicycle. According to the (11) of Appendex C.1, when $\cos(\alpha) > 0$, safety index should satisfy the following condition:

$$\forall (\alpha,v), \exists (a,w), \text{ s.t. } -\frac{a}{v} + \tan(\alpha) w \geq \frac{n d^{n-1}}{k},$$

which indicates setting $n=1$, and take the maximizer of acceleration and angular velocity, the following condition should hold

$$\forall (\alpha,v), k + \tan(\alpha)vk - v \geq 0.$$

Therefore, we can then following the paper and construct the refute problem and associated SOSP as documented in experiments/unicycle_SI_optimization.m.