---
readstatus: false
dateread:
title: "Robust Control Synthesis and Verification for Wire-Borne Underactuated Brachiating Robots Using Sum-of-Squares Optimization"
year: 2020
authors:

  
  - "Farzan, Siavash"
  
  - "Hu, Ai-Ping"
  
  - "Bick, Michael"
  
  - "Rogers, Jonathan"
  

citekey: "farzanRobustControlSynthesis2020"







pages: 7744-7751

---
# Indexing Information
## DOI
[10.1109/IROS45743.2020.9341348](https://doi.org/10.1109/IROS45743.2020.9341348)
## ISBN
[](https://www.isbnsearch.org/isbn/)
## Tags:
#Trajectory, #Uncertainty, #Actuators, #Cable-TV, #Feedback-control, #Optimization, #Parametric-statistics, #read

>[!Abstract]
>Control of wire-borne underactuated brachiating robots requires a robust feedback control design that can deal with dynamic uncertainties, actuator constraints and unmeasurable states. In this paper, we develop a robust feedback control for brachiating on flexible cables, building on previous work on optimal trajectory generation and time-varying LQR controller design. We propose a novel simplified model for approximation of the flexible cable dynamics, which enables inclusion of parametric model uncertainties in the system. We then use semidefinite programming (SDP) and sum-of-squares (SOS) optimization to synthesize a time-varying feedback control with formal robustness guarantees to account for model uncertainties and unmeasurable states in the system. Through simulation, hardware experiments and comparison with a time-varying LQR controller, it is shown that the proposed robust controller results in relatively large robust backward reachable sets and is able to reliably track a pre-generated optimal trajectory and achieve the desired brachiating motion in the presence of parametric model uncertainties, actuator limits, and unobservable states.

>[!note] Markdown Notes
>None!

>[!seealso] Related Papers
>

# Annotations

>[!fail] Possibly Incorrect
> *A. Robust Control Synthesis and Verification Results  The iterative optimization algorithm described in (15) to (17) was carried out for the brachiating robot system detailed above. We used polynomials of degree 4 for the Lagrange multipliers L, Lu, Lw and Lt, while the degree of the controller polynomial u ̄ is set to 1. The computing time required for the offline optimization convergence was approximately 4 hours. The long time required for convergence is not an issue for practical implementation of the controller, as the resulting feedback control policy u ̄(y ̄, t) (represented by time-varying gains on measurable states) will be hard-coded into the robot. To visualize the resulting robust backward reachable set, we project its 2-dimensional subspaces (out of the full 6dimensional state-space) on 2D plots. Fig. 4 shows the projections of each state vs. θ1, and compares the innerapproximation of the robust backward reachable sets for both the SOS-based controller and the time-varying LQR controller. As shown on the plots, the resulting invariant sets for the SOS-based controller cover a larger part of the statespace compared to TVLQR. The inner-approximation of the backward reachable set for the TVLQR controller is computed by solving the SOS program in (14a) without including the controller u ̄ in the optimization decision variables, eliminating the need for the second step optimization in (16). Furthermore, as depicted in Fig. 5, the verified set of initial conditions X0 which is driven to the desired set Xf by the SOS-based controller is larger in every dimension compared to the corresponding set for the TVLQR controller.*
> 
> >[!note] Note
> >I don't think any of this actually means anything about being 'Robust'. They're saying the backwards sets are 'robust' but what does that even mean..? There is no mention of plant perturbation at all.

>[!done] Quote
> *The performance of the robust SOS-based controller as well as the inner-approximation of its backward reachable set are validated by 20 simulation trials of the brachiating robot attached to the full-cable model. The stiffness of the cable is set to 20% less than the nominal value.*
> 
> >[!note] Note
> >Only one plant perturbation and it's structured...

>[!attention] Highlight
> *The robot starts from random initial conditions on the cable within the verified set of initial condition*
> 
> >[!note] Note
> >Robust to initial conditions, but not to actual plant perturbations.

### Imported: 2024-10-22 9:12 am


