Co-optimization of Acrobot Design and Controller for Increased Certifiable Stability

Lasse Maywald 1 Felix Wiebe 1 Shivesh Kumar 1 Mahdi Javadi 1 Frank Kirchner 1 2

German Research Center for Artificial Intelligence 1
University of Bremen 2

Presentation at IROS 2022, Kyoto, Japan

Introduction

Recent research on control of walking robots focuses on the study of undercatuated systems, where taking into account the systems passive dynamics is required to achieve truly dynamic and energy efficient behaviours. Many approaches to controlling such systems incorporate offline trajectory optimization and stabilization using online optimal control. Furthermore, design optimization can be carried out to optimize the physical parameters such as masses and link lengths. In this work we propose a novel a novel approach to co-optimzation of design and control parameters. The goal of this optimization is maximizing the Region of attraction (ROA) of a desired state that is associated to a fixed point of the closed loop dynamics. For a first case study, we consider a scenario that involves stabilizing an acrobot at its unstable upright posture using infinite horizon LQR control.

Shared optimality as a result of offline trajectory optimization, online stabilization and design optimization. This work focuses on the interplay of stabilization and design optimization.

Methodology

We employ two black box optimization algorithms, namely CMA-ES and Nelder-Mead to maximize the volume of the ROA, which is estimated using a probabilistic method based on najafi. For this, we consider a quadratic Lyapunov function that is constructed using the optimal cost-to-go obtained by solving the LQR problem for the linearized dynamics around the upright pose of the acrobot. In order to reduce the dimensionality of the problem, we optimize over the physical design variables and the control variables separately.

Overview of the cooptimization methodology.

Case Study

For the acrobot case study, we optimized over both link-lengths, a point mass at the end of the second link and the diagonal elements of the \(\mathbf{Q}\) and \(\mathbf{R}\) matrices. We intialized the optimization with \(l_1=0.3m\), \(l_2=0.2m\) and \(m_2=0.63kg\) and identity matrices for \(\mathbf{Q}\) and \(\mathbf{R}\).

a) Estimated ROA of the initial (red) and optimized (blue) design projected into the q_1 vs. q_2 plane. Estimated ROA Volume as a function of b) design- and c) control parameters. The green dot marks the optimal solution found by CMA-ES.

After multiple simulations using two different optimizers and different strategies, the parameters for the best performing design were found to be \(l_1=0.2m\), \(l_2=0.4m\), \(m_2=0.22kg\), \(q_{11}=2.08\), \(q_{22}=0.15\), \(q_{33}=0.99 = q_{44}=0.99\) and \(r_{11}=0.62\). For these parameters the estimated ROA is significantly larger than for the initial ones as can be seen in the figure above that also shows the objective function in different projections of the state space.

Citation

Maywald, Lasse & Wiebe, Felix & Kumar, Shivesh & Javadi, Mahdi & Kirchner, Frank. (2022). Co-optimization of Acrobot Design and Controller for Increased Certifiable Stability. 10.13140/RG.2.2.36436.07043.

@misc{maywald2022,
  author = {Maywald, Lasse and Wiebe, Felix and Kumar, Shivesh and Javadi, Mahdi and Kirchner, Frank},
  year = {2022},
  month = {07},
  pages = {},
  title = {Co-optimization of Acrobot Design and Controller for Increased Certifiable Stability},
  doi = {10.13140/RG.2.2.36436.07043}
}