The Painlev\'{e}-type asymptotics of defocusing complex mKdV equation with finite density initial data

We consider the Cauchy problem for the defocusing complex mKdV equation with a finite density initial data \begin{align*} &q_t+\frac{1}{2}q_{xxx}-\left(|q|^2q\right)_{x}=0,\\ &q(x,0)=q_{0}(x) \sim \pm 1, \ x\to \pm\infty, \end{align*} which can be formulated into a Riemann-Hilbert(RH) problem. With a $\bar\partial$-generation of the nonlinear steepest descent approach and a double scaling limit technique, in the transition region $$\mathcal{D}:=\left\{(x,t)\in\mathbb{R}\times\mathbb{R}^+\big|-C< \left(\frac{x}{2t}+\frac{3}{2}\right) t^{2/3}<0, C\in\mathbb{R}^+\right\},$$ we find that the long-time asymptotics of the solution $q(x,t)$ to the Cauchy problem is associated with the Painlev\'{e}-II transcendents

Learning Motion Predictors for Smart Wheelchair using Autoregressive Sparse Gaussian Process

Constructing a smart wheelchair on a commercially available powered wheelchair (PWC) platform avoids a host of seating, mechanical design and reliability issues but requires methods of predicting and controlling the motion of a device never intended for robotics. Analog joystick inputs are subject to black-box transformations which may produce intuitive and adaptable motion control for human operators, but complicate robotic control approaches; furthermore, installation of standard axle mounted odometers on a commercial PWC is difficult. In this work, we present an integrated hardware and software system for predicting the motion of a commercial PWC platform that does not require any physical or electronic modification of the chair beyond plugging into an industry standard auxiliary input port. This system uses an RGB-D camera and an Arduino interface board to capture motion data, including visual odometry and joystick signals, via ROS communication. Future motion is predicted using an autoregressive sparse Gaussian process model. We evaluate the proposed system on real-world short-term path prediction experiments. Experimental results demonstrate the system's efficacy when compared to a baseline neural network model.Comment: The paper has been accepted to the International Conference on Robotics and Automation (ICRA2018