Parameter Refinement of a Ballbot and Predictive Control for Reference Tracking with Linear Parameter-Varying Embedding

TL;DR

The LPV-MPC control method can be solved efficiently as a quadratic program (QP) that provides timing that supports real-time implementation and theoretical guarantees such as stability and recursive feasibility are provided for a single-set point reference.

Abstract

In this study, we implement a control method for stabilizing a ballbot that simultaneously follows a reference. A ballbot is a robot balancing on a spherical wheel where the single point of contact with the ground makes it omnidirectional and highly maneuverable but with inherent instability. After introducing the scheduling parameters, we start the analysis by embedding the nonlinear dynamic model derived from first principles to a linear parameter-varying (LPV) formulation. Continuously, and as an extension of a past study, we refine the parameters of the nonlinear model that enhance significantly its accuracy. The crucial advantages of the LPV formulation are that it consists of a nonlinear predictor that can be used in model predictive control (MPC) by retaining the convexity of the quadratic optimization problem with linear constraints and further evades computational burdens that appear in other nonlinear MPC methods with only a slight loss in performance. The LPVMPC control method can be solved efficiently as a quadratic program (QP) that provides timing that supports real-time implementation. Finally, to illustrate the method, we test the control designs on a two-set point 1D non-smooth reference with sudden changes, to a 2D nonstationary smooth reference known as Lissajous curves, and to a single-set point 1D non-smooth reference where for this case theoretical guarantees such as stability and recursive feasibility are provided.

Figures (6)

• Figure 1: The Ballbot constructed at the Institute for Electrical Engineering in Medicine (IME) and side view on the $xz$-plane.
• Figure 2: Convergence of the Newton scheme with residual error $\Vert F(b^*)\Vert=0.4330$. The optimal parameter vector $b^*$ is shown in Tab. \ref{['tab:bparameters']}.
• Figure 3: Models simulations; the linearized model given in dashed blue line; non-linear model simulation given in black; LPV in dashed red lines; all done using ode45 on MATLAB. Finally, with green is the RK4 method utilizing \ref{['eq:dlpv']}. The input $\tau_y$ is a multiharmonic signal.
• Figure 4: Ballbot reference tracking with the angular displacement $\phi$ traveling from the origin to $2\pi$ rad and back. A comparison between the linear and LPV MPC frameworks is illustrated. $\theta$ measuring balancing of ballbot, $\dot{\phi}$, $\dot{\theta}$ measuring the angular velocities, $\tau_y$ is the virtual torque, $J$ is the energy cost function.
• Figure 5: Stabilization and Lissajous curve trajectory tracking of the ballbot in 2D using the LPVMPC algorithm solved in both $xz$ and $yz$ planes.

Theorems & Definitions (1)

• Remark 1: The operator $f_{\text{LPV}}$ with a given scheduling variable $\rho$ is linear