Band-Limited Trajectory Planning and Tracking for Certain Dynamically Stabilized Mobile Systems

[+] Author and Article Information
Kaustubh Pathak

Mechanical Systems Lab, Department of Mechanical Engineering, University of Delaware, Newark, DE 19716pathakk1@asme.org

Sunil K. Agrawal

Mechanical Systems Lab, Department of Mechanical Engineering, University of Delaware, Newark, DE 19716agrawal@me.udel.edu

J. Dyn. Sys., Meas., Control 128(1), 104-111 (Nov 21, 2005) (8 pages) doi:10.1115/1.2168158 History: Received March 17, 2005; Revised November 21, 2005

In this paper, a general framework for trajectory planning and tracking is formulated for dynamically stabilized mobile systems, e.g., inverted wheeled pendulums and autonomous helicopters. Within this framework, the system state is divided into slow and fast substates. The fast substate is used as a pseudocontrol for tracking a desired slow substate trajectory. First, an exponential fast substate controller is designed to track a fast substate reference trajectory. This fast substate reference trajectory is, in turn, planned so that the slow substate follows its desired trajectory. To ensure that the fast substate reference trajectory is feasible for the exponential controller, it is designed using band-limited “Sinc” functions whose maximum frequency is less than the inverse of the time constant of the exponential controller. To illustrate the procedure, the dynamic model of an inverted wheeled pendulum is reformulated by a partial feedback linearization such that it is amenable to the separation into slow and fast components. The planning and tracking controller design is explained using simulation results. This technique is shown to be easily embedded inside a modified nonlinear model predictive control framework for the slow subsystem. This framework tries to explicitly take the computational delay into account. The computation time required for this technique is encouraging from a real-world implementation perspective.

Copyright © 2006 by American Society of Mechanical Engineers
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Figure 1

Schematic of the planner/control strategy. The model predictive control (MPC) loop is shown in dashed lines. The planner computes a band-limited trajectory for xs with xfr(t) as pseudoinput. xfr(t) is then tracked by the exponential controller Cf.

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Figure 2

Comparison between Sinc and Lagrange polynomial approximation. The upper figure shows the function f(x) being interpolated with the samples shown by circles. The lower figure shows the error in approximation f(x)−fa(x). The polynomial approximation has big oscillations near the boundary points. The Sinc approximation error is close to zero.

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Figure 3

Geometric parameters and coordinate systems for the mobile inverted wheeled pendulum

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Figure 4

The planned x−y path and the time trajectories of α (in rads), θ (in rads), and v (in millimeters per second). The circled points show the collocation locations, which are joined smoothly by the Sinc interpolant.

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Figure 5

The tracking-error performance of the controller given in Eq. 31 corresponding to Fig. 4. The last figure shows the normalized planned and actual controller generated motor torques ∥τ∥∕τm, τm=0.7Nm. The planned samples are below unity, whereas the actual controller torque is seen to remain less than unity except for a short wiggle, which is caused by the interpolation.

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Figure 6

The first few steps of MPC shown for a particular component x of the state vector. The computation time required is denoted by ΔTc. A new plan Pi is computed between iΔTc to (i+1)ΔTc, to run the system from (i+1)ΔTc to (i+1)ΔTc+ΔTp, where ΔTp is the planning horizon. In the figure, the system has run until 2ΔTc. The system was kept stationary from t=0 to ΔTc by setting the reference of the tracking controller Cf to 0. Every plan computed at and after 2ΔTc requires an estimation of the state at the start of the next planning instant. This estimation is shown by dotted lines xest.

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Figure 7

The first four steps shown for the NMPC applied to the inverted-wheeled-pendulum example. ΔTc=7.5s and ΔTp=15s. The solid curve is the actual system trajectory until t=32.5s. The four plans' sample points are depicted by variously shaped markers to distinguish them. The start time for the plans are depicted by vertical lines. The discarded portion of each plan is also shown. Note that the end point for a plan slides along a user-specified waypoint trajectory. α is shown to follow the plan closely. The simulation parameters are as shown in Table 1, except that the values of the model Iyy, Izz, and cz have been perturbed by 1.5%.




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