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.

1.
Segway LLC
, 2005, Segway Human Transporter, http://www.segway.comhttp://www.segway.com
2.
Salerno
,
A.
, and
Angeles
,
J.
, 2003. “
On the Nonlinear Controllability of a Quasiholonomic Mobile Robot
,”
Proc. IEEE Conf. Robot. Autom.
,
IEEE
, New York, pp.
3379
3384
.
3.
Grasser
,
F.
,
D'Arrigo
,
A.
,
Colombi
,
S.
, and
Rufer
,
A.
, 2002, “
Joe: A Mobile, Inverted Pendulum
,”
IEEE Trans. Ind. Electron.
0278-0046,
IE-49
(
1
), pp.
107
114
.
4.
Pathak
,
K.
, and
Agrawal
,
S. K.
, 2005, “
An Integrated Spatial Path-Planning and Controller Design Approach for a Hover-Mode Helicopter Model
,”
Proc. IEEE Conf. Robot. Autom.
, Barcelona, IEEE, New York, pp.
1902
1907
.
5.
Pathak
,
K.
,
Franch
,
J.
, and
Agrawal
,
S. K.
, 2005, “
Velocity and Position Control of a Wheeled Inverted Pendulum by Partial Feedback Linearization
,”
IEEE Trans. Rob. Autom.
1042-296X,
21
(
3
), pp.
505
513
.
6.
Pathak
,
K.
,
Franch
,
J.
, and
Agrawal
,
S. K.
, 2004, “
Velocity Control of a Wheeled Inverted Pendulum by Partial Feedback Linearization
,”
IEEE Conf. on Decision and Control, Atlantis, Bahamas
,
IEEE
, New York, pp.
3962
3967
.
7.
Biegler
,
L. T.
, 2000, “
Efficient Solution of Dynamic Optimization and NMPC Problems
,”
Progress in Systems and Control Theory
,
Birhäuser Verlag
, Basel, Switzerland, Vol.
26
.
8.
Biegler
,
L. T.
, 1984, “
Solution of Dynamic Optimization Problems by Successive Quadratic Programming and Orthogonal Collocation
,”
Comput. Chem. Eng.
0098-1354,
8
(
3
), pp.
243
247
.
9.
Findeisen
,
R.
,
Imsland
,
L.
,
Allgöwer
,
F.
, and
Foss
,
B. A.
, 2004, “
Towards a Sampled-Data Theory for Nonlinear Model Predictive Control
,”
New Trends in Nonlinear Dynamics and Control, and their Applications
,
Springer-Verlag
, Berlin, pp.
295
313
.
10.
Findeisen
,
R.
, and
Allgöwer
,
F.
, 2002, “An Introduction to Nonlinear Model Predictive Control,” 21st Benelux Meeting on Systems and Control.
11.
Sutton
,
G. J.
, and
Bitmead
,
R. R.
, 2000, “
Performance and Computational Implementation of Nonlinear Model Predictive Control on a Submarine
,”
Progress in Systems and Control Theory
,
Birhäuser Verlag
, Basel, Switzerland, Vol.
26
.
12.
Khalil
,
H. K.
, 1996,
Nonlinear Systems
,
3rd ed.
,
Prentice-Hall
, Englewood Cliffs, NJ.
13.
Pathak
,
K.
,
Agrawal
,
S. K.
, and
Messina
,
E.
, 2003, “
A Computationally Efficient Scheme for Hierarchical Predictive Control
,”
Performance Metrics for Intelligent Systems PerMIS
,
NIST
.
14.
Mayne
,
D. Q.
,
Rawlings
,
J. B.
,
Rao
,
C. V.
, and
Scokaert
,
P. O. M.
, 2000, “
Constrained Model Predictive Control: Stability and Optimality
,”
Automatica
0005-1098,
36
, pp.
789
814
.
15.
Trefethen
,
L. N.
, 1996, “Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations,” http://web.comlab.ox.ac.uk/oucl/work/nick.trefethen/pdetext.htmlhttp://web.comlab.ox.ac.uk/oucl/work/nick.trefethen/pdetext.html
16.
Longo
,
E.
,
Teppati
,
G.
, and
Bellomo
,
N.
, 1996, “
Discretization of Nonlinear Models by Sinc Collocation-Interpolation Methods
,”
Comput. Math. Appl.
0898-1221,
32
(
4
), pp.
65
81
.
17.
Meijering
,
E. H. W.
,
Niessen
,
W. J.
, and
Viergever
,
M. A.
, 1999, “
The Sinc-Approximating Kernels of Classical Polynomial Interpolation
,”
IEEE Int. Conf. on Image Processing
,
IEEE
, New York, Vol.
3
, pp.
652
656
.
18.
Proakis
,
J. G.
, and
Manolakis
,
D. G.
, 1996,
Digital Signal Processing
.
Prentice-Hall
, Englewood Cliffs, NJ.
19.
Press
,
W. H.
, 1993,
Numerical Recipes in C: The Art of Scientific Computing
,
2nd ed.
,
Cambridge University Press
, Cambridge, England.
20.
Pathak
,
K.
, and
Agrawal
,
S. K.
, 2004, “Optimal Band-Limited Trajectory Planning and Control for an Inverted Wheeled Pendulum,” ASME Int. Mech. Eng. Congr. and RD&D Expo, Anaheim, November.
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