Trajectory Generation for the N-Trailer Problem using Goursat
Normal Form
D. Tilbury, R. Murray, S. Sastry
UC Berkeley ERL Memo Number M93/12.
IEEE Conference on Decision and Control, 1993.
To appear in IEEE Transactions on Automatic Control, 1995.
In this paper, we develop the machinery of exterior differential
forms, more particularly the Goursat normal form for a Pfaffian
system, for solving nonholonomic motion planning problems, ie
planning problems with non-integrable velocity constraints. We apply
this technique to solving the problem of steering a mobile robot with
n trailers. We present an algorithm for finding a family of
transformations which will display the given system of rolling
constraints on the wheels of the robot with n trailers in the
Goursat canonical form. Two of these transformations are studied in
detail. The Goursat normal form for exterior differential systems is
dual to the so-called chained form for vector fields that we have
studied in our earlier work. Consequently, we are able to give the
state feedback law and change of coordinates to convert the
N-trailer system into chained form. Three methods for steering
chained form systems using sinusoids, piecewise constants and
polynomials as inputs are presented.
The motion planning strategy is therefore to first convert the
N-trailer system into chained form, steer the corresponding chained
form system, then transform the resulting trajectory back into the
original coordinates. Simulations and frames of movie animations of
the N-trailer system for parallel parking and backing into a loading
dock using this strategy are also included.
Steering Three-Input Chained Form Nonholonomic Systems: The Fire
Truck Example
L. G. Bushnell, D. M. Tilbury and S. S. Sastry
1993 European Control Conference.
To appear in the International Journal of Robotics Research, 1995.
In this paper, we steer nonholonomic systems with linear velocity
constraints represented mathematically in a special form, called
chained form. We observe that chained form systems can be steered
from an initial configuration to a final configuration with sinusoidal
inputs. The controller we use is open loop and no special provisions
are made for obstacle avoidance. Sufficient conditions are presented
for converting a three-input system with nonholonomic velocity
constraints into a ``two-chain, single-generator chainedform.'' An
algorithm is stated that constructs the sinusoidal control inputs to
steer this system from any initial configuration to any desired final
point. Our example of a three-input nonholonomic system is a
firetruck, or tiller truck. In this three-axle system, the control
inputs are the steering velocities of both the front and rear wheels
of the truck and the driving velocity of the truck. Simulation
results are given for the familiar parallelparking problem and other
trajectories.In tIn this paper we examine a multi-rate control scheme
for nonholonomic path planning using constant control inputs over
different time periods. For chained systems, an exact point-to-point
trajectory is generated. Simulation results are presented for a
three-input system, and comparisons are made with a sinusoidal method
for path planning.
Steering a Three-Input Nonholonomic System Using Multirate
Controls
D. Tilbury and A. Chelouah
1993 European Control Conference.
In this paper we examine a multi-rate control scheme for nonholonomic
path planning using constant control inputs over different time
periods. For chained systems, an exact point-to-point trajectory is
generated. Simulation results are presented for a three-input system,
and comparisons are made with a sinusoidal method for path planning.
Stabilization of Trajectories for Systems with
Nonholonomic Constraints
G. Walsh, D. Tilbury, S. Sastry, R. Murray, and J-P. Laumond
IEEE Transactions on Automatic Control, January, 1993.
IEEE International Conference on Robotics and Automation, 1991.
A new technique for stabilizing
nonholonomic systems to trajectories
is presented. It is well
known that such systems cannot
be stabilized to a point using smooth static state feedback.
In this paper we suggest the use of control laws
for stabilizing a system about a trajectory, instead of a point.
Given a nonlinear system and a desired (nominal)
feasible trajectory, the paper gives an explicit control
law which will locally exponentially stabilize
the system to the desired trajectory.
The theory is applied to several examples, including a car-like robot.