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Nonholonomic motion planning: steering using sinusoids

TL;DR: Methods for steering systems with nonholonomic c.onstraints between arbitrary configurations are investigated and suboptimal trajectories are derived for systems that are not in canonical form.
Abstract: Methods for steering systems with nonholonomic c.onstraints between arbitrary configurations are investigated. Suboptimal trajectories are derived for systems that are not in canonical form. Systems in which it takes more than one level of bracketing to achieve controllability are considered. The trajectories use sinusoids at integrally related frequencies to achieve motion at a given bracketing level. A class of systems that can be steered using sinusoids (claimed systems) is defined. Conditions under which a class of two-input systems can be converted into this form are given. >

Summary (3 min read)

I. INTRODUCTION

  • OTION planning for robots has a rich history.
  • The authors interests in motion planning are not along the lines of the aforementioned approaches, but are complemen-Manuscript received May 22, 1991; revised April 5, 1992 tary: they involve motion planning in the presence of nonholonomic or nonintegrable constraints.
  • If an object is twirled through a cyclic motion that returns the object to its initial position and orientation, and the fingers roll without slipping on the surface of the object, the fingers do not necessarily return to their initial configurations.
  • Nonholonomic constraints arise either from the nature of the controls that can be physically applied to the system or from conservation laws which apply to the system.
  • In Section 111, using some outstanding results of Brockett on optimal steering of certain classes of systems as motivation [6], the authors discuss the use of sinusoidal inputs for steering systems of first order, i.e., systems where controllability is achieved after just one level of Lie brackets of the input vector fields.

11. MATHEMATICAL PRELIMINARIES

  • This section describes the notation to be used throughout the paper and collects a variety of results from nonlinear control theory and Lie algebras which will be used in the sequel.
  • For basic definitions and concepts in differential geometry, see Boothby [5] or Spivak [48] .
  • A good introduction to nonlinear control theory which includes many of the necessary differential geometric concepts can be found in Isidori [21] or Nijmeijer and van der Schaft M11.

A. Nonlinear Control Theory

  • The net motion consists of flowing along g,,g,, -gl, -g, for time E and can be shown to satisfy (3) Thus, the Lie bracket is the infinitesimal motion that results from flowing around a square defined by two tangent vectors.
  • Roughly speaking, Chow's theorem states that if the authors can move in every direction using Lie bracket motions (possibly of higher order than one), then the system is controllable.

C. Examples of Nonholonomic Systems

  • The robot consists of a leg which can both rotate and extend.
  • The configuration of the mechanism is given by the angle of the body and the angle and length of the leg.

These algorithms.

  • This robot consists of a body with an actuated leg that can rotate and extend; the "constraint" on the system is conservation of angular momentum.
  • For simplicity, the authors take the body mass to be one and concentrate the mass of the leg, m,, at the foot.
  • The constraints for the front and rear wheels are formed by writing the sideways velocity of the wheels:.
  • The wheels of the individual trailers are aligned with the body of the trailer.

A. First-Order Systems

  • This problem is related to finding the geodesics associated with a singular Riemannian metric (Camot-Caratheodory metric).
  • To solve the problem, Brockett considers a class of systems which have a special canonical form.
  • Using this as justification, the authors attempt to use their proposed algorithm to steer the full nonlinear system.
  • Since the authors control the @ and 1 states directly, they first steer them to their desired values.

B. Second-Order Systems

  • The authors next consider systems in which the first level of bracketing is not enough to span R".
  • The authors begin by trying to extend the previous canonical form to the next higher level of bracketing.
  • Consider a system which can be expressed as EQUATION Because Jacobi's identity imposes relations between certain brackets, not all xlJk combinations are permissible if the system (10) is to be completely controllable.
  • Constructing the Lagrangian (with the same integral cost function) and substituting into the Euler-Lagrange equations does not in general result in a constant set of Lagrange multipliers, although Brockett and Dai have shown that for m = 2 the optimal inputs are elliptic functions [9] .
  • The authors can extend and apply their previous algorithm as follows.

Iv. CHAINED SYSTEMS

  • The authors now study more general examples of nonholonomic systems and investigate the use of sinusoids for steering such systems.
  • As in the previous section, the authors try to generate canonical classes of higher order systems, i.e., systems where more than one level of Lie brackets is needed to span the tangent space to the configuration space.
  • The authors give sufficient conditions under which systems can be transformed into a chained form and show the procedure applied to several illustrative examples.

A. Maximum Growth Canonical Systems

  • Hall basis, it is possible to construct vector fields which have maximum growth; at each level of bracketing the dimension of the filtration grows by the maximum possible amount.
  • Hall basis elements, the resulting set of vector fields is linearly independent.
  • Thus, the authors can associate with each such basis element a well-defined vector a E B" which indicates the number of times each basis element occurs in the expansion (12); i.e., ai(k) is the number of times B, appears in the expansion for Bi.
  • Then nents in the derivatives fo the dynamic system: Example 8: Consider the two input example given previously, but with order of nilpotency 4 instead of 5. sions of the canonical forms the authors have seen for degree of nonholonomy and 3.
  • It may still be possible to steer the system using combinations of sinusoids at different frequencies for each input or using more complicated periodic functions (such as elliptic functions, see [91).

B. Chained Systems

  • Rather than explore the use of more complicated inputs for steering nonholonomic systems, the authors consider instead a simpler class of systems.
  • The justification for changing the class of systems is simple-most of the systems encountered as examples do not have the complicated structure of their canonical example.
  • Thus there may be a simpler class of systems which is both steerable using simple sinusoids and representative of systems in which the authors are interested.

is, in

  • The same approach can be used to steer x , through x5.the authors.
  • The first item is to check the controllability of these systems.
  • To steer this system, the authors use sinusoids at integrally related frequencies.
  • Direct computation starting from the origin yields.
  • Let hfj represent the motion corresponding to the Lie product Xi.

C. Noncanonical Chained Systems

  • Hence, the authors cannot use simple sinusoids to steer this system as before.
  • Since the automobile had degree of nonholonomy 3, the problems present in the previous example do not occur.
  • Similar efforts have been used by Lafferriere and Sussmann [261 to convert systems into nilpotent form for use with their planning algorithm.
  • Finally, sinusoids may be useful for steering systems.
  • The following example illustrates the limitations of using sinusoidal which ake not locally in canonical form.

D. Converting Systems to Chained Form

  • This set of conditions gives a constructive method for building a feedback transformation which accomplishes the conversion.
  • The motion is implemented as a feedback precompensator which converts the U inputs into the actual system input, U. This feedback transformation agrees the that used in Lafferriere and Sussmann to nilpotentize the kinematic car example.
  • The results of Section I11 are somewhat complimentary-the methods can easily be applied to certain systems which are not nilpotent, but the general case requires a restrictive canonical form.
  • The work in nonholonomic motion planning thus far has been primarily in the generation of open-loop trajectories.
  • Recent work by Coron has shown that it is possible to stabilize a nonholonomic system using smooth, time-varying state feedback McClamroch have studied problems related to stabilization to a manifold instead of a point [31, [41.

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700
IEEE
TRANSACTIONS ON AUTOMATIC CONTROL, VOL.
38,
NO.
5, MAY 1993
Nonholonomic Motion Planning:
Steering Using Sinusoids
Richard
M. Murray,
Member, IEEE,
and
S.
Shankar Sastry,
SeniorMember, IEEE
Abstract--In this paper, we investigate methods for steering
systems with nonholonomic constraints between arbitrary con-
figurations. Early work by Brockett derives the optimal controls
for
a set of canonical systems in which the tangent space to the
configuration manifold is spanned by the input vector fields and
their first order Lie brackets. Using Brockett’s result as motiva-
tion, we derive suboptimal trajectories for systems which are not
in canonical form and consider systems in which it takes more
than one level of bracketing to achieve controllability. These
trajectories use sinusoids at integrally related frequencies to
achieve motion at a given bracketing level. We define a class
of
systems which can be steered using sinusoids (chained systems)
and give conditions under which a class
of
two-input systems
can be converted into this form.
I.
INTRODUCTION
OTION planning for robots has a rich history. The
M
traditional difficulty in planning robot trajectories
is the avoidance of obstacles, often referred to as the
piano mover’s problem, in which we attempt to move an
object (the piano) through a cluttered environment. This
problem is solved by investigating the free configuration
space of the piano-all configurations for which the piano
does not intersect an obstacle. If the start and goal
locations of the piano lie in the same connected compo-
nent of the free configuration space, the motion planning
problem is solvable.
In recent years, there has been a great deal of activity
in the generation
of
efficient motion planning algorithms
for robots. Most of this work has concentrated on the
global problem of determining a path when the obstacle
positions are known and dynamic constraints on the robot
are not considered. This has resulted in a rather complete
understanding of the complexity of the computational
effort required to plan the trajectories of robots to avoid
both fixed and moving obstacles [lo], [22], [281. Other
approaches include the use of potential functions for
navigating in cluttered environments [24], [25] and compli-
ant motion planning for navigating in the presence of
uncertainty [131, [141, [371.
Our interests in motion planning are not along the lines
of the aforementioned approaches, but are complemen-
Manuscript received May 22, 1991; revised April
5,
1992. Paper
recommended by Associate Editor, A.
H.
Abed. This work was sup-
ported
in
part by the
NSF
under Grant IRI-90-14490.
R.
M.
Murray is with the Department of Mechanical Engineering,
California Institute of Technology, Pasadena, CA 91125.
S.
Shankar Sastry is with Electronics Research Laboratory, University
of
California, Berkeley, CA 94720.
IEEE
Log Number 9208451.
tary: they involve motion planning in the presence of
nonholonomic or nonintegrable constraints. That is, we
consider systems in which there are constraints on the
velocities of the robots which cannot be integrated to give
constraints which are exclusively a function of the config-
uration variables. These situations arise in a number of
different ways and we describe a few of the sources of
their origin:
1)
Mobile Robots Navigating
in
a Cluttered Environment:
The kinematics of the drive mechanisms
of
robot carts
results in constraints on the instantaneous velocities that
can be achieved. For instance, a cart with
two
forward
drive wheels and
two
back wheels cannot move sideways.
This was first pointed out by Laumond in the context of
motion planning for the Hilare mobile robot 1291, [301.
2)
Multifngered Hands Manipulating a Grasped Object:
If an object is twirled through a cyclic motion that returns
the object to its initial position and orientation, and the
fingers roll without slipping on the surface of the object,
the fingers do not necessarily return to their initial con-
figurations. This feature can be used to plan the regrasp
of a poorly grasped object or to choose the nature of this
grasp. This application of nonholonomic motion planning
was first pointed out by Li [341, [351 (see also [391).
3)
Space Robotics:
Unanchored robots in space are
difficult to control with either thrusters or internal motors
since they conserve total angular momentum. This
is
a
nonintegrable constraint. The motion of astronauts on
space walks is of this ilk,
so
that planning a strategy to
reorient an astronaut is a nonholonomic motion planning
problem [55]. Other examples of this effect include
gym-
nasts and springboard divers.
Nonholonomic constraints arise either from the nature
of the controls that can be physically applied to
the system or from conservation laws which apply to the
system. Conventional path planners implicitly assume that
arbitrary motion in the configuration space is allowed as
long as obstacles are avoided. If a system contains
nonholonomic constraints, many of these path planners
cannot be directly applied. If we attempt to ignore the
constraint, the paths generated by a path planner may
not be feasible (see Fig. 1). For this reason, it is important
to understand how to efficiently compute paths for
nonholonomic systems.
To
be more specific, we are interested in mechanical
systems with linear velocity constraints of the form
wi(x)i
=
0
i
=
l,...,
k.
(1)
0018-9286/93$03.00
0
1993 IEEE

MURRAY
AND
SASTRY: STEERING USING
SINUSOIDS
701
Path satisfying
-I
-
velocity constraints
-
Path satisfying
Fig.
1.
Paths generated by conventional path planners may ignore
nonholonomic constraints. The straight line path in the figure indicates
the path that a conventional path planner might generate. The curved
path is one which satisfies the nonholonomic constraints
on
the car's
motion.
Here,
x
E
R"
is the configuration of the system being
controlled and
q(x)
is a row vector in
R".
These are
constraints on the
velocities
of the system. In some cases,
the constraints may be explicitly integrable, giving con-
straints of the form
hi(x)
=
ci
for some constant
ci.
If this is possible, motion of the
system is restricted to a level surface of
hi.
Such a
constraint is said to be
holonomic.
By choosing coordi-
nates for the surface, configuration space methods can be
applied. In the instance that there is only one constraint
on the velocity of the system, its integrability may be
determined by checking the symmetry of the Jacobian
matrix of
ol(x).
There is no easy extension of this charac-
terization to the case of multiple constraints.
A
constraint is said to be
nonholonomic
if it cannot be
written as an algebraic constraint in the configuration
space. There are many types of nonholonomic constraints,
corresponding to different physical situations.
It will be convenient for us to convert problems with
nonholonomic constraints into steering problems for con-
trol systems. Consider the problem of constructing a path
x(t)
E
R"
between a given
xo
and
x1
subject to the
k
constraints given in equation (1). We assume the
mi's
are
smooth and linearly independent. Specific examples of
such systems are given in Section
11.
Roughly speaking, we
would like to convert the constraint specification from
describing the directions in which the system cannot move
to those in which it can. Formally, we choose a basis for
the right null space of the constraints, denoted by
gi(x)
E
R",
i
=
l,...,
n
-
k.
The path planning problem can be
restated as finding an input function,
u(t)
E
IWnPk,
such
that the control system
i
=
g,(x)u,
+
..*
+gn-k(x)Un-k
is driven from
x,,
to
xl.
It can be shown that if the
wi7s
are smooth and linearly independent, then the
gi's
inherit
these properties.
The outline of this paper is as follows: in Section 11, we
collect some mathematical preliminaries from the liter-
ature on controllability of nonlinear systems and on clas-
sification of free Lie algebras. These are drawn from
classical references
in
control theory [71, [201, [211, [411,
[49] and Lie algebras
[181,
[53]. In Section 111, using some
outstanding results of Brockett on optimal steering of
certain classes of systems as motivation
[6],
we discuss the
use of sinusoidal inputs for steering systems of first order,
i.e., systems where controllability is achieved after just
one level of Lie brackets of the input vector fields. Section
IV attempts to expand the domain
of
applicability
of
these
results to more complex systems, where several orders of
Lie brackets are needed to obtain the full Lie algebra
associated with the input distribution. The style of the
paper is self-contained
so
as to make it accessible to both
robotics and control researchers and several examples are
sustained through the paper.
A
target problem which we set ourselves at the start of
this research was that of parking
of
a car with
N
trailers.
This problem remains unsolved and indeed has generated
some fascinating new ideas in the field. It is not a "toy
problem" since efforts are underway to automate baggage
handling by carts with multiple trailers in airports (not to
mention trucks with multiple trailers). It is fair to say that
the study of nonholonomic motion planning is
in
its
infancy. There have, however, been notable contributions
by Laumond
et al.
[231, [29], [31]-[33] and by Barraquand
and Latombe [2] on motion planning for mobile robots in
a cluttered field. While this work represents important
initial progress, we feel that less computationally intensive
and more insightful approaches are possible by conduct-
ing a systematic research program on motion planning of
dynamical systems with nonholonomic constraints. We are
joined by several complementary efforts, notably those of
Li and co-workers [15], 2351 and Sussmann and co-workers
[27],
[SI.
We have also applied the techniques of this
paper to steering of space robots using sinusoids in [551.
11.
MATHEMATICAL
PRELIMINARIES
This section describes the notation to be used through-
out the paper and collects a variety of results from nonlin-
ear control theory and Lie algebras which
will
be used in
the sequel. For basic definitions and concepts in differen-
tial geometry, see Boothby [5] or Spivak [48].
A
good
introduction to nonlinear control theory which includes
many of the necessary differential geometric concepts can
be found in Isidori [21] or Nijmeijer and van der Schaft
M11.
A.
Nonlinear
Control
Theory
We consider the problem of steering a control system
c:
i
=
g,(x)u,
+
**-
+g,(x)u,
x
E
U
c
R"
U
E
R"
(2)

702
IEEE
TRANSACTIONS
ON
AUTOMATIC CONTROL,
VOL. 38,
NO.
5,
MAY
1993
from an initial state
xo
E
U
to a final state
x1
E
U
by
appropriate choice of a control
U:
[O,T]
+
R".
For sim-
plicity, we assume
U
to be an open neighborhood of the
origin and
{gi}
to be a collection of smooth, linearly
independent vector fields defined on
U.
Associated with
the system
8
is a distribution
A
=
span
{g,,...,
g,l
where we take the span over the set of smooth real-valued
functions on
U.
A,
c
R"
denotes the subspace defined by
evaluating
A
at a point
x.
Controllability of the system
I:
can be characterized
in
terms of the Lie algebra generated by the vector fields
gi.
Define the Lie bracket between two vector fields
f,
g
as
A
straightforward calculation shows that the Lie
has the following properties:
[f,gl
=
-
[g,fl
(skew-symmetry)
[f,[g,hll+ [g,[h,fll+ [h,[f,gll
=o
(Jacobi identity).
Given a distribution
A,
the involutive closure
bracket
of the
distribution, denoted
x,
is the closure of
A
under Lie
bracketing.
A
system
8
is controllable
if
for any
xo,
x1
E
U
there
exists a
T
>
0
and
U:
[0,
TI
+
R"
such that
I:
satisfies
x(0)
=xo
and
x(T)
=xl.
For a control system which is
linear in the input, the time interval
T
is arbitrary since
we can scale the inputs (and hence time) as needed. The
conditions for controllability are given by Chow's theorem
(see
[20]).
Theorem
I
(Chow):
If
x,
=
R"
for all
x
E
U
then the
system
8
is controllable on
U.
A
useful interpretation of Chow's theorem can be
obtained by using the following characterization of the
Lie bracket. Let
4[:
U
+
R"
denote the flow of a vector
field
f
for time
t
and consider the sequence of flows
depicted
in
Fig.
2.
The net motion consists of flowing
along
g,,g,,
-gl,
-g,
for time
E
and can be shown to
satisfy
4;gzo
4;glo
@20
4,gl<x0>
=
E2[g,,
g21(x0)
+
o(E~>.
(3)
Thus, the Lie bracket is the infinitesimal motion that
results from flowing around a square defined by two
tangent vectors. If
[gl,g,l
=
0
then
g,
and
g,
commute
and it can be shown that the right-hand side of
(3)
is
identically zero; i.e., we return to the starting point.
Roughly speaking, Chow's theorem states that if we
can move in every direction using Lie bracket motions
(possibly of higher order than one), then the system is
controllable.
A
nonzero
net motion
&
/
Eg2
Fig.
2.
A
Lie
bracket
motion.
B.
Classification
of
Lie Algebras
We now develop some concepts which allow us to
classify nonholonomic systems.
A
more complete treat-
ment can be found in the work of Vershik
[
161, [54].
Basic
facts concerning Lie algebras are taken from Varadarajan
[531.
Let
A
=
span{g,;..,g,} be the distribution associ-
ated with the control system
(2).
Define
G,
=
A
and
Gi
=
Gi-,
+
[Gl,Gi-,1
where
[G,,Gi-,1
=
span{[g,hl:
g
E
G,,h
E
Gi-l).
The set of all
Gi's
defines the
filtration
associated with a
distribution. Each
Gi
is defined to be spanned by the
input vector fields plus the vector fields formed by taking
up to
i
-
1
Lie brackets. The Jacobi identity implies
[Gi,
GjI
C
[GI, Gi+j-
11
C
Gi+j.
A
filtration is
regular
in a neighborhood
U
of
xo
if
rank
Gib)
=
rank
Gi(xo)
Vx
E
U.
We say a system is regular if the corresponding filtration
is regular. If a filtration is regular, then at each step of its
construction,
Gi
either gains dimension or the construc-
tion terminates. If rank
Gi+
,
=
rank
Gi
then
Gi
is involu-
tive and hence
Gi+j
=
Gi
for all
j
2
0.
Clearly, rank
Gi
I
n
and hence
if
a filtration is regular, then there exists an
integer
p
<
n
such that
G,
=
Gp
for all
i
2
p.
We refer to
p
as the
degree
of
nonholonomy
of the distribution.
For a regular system, Chow's theorem states that a path
exists between two arbitrary points in an open set
U
c
R"
if
and only if
G,,(x)
=
R"
for all
x
E
U.
A
system (or
distribution) satisfying the conditions of this theorem is
said to be
maximally nonholonomic.
If a regular system is
not maximally nonholonomic, then by Frobenius' theorem
we can restrict ourselves to a manifold on which the
system is maximally nonholonomic.
It is also useful to record the dimension of each
Gi.
For
a regular system, we define the
growth vector r
E
ZP
as
ri
=
rank
Gi.
We define the
relative growth vector
U
E
ZP
as
ai
=
ri
-
ri-,
and
ro
:=
0.
The growth vector for a system
is
a
convenient way to represent information about the associ-

MURRAY
AND
SASTRY.
STEERING
USING
SINUSOIDS
103
ated control Lie algebra. For a distribution with finite
rank, the growth vector is bounded from above at each
step.
To
properly determine this bound, we must deter-
mine the maximal rank of
Gi
taking into account skew-
symmetry and the Jacobi identity.
A
careful calculation
[46]
gives
where
ai
is the maximum relative growth at the ith stage
and
jli
means all integers
j
such that
j
divides
i.
If
ui
=
ai
for all
i,
we say
A
has
maximum growth.
C.
Examples
of
Nonholonomic Systems
To
illustrate the classification of nonholonomic systems,
we
present
are
used in later sections as a basis for testing planning
Fig.
3.
A
simple hopping robot. The robot consists
of
a leg which
can
both rotate and extend. The configuration
of
the mechanism
is
given by
the angle
of
the body and the angle and length (extension)
of
the leg.
These
algorithms.
Example
I
(Hopping Robot):
As
our first example, we
consider the dynamics of a hopping robot in flight phase
[36],
as shown in Fig.
3.
This robot consists of a body
with an actuated leg that can rotate and extend; the
“constraint” on the system is conservation of angular
momentum.
Let
(9,
I,
8)
be the body angle, leg extension, and leg
angle of the robot. For simplicity, we take the body mass
to be one and concentrate the mass of the leg,
m,,
at the
foot. The upper leg length is also taken to be one, with
1
representing the extension of the leg past this point. Since
we control the leg angle and extension directly, we choose
their velocities as our inputs. The angular momentum of
the robot is given by
Example
2
(Enematic
Car):
Consider a simple kine-
matic model for an automobile with front and rear tires
[40],
as shown in Fig.
4.
The rear tires are aligned with
the car while the front tires are allowed to spin about the
vertical axes.
To
simplify the derivation, we model
the front and rear pairs
of
wheels as single wheels at the
midpoints of the axles. The constraints on the system arise
by allowing the wheels to roll and spin, but not slip.
Let
(x,
y,
+,e)
denote the configuration of the car,
parameterized by the location of the rear wheel(s), the
angle of the car body with respect to the horizontal
(e),
and the steering angle with respect to the car body
(4).
The constraints for the front and rear wheels are formed
by writing the sideways velocity of the wheels:
e
+
m,(l
+
112(
e
+
4)
=
0.
d
-(x
+
1
cos
8).
sin(8
+
4)
-
-(y
+
1
sin
e)
cos(8
+
4)
=
0
i
sin
8
-
y
cos
8
=
0.
Thus, our equations become
dt
d
*=U,
1
=
U2
dt
Written as one forms we have
w1
=sin(8+4)dx-cos(8+4)dy-lcos+dO
w2
=
sin
8dx
-
cos
8dy.
Converting this to a control system gives
i
=
COS
eu,
y
=
sin
Bu,
4
=
U2
.1
8
=
-
tan
+U,.
1
In a neighborhood of
1
=
0,{g,,g2,g3}
is full rank and
hence the hopping robot has degree of nonholonomy
2
with growth vector
(2,3).
For this choice of vector fields,
U,
corresponds to the
forward velocity of the rear wheels of the car and
u2
corresponds to the velocity
of
the steering wheel.

704
IEEE
TRANSACTIONS
ON
AUTOMATIC
CONTROL,
VOL.
38,
NO.
5,
MAY
1993
Y
'5
Fig.
4.
Kinematic model of an automobile. The configuration
of
the car
is
determined by the Cartesian location
of
the back wheels, the angle the
car makes with the horizontal and the steering wheel angle relative to
the car body. The
two
inputs are the velocity of the rear wheels and the
steering velocity.
To calculate the growth vector, we compute the control
Lie algebra:
d
dl
d
g,
=
cos
8-
+
sin
8-
+
-
tan
4-
dX
dY
1
de
Fig.
5.
Kinematic car with trailers. The trailer configuration is described
the angle the trailer makes with the horizontal,
0,.
The rear wheels of
the trailer are fixed and constrained to move along the line in which they
point
or
rotate about their center. The inputs to the system are the
inputs to the
tow car: the driving velocity
(of
the front wheels) and the
steering velocity.
trailer is pointing and its perpendicular. The perpendicu-
lar component causes the trailer to spin. Letting
vi-,
be
the forward velocity of the previous trailer, we have
.1
ei
=
-
sin(8,-,
-
ei)Vi-,
4
vi
=
cos(ei-,
-
ei)vi-,.
d
g2
=
3
Aggregating these equations gives
-1
d
g3
=
[g,,g,I
=
--
i
COS~
4
de
{g,,
g,,
g,,
g4}
are linearly independent when
4
.fL
*
~/2.
Thus, the system has degree of nonholonomy
3
with
growth vector
r
=
(2,3,4) and relative growth vector
U
=
(2,1,1). The system is regular away from
4
=
f
~/2,
where
g,
is undefined.
Example
3
(Car with
N
Trailers):
Fig.
5
shows a car with
N
trailers attached. We attach the hitch of each trailer to
the center of the rear axle of the previous trailer. The
wheels of the individual trailers are aligned with the body
of the trailer. The constraints are again based on allowing
the wheels only to roll and spin, but not slip. The dimen-
sion of the state space is 4
+
N
with 2 controls.
We parameterize the configuration by the states of the
automobile plus the angles of each of the trailers with
respect to the horizontal. For consistency we will write
8,
for the angle of the car. Calculation of the constraints
becomes tedious since we have to write the velocity of the
wheels of each trailer, which depend on all previous
trailers. Instead. we choose to use the same inmts as the
i
=
COS
e,u,
4
=
u2
y
=
sin
B,u,
1
.I
8,
=
-
tan
4ul
1
cos(ej-,
-
ej)
sin(8,-,
-
e&,.
1
e+
1
i-1
di
j=1
The filtration corresponding to the
N
trailer problem is
very complex. For small values of
N,
controllability can be
verified directly. For the general case, a very detailed and
well-organized calculation by Laumond [31] shows that
the system is controllable with degree of nonholonomy
N
+
3
and relative growth vector
U
=
(2,
l,...,
11.'
D.
Philip Hall Bases for Lie Algebras
We
will
be interested in the sequel in constructing
nonholonomic systems which are canonical
in
the sense
that they allow for the maximal growth of the filtration
associated with a set of vector fields
A
=
span
{g,,..., g,}.
To construct such systems with a given number of
inputs and degree of nonholonomy, it is necessary to
introduce some additional machinery. In constructing
canonical nonholonomic systems we must observe the
automobile and calculate the effect on the trailer angles.
sum of
two
components: the velocity in the direction the
'Laumond
uses
a slightly different system, obtained by ignoring
Q,
and
steer
I$
independently, controllability for the system given here follows
from
Laumond's result.
At each trailer,
we
can
write the hitch velocity
as
the
choosing
U1
and
U1
tan
'$
as inputs. sine Setting
U1
=
0
alhs
us
to

Citations
More filters
Book
22 Mar 1994
TL;DR: In this paper, the authors present a detailed overview of the history of multifingered hands and dextrous manipulation, and present a mathematical model for steerable and non-driveable hands.
Abstract: INTRODUCTION: Brief History. Multifingered Hands and Dextrous Manipulation. Outline of the Book. Bibliography. RIGID BODY MOTION: Rigid Body Transformations. Rotational Motion in R3. Rigid Motion in R3. Velocity of a Rigid Body. Wrenches and Reciprocal Screws. MANIPULATOR KINEMATICS: Introduction. Forward Kinematics. Inverse Kinematics. The Manipulator Jacobian. Redundant and Parallel Manipulators. ROBOT DYNAMICS AND CONTROL: Introduction. Lagrange's Equations. Dynamics of Open-Chain Manipulators. Lyapunov Stability Theory. Position Control and Trajectory Tracking. Control of Constrained Manipulators. MULTIFINGERED HAND KINEMATICS: Introduction to Grasping. Grasp Statics. Force-Closure. Grasp Planning. Grasp Constraints. Rolling Contact Kinematics. HAND DYNAMICS AND CONTROL: Lagrange's Equations with Constraints. Robot Hand Dynamics. Redundant and Nonmanipulable Robot Systems. Kinematics and Statics of Tendon Actuation. Control of Robot Hands. NONHOLONOMIC BEHAVIOR IN ROBOTIC SYSTEMS: Introduction. Controllability and Frobenius' Theorem. Examples of Nonholonomic Systems. Structure of Nonholonomic Systems. NONHOLONOMIC MOTION PLANNING: Introduction. Steering Model Control Systems Using Sinusoids. General Methods for Steering. Dynamic Finger Repositioning. FUTURE PROSPECTS: Robots in Hazardous Environments. Medical Applications for Multifingered Hands. Robots on a Small Scale: Microrobotics. APPENDICES: Lie Groups and Robot Kinematics. A Mathematica Package for Screw Calculus. Bibliography. Index Each chapter also includes a Summary, Bibliography, and Exercises

6,592 citations

MonographDOI
01 Jan 2006
TL;DR: This coherent and comprehensive book unifies material from several sources, including robotics, control theory, artificial intelligence, and algorithms, into planning under differential constraints that arise when automating the motions of virtually any mechanical system.
Abstract: Planning algorithms are impacting technical disciplines and industries around the world, including robotics, computer-aided design, manufacturing, computer graphics, aerospace applications, drug design, and protein folding. This coherent and comprehensive book unifies material from several sources, including robotics, control theory, artificial intelligence, and algorithms. The treatment is centered on robot motion planning but integrates material on planning in discrete spaces. A major part of the book is devoted to planning under uncertainty, including decision theory, Markov decision processes, and information spaces, which are the “configuration spaces” of all sensor-based planning problems. The last part of the book delves into planning under differential constraints that arise when automating the motions of virtually any mechanical system. Developed from courses taught by the author, the book is intended for students, engineers, and researchers in robotics, artificial intelligence, and control theory as well as computer graphics, algorithms, and computational biology.

6,340 citations


Cites background or methods from "Nonholonomic motion planning: steer..."

  • ...The model given here is adapted from [730]....

    [...]

  • ...The steering method presented in this section is based on initial work by Brockett [144] and a substantial generalization of it by Murray and Sastry [730]....

    [...]

  • ...The presentation given here is based on [730, 848]....

    [...]

  • ...For more details beyond the presentation here, see [599, 728, 730, 848]....

    [...]

  • ...For a proof of the correctness of the second phase, and more information in general, see [730, 848]....

    [...]

Journal ArticleDOI
TL;DR: In this paper, the authors introduce flat systems, which are equivalent to linear ones via a special type of feedback called endogenous feedback, which subsumes the physical properties of a linearizing output and provides another nonlinear extension of Kalman's controllability.
Abstract: We introduce flat systems, which are equivalent to linear ones via a special type of feedback called endogenous. Their physical properties are subsumed by a linearizing output and they might be regarded as providing another nonlinear extension of Kalman's controllability. The distance to flatness is measured by a non-negative integer, the defect. We utilize differential algebra where flatness- and defect are best defined without distinguishing between input, state, output and other variables. Many realistic classes of examples are flat. We treat two popular ones: the crane and the car with n trailers, the motion planning of which is obtained via elementary properties of plane curves. The three non-flat examples, the simple, double and variable length pendulums, are borrowed from non-linear physics. A high frequency control strategy is proposed such that the averaged systems become flat.

3,025 citations


Cites background from "Nonholonomic motion planning: steer..."

  • ...Novel and Lévine 1990, Marttinen et al. 1990) and the car with n trailers (Murray and Sastry 1993, Rouchon et al. 1993a)....

    [...]

  • ...This implies the existence of non-flat systems which verify the strong accessibility property (Sussmann and Jurdjevic 1972)....

    [...]

  • ...The motion planning of the car with n-trailer is perhaps the most popular example of path planning of nonholonomic systems (Laumond 1991, Murray and Sastry 1993, Monaco and Normand-Cyrot 1992, Rouchon et al. 1993a, Tilbury et al. 1993, Martin and Rouchon 1993, Rouchon et al. 1993b)....

    [...]

  • ...The resulting dynamics are described by the following equations (the notations are those of (Murray and Sastry 1993) and summarized on figure 3): ẋ0 = u1 cos θ0 ẏ0 = u1 sin θ0 φ̇ = u2 θ̇0 = u1 d0 tan φ θ̇i = u1 di ( i−1∏ j=1 cos(θj−1 − θj) ) sin(θi−1 − θi) for i = 1, . . . , n (18) where (x0,…...

    [...]

  • ...Steering a car with n trailers is now the object of active researches (Laumond 1991, Murray and Sastry 1993, Monaco and Normand-Cyrot 1992, Rouchon et al. 1993a, Tilbury et al. 1993)....

    [...]

Journal ArticleDOI
TL;DR: In this paper, the authors presented the first randomized approach to kinodynamic planning (also known as trajectory planning or trajectory design), where the task is to determine control inputs to drive a robot from an unknown position to an unknown target.
Abstract: This paper presents the first randomized approach to kinodynamic planning (also known as trajectory planning or trajectory design). The task is to determine control inputs to drive a robot from an ...

2,993 citations


Cites background from "Nonholonomic motion planning: steer..."

  • ...Techniques also exist for general system classes, such as nilpotent (Laffierriere and Sussman 1991), differentially flat (Murray, Rathinam, and Sluis 1995; Fliess et al. 1993), and chained form (Bushnell, Tilbury, and Sastry 1995; Murray and Sastry 1993; Struemper 1997)....

    [...]

  • ...For more complicated kinematic models, nonoptimal steering techniques have been introduced, which include for example a car pulling trailers (Murray and Sastry 1993) and fire trucks (Bushnell, Tilbury, and Sastry 1995)....

    [...]

Journal ArticleDOI
Arie Levant1
TL;DR: In this article, the authors proposed arbitrary-order robust exact differentiators with finite-time convergence, which can be used to keep accurate a given constraint and feature theoretically-infinite-frequency switching.
Abstract: Being a motion on a discontinuity set of a dynamic system, sliding mode is used to keep accurately a given constraint and features theoretically-infinite-frequency switching. Standard sliding modes provide for finite-time convergence, precise keeping of the constraint and robustness with respect to internal and external disturbances. Yet the relative degree of the constraint has to be 1 and a dangerous chattering effect is possible. Higher-order sliding modes preserve or generalize the main properties of the standard sliding mode and remove the above restrictions. r-Sliding mode realization provides for up to the rth order of sliding precision with respect to the sampling interval compared with the first order of the standard sliding mode. Such controllers require higher-order real-time derivatives of the outputs to be available. The lacking information is achieved by means of proposed arbitrary-order robust exact differentiators with finite-time convergence. These differentiators feature optimal asymptot...

2,954 citations


Cites background from "Nonholonomic motion planning: steer..."

  • ...Output-feedback control simulation Consider a simple kinematic model of car control (Murray and Sastry 1993) _x ¼ v cos’; _y ¼ v sin’ _’ ¼ ðv=lÞ tan _ ¼ u where x and y are Cartesian coordinates of the rear-axle middle point, ’ is the orientation angle, v is the longitudinal velocity, l is the…...

    [...]

  • ...Consider a simple kinematic model of car control ( Murray and Sastry 1993 )...

    [...]

References
More filters
Book
01 Jan 1974
TL;DR: In this paper, Newtonian mechanics: experimental facts investigation of the equations of motion, variational principles Lagrangian mechanics on manifolds oscillations rigid bodies, differential forms symplectic manifolds canonical formalism introduction to pertubation theory.
Abstract: Part 1 Newtonian mechanics: experimental facts investigation of the equations of motion. Part 2 Lagrangian mechanics: variational principles Lagrangian mechanics on manifolds oscillations rigid bodies. Part 3 Hamiltonian mechanics: differential forms symplectic manifolds canonical formalism introduction to pertubation theory.

11,008 citations

Book
01 Jan 1985
TL;DR: In this paper, a systematic feedback design theory for solving the problems of asymptotic tracking and disturbance rejection for linear distributed parameter systems is presented, which is intended to support the development of flight controllers for increasing the high angle of attack or high agility capabilities of existing and future generations of aircraft.
Abstract: : The principal goal of this three years research effort was to enhance the research base which would support efforts to systematically control, or take advantage of, dominant nonlinear or distributed parameter effects in the evolution of complex dynamical systems. Such an enhancement is intended to support the development of flight controllers for increasing the high angle of attack or high agility capabilities of existing and future generations of aircraft and missiles. The principal investigating team has succeeded in the development of a systematic methodology for designing feedback control laws solving the problems of asymptotic tracking and disturbance rejection for nonlinear systems with unknown, or uncertain, real parameters. Another successful research project was the development of a systematic feedback design theory for solving the problems of asymptotic tracking and disturbance rejection for linear distributed parameter systems. The technical details which needed to be overcome are discussed more fully in this final report.

8,525 citations

Book
01 Jan 1990
TL;DR: This chapter discusses the configuration space of a Rigid Object, the challenges of dealing with uncertainty, and potential field methods for solving these problems.
Abstract: 1 Introduction and Overview.- 2 Configuration Space of a Rigid Object.- 3 Obstacles in Configuration Space.- 4 Roadmap Methods.- 5 Exact Cell Decomposition.- 6 Approximate Cell Decomposition.- 7 Potential Field Methods.- 8 Multiple Moving Objects.- 9 Kinematic Constraints.- 10 Dealing with Uncertainty.- 11 Movable Objects.- Prospects.- Appendix A Basic Mathematics.- Appendix B Computational Complexity.- Appendix C Graph Searching.- Appendix D Sweep-Line Algorithm.- References.

6,186 citations


"Nonholonomic motion planning: steer..." refers background in this paper

  • ...This has resulted in a rather complete understanding of the complexity of the computational effort required to plan the trajectories of robots to avoid both fixed and moving obstacles [8, 25, 19]....

    [...]

Book
01 Dec 1979
TL;DR: Spivak's comprehensive introduction to differential geometry as discussed by the authors takes as its theme the classical roots of contemporary differential geometry, and explains why it is absurdly inefficient to eschew the modern language of manifolds, bundles, forms, etc., which was developed precisely to rigorize the concepts of classical differential geometry.
Abstract: Spivak's Comprehensive introduction takes as its theme the classical roots of contemporary differential geometry. Spivak explains his Main Premise (my term) as follows: "in order for an introduction to differential geometry to expose the geometric aspect of the subject, an historical approach is necessary; there is no point in introducing the curvature tensor without explaining how it was invented and what it has to do with curvature". His second premise concerns the manner in which the historical material should be presented: "it is absurdly inefficient to eschew the modern language of manifolds, bundles, forms, etc., which was developed precisely in order to rigorize the concepts of classical differential geometry". Here, Spivak is addressing "a dilemma which confronts anyone intent on penetrating the mysteries of differential geometry". On the one hand, the subject is an old one, dating, as we know it, from the works of Gauss and Riemann, and possessing a rich classical literature. On the other hand, the rigorous and systematic formulations in current use were established relatively recently, after topological techniques had been sufficiently well developed to provide a base for an abstract global theory; the coordinate-free geometric methods of E. Cartan were also a major source. Furthermore, the viewpoint of global structure theory now dominates the subject, whereas differential geometers were traditionally more concerned with the local study of geometric objects. Thus it is possible and not uncommon for a modern geometric education to leave the subject's classical origins obscure. Such an approach can offer the great advantages of elegance, efficiency, and direct access to the most active areas of modern research. At the same time, it may strike the student as being frustratingly incomplete. As Spivak remarks, "ignorance of the roots of the subject has its price-no one denies that modern formulations are clear, elegant and precise; it's just that it's impossible to comprehend how any one ever thought of them." While Spivak's impulse to mediate between the past and the present is a natural one and is by no means unique, his undertaking is remarkable for its ambitious scope. Acting on its second premise, the Comprehensive introduction opens with an introduction to differentiable manifolds; the remaining four volumes are devoted to a geometric odyssey which starts with Gauss and Riemann, and ends with the Gauss-Bonnet-Chern Theorem and characteristic classes. A formidable assortment of topics is included along the way, in which we may distinguish several major historical themes: In the first place, the origins of fundamental geometric concepts are investigated carefully. As just one example, Riemannian sectional curvature is introduced by a translation and close exposition of the text of Riemann's remarkable paper, Über die Hypothesen, welche der Geometrie zu Grunde

3,840 citations

Book
01 Jul 1990
TL;DR: This paper reformulated the manipulator control problem as direct control of manipulator motion in operational space-the space in which the task is originally described-rather than as control of the task's corresponding joint space motion obtained only after geometric and kinematic transformation.
Abstract: This paper presents a unique real-time obstacle avoidance approach for manipulators and mobile robots based on the artificial potential field concept. Collision avoidance, tradi tionally considered a high level planning problem, can be effectively distributed between different levels of control, al lowing real-time robot operations in a complex environment. This method has been extended to moving obstacles by using a time-varying artificial patential field. We have applied this obstacle avoidance scheme to robot arm mechanisms and have used a new approach to the general problem of real-time manipulator control. We reformulated the manipulator con trol problem as direct control of manipulator motion in oper ational space—the space in which the task is originally described—rather than as control of the task's corresponding joint space motion obtained only after geometric and kine matic transformation. Outside the obstacles' regions of influ ence, we caused the end effector to move in a straight line with an...

3,063 citations


"Nonholonomic motion planning: steer..." refers background in this paper

  • ...Other approaches include the use of potential functions for navigating in cluttered envi ronments [22, 21] and compliant motion planning for navigating in the presence of uncertainty [10, 11, 34]....

    [...]