A Harmonic Potential Field Approach for Navigating a Rigid, Nonholonomic Robot in a Cluttered Environment

Abstract

This paper demonstrates the ability of the harmonic
potential field (HPF) planning approach to generate a
provably-correct, constrained, well-behaved trajectory for a
rigid, nonholonomic robot (a tractor-trailer robot is not rigid)
in a stationary, cluttered environment. This is accomplished
using a closed loop control scheme that is inspired by model
predictive control (MPC). The scheme is realized using a
synchronizing signal derived from the HPF along with a
procedure for inverting the process the robot is using for
actuating motion. Performance proofs as well as simulation
results of the suggested planner are supplied.

Full text

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A Harmonic Potential Field Approach for Navigating a
Rigid, Nonholonomic Robot in a Cluttered Environment
Ahmad A. Masoud
Electrical Engineering Department, KFUPM, P.O. Box 287, Dhaharan 31261, Saudi Arabia, masoud@kfupm.edu.sa

Abstract: This paper demonstrates the ability of the harmonic
potential field (HPF) planning approach to generate a
provably-correct, constrained, well-behaved trajectory for a
rigid, nonholonomic robot (a tractor-trailer robot is not rigid)
in a stationary, cluttered environment. This is accomplished
using a closed loop control scheme that is inspired by model
predictive control (MPC). The scheme is realized using a
synchronizing signal derived from the HPF along with a
procedure for inverting the process the robot is using for
actuating motion. Performance proofs as well as simulation
results of the suggested planner are supplied.
I. Introduction and Background
A planner is an interface between an operator and a servo
process whose function is to interpret the commands and
constraints on the process within the confines of the
environment which the process is situated in. Despite the
diversity of planning methods [1,2] they may all be divided
into two classes: a class that separates a planner into two
modules one called the high level controller (HLC) and the
other is called the low level controller (LLC). The first is
responsible for converting the command, constraints and
environment feed into a desired behavior which the process
must find a way to actualize if the task is to be accomplished
(a know-what-to-do guidance signal). On the other hand, the
second module determines what actions the process actuators
of motion should release in order to actualize the desired
behavior (a know-how-to-do control signal). Although this
division of role in building planners is widely accepted by
researchers in the area, it is believed to be a source of several
problems. It is well-known in practice that processes using the
HLC-LLC paradigm are relatively slow. Incompatibilities
between the guidance and control signals could lead to
unwanted artifacts in the behavior and undesirable control
effort that consumes too much energy or put too much strain
on the actuators. Jointly designing the guidance and control
modules is expected to yield a simpler and more efficient
planner compared to a design that treats the two modules
separately.
Simultaneous consideration of the guidance and control signals
in the design of a planner is a challenging task. While limited
success was achieved in designing controllers that can
incorporate simple avoidance regions with convex geometry
in state-space [3,4], imposing general, nonconvex avoidance
regions in the state-space of a dynamical system is difficult
[5,6]. The task is further complicated when state-space
constraints have to be implemented along with constraints in
the control space as is the case with dynamical, nonholonomic
systems.
Instead of using the relatively simple, two-tier approach to
planner design or the excessively complex joint state-space
control space approach, an approach in the middle is adopted.
Here the capabilities of a carefully selected planner that can
only generate a guidance signal (i.e. deals only with the
kinematic aspects of motion) are augmented to generate also
the needed control signal. The guidance field from the
kinematic planner is left unchanged. However, instead of the
control component of the planner being designed to enforce
strict compliance of motion with the guidance field, we only
require that the control component strongly discourages
motion from deviating from the course set by the guidance
field (effective compliance with guidance).
As far as this work is concerned, the extremely rich variety of
kinematic motion planners may be categorized in one of two
classes: path tracking planners and goal seeking planners. A
path tracking planner provides a sequence of guidance
instructions that mark one and only one path from an initial
state to a target state. If an unexpected event occur throwing
the state away from the guidance path, it must find its way
back to the path in order to proceed to the target. On the other
hand, a goal seeking planner supplies a guidance instruction at
every possible state the system may exist in. Therefore, a
disruption caused by an influence external to the system will
not cause a halt in the effort to drive the state closer to the
target.
The HPF approach is an excellent goal-seeking planner. It
works by inducing, using a potential field, a dense collective
of guidance vectors on the admissible space of the robot (S).
A group structure is then evolved on this collective to generate
a macro template encoding the guidance information the
process needs to execute. The action selection mechanism the
approach utilizes for generating the structure is in conformity
with the artificial life (AL) method [7]. The HPF approach
offers a solution to the local minima problem faced by the
potential field approach Khatib suggested in [8]. It was
simultaneously and independently proposed by several
researchers [9-12] of whom the work of Sato in 1987 may be
regarded as the first on the subject [13]. An HPF is generated
using a Laplace boundary value problem (BVP) configured
using a properly chosen set of boundary conditions. There are
several settings one may use for a Laplce BVP (LBVP) in
order to generate a navigation potential [14-16]. Each one of
these settings possesses its own, distinct, topological
properties [12]. An example is shown below of an LBVP that
is configured using the homogeneous Neumann boundary
conditions and encodes region avoidance constraints and target
location:
L2V(X)/0 X0S (1)
subject to: V(XS) = 1, V(XT) = 0 , and at X = ',
∂
∂
V
n
= 0
where S is the workspace, ' is its boundary, n is a unit vector
normal to ', Xs is the start point, and XT is the target point.
The trajectory to the target (X(t)) is generated using the HPF-

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based, gradient dynamical system:

The trajectory is guaranteed to:
(2)
?XV(X)X(0)X0
= −∇= ∈ Ω
1-
Below is also a BVP similar to (1) that adds motion arrival
orientation to the target to the set of encoded features:
L2V(X)/0
subject to: V(XS) = 1, V(XT) = 0, V(XT+,Ah)=1 , and
∂
∂
n
the desired target direction..
2-
limX(t)
t
→∞
X
T
→
X(t) ∈∀Ω
t
X0S (3)
at X = ', where 1>>,>0 and h is a unit vector in
V
= 0
Harmonic functions have many useful properties[17] for
motion planning. Most notably, a harmonic potential is also a
Morse function [21] and a general form of the navigation
function suggested in [18]. The HPF approach may be
configured to operate in a model-based and/or sensor-based
mode. It can also be made to accommodate a variety of
differential and state constraints [16]. It ought to be mentioned
that the HPF approach is only a special case of a much larger
class of planners called: evolutionary, pde-ode motion
planners [14], figure-1.

Figure-1: Block diagram of an evolutionary PDE-ODE planner.
Figures-2 shows the guidance fields and paths generated by a
special type of HPF planners [16] called nonlinear,
anisotropic HPF planner (NAHPF). In addition to enforcing
regional avoidance constraints, NAHPF planners can also
enforce directional constraints in S.

Figure-2: Output from a directional sensitive, kinematic, HPF planner.
Up untill now HPF planners can only deal with holonomic
robots. In general, extending a holonomic planner to work
under nonholonomic constraints is not always possible.
However, due to the properties HPF planners enjoy, the
situation is different. A planner for a nonholonomic robot
whose points satisfy the rigidity constraints may be
constructed by utilizing the gradient guidance field from an
HPF as a motivator of motion. This requires that the
nonholonomic robot be described using a two-stage model
(figure-3). The first stage models the manner in which the
robot converts the control variables used to actuate motion in
its local coordinates. The second stage is concerned with
transforming the motion from the local coordinates into one
that is global coordinates-centered (namely, position and
orientation).

ν
ω
⎡
⎣⎢
⎤
⎦⎥
u
u
1
2
⎡
⎣⎢
⎤
⎦⎥
?
?
?
x
y
θ
⎣
⎡
⎢
⎢
⎢
⎤
⎥
⎥
⎥
⎦
Figure-3: Two-stage model of a rigid, nonholonomic robot.
The nonholonomic planner is constructed as follows: at each
point in S (Xi) a reference motion dXri(Xi)/dt is selected as
the negative gradient of the HPF. It is required that the robot’s
motion in its local coordinates be equal to the reference
motion. To achieve this, an inverse of the motion actuation
stage of the robot is applied to the field of reference
motions. The field of reference motions marks the solution
trajectories which the robot can proceed along to the target.
The inverse process, in effect, attaches to each solution
trajectory a dense sequence of control vectors. Due to
limitations on the inversion process and (or) the initial state the
robot is in, the robot will not remain on the solution trajectory
it is currently situated at and will transit to a new one. In a
manner similar to MPC, [22] the robot will only use the first
control instruction in the sequence associated with a solution
trajectory and discard the remaining ones. This is repeated till
the robot finally reaches its target (figure-4).

X1
X2
X3
X4
X5
Xi
?(X X
i
)
i
?
()() Xr X
i
V X
ii
= −∇
Figure-4: The MPC paradigm for the nonholonomic planner.
It is shown in the sequel that the above paradigm does provide
good basis for building a provably-correct nonholonomic
motion planner for rigid robots. It ought to be mentioned that
the nature of the paradigm does not limit the construction of
planners for planar robots and makes it possible to deal with
three dimensional even N-D spaces.
This paper is organized as follows: in section II the suggested
HPF-based, nonholonomic controller for the massless robot
case is presented. Section III provides performance analysis
for the massless controller. Section IV suggests an extension
to the case where the robot has second order dynamics.
Simulation results are in section V and conclusions are placed
in section VI.

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II. The Kinematic Planner
Adapting the above approach to planning hinges on the ability
to find a realization that can make the nonholonomic path of
the robot as close as possible to the holonomic path generated
by the gradient of an HPF. Achieving this, ensures that all the
provably-correct properties of a path generated by a
holonomic HPF planner are migrated to the corresponding
nonholonomic path. A realization that can accomplish the
above has two stages (figure-5): a stage that generates an HPF-
based, synchronizing signal that attempts to align the velocity
vector of the robot with the reference velocity vector selected
as the negative gradient of the HPF. This synchronizing signal
has the form:
ν
θ
∆
−∇
⎣
arg(
where " is a non-negative integer, 2 is the orientation of the
robot in its world coordinates, )2 is the difference between the
orientation of the robot and the desired one and <r is the
reference radial speed the robot is required to assume. The
second stage operates on the synchronizing signal in (4) with
an operator that attempts to invert the motion actuation process
of the robot in order to yield the control signal u=[u1 u2]T:
u
u
2
θ∆
where F is an inverse operator (linear or nonlinear) derived
from the motion actuation stage of the robot’s model.

(4)
θ
θ
α
r
⎡
⎣⎢
⎤
⎦⎥=
− ∇⋅−
V
−∇
−
⎡
⎢
⎤
⎥
⎦
V arg
)
Vcos (( ))
(5)
F
1
⎡
⎣⎢
⎤
⎦⎥=
⎡
⎣⎢
⎤
⎦⎥
()
ν
ν
∆
θ
r
⎡
⎣⎢
⎤
⎦⎥
u
u
1
2
⎡
⎣⎢
⎤
⎦⎥
Figure-5: the nonholonomic planner - The massless case.
Below are the two-stage models and the inverse operator for
two popular mobile robots, the differential drive robot and the
front wheel steered car (the slow steering case):
1- The differential drive robot (DDR) (figure-6):

Figure-6: A differential drive mobile robot.

The equations describing motion for such a robot are:
?
?
?
0
θ
⎣
⎢
⎦
⎥
⎣
⎢
(6)
( )
( )
θ
x
y
cos
sin
0
0
1
θ
ν
ω
⎡
⎢
⎢
⎤
⎥
⎥
=
⎡
⎢
⎢
⎤
⎥
⎥
⎥
⎦
⎡
⎣⎢
⎤
⎦⎥
and (7)
ν
ω
ω
ω
⎡
⎣⎢
⎤
⎦⎥=
−
W
⎡
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎦
⎡
⎣⎢
⎤
⎦⎥
r
2
r
r
2
W
r
R
L
where r is the radius of the robot’s wheels, W is the width of
the robot, TR and TL are the angular speeds of the right and
left wheels of the robot respectively. The inverse operator is:
1
r
1
r
⎣
2-The Front wheel Steered Robot (FSR) (figure-7):

(8)
ω
ω
ν
∆
θ
R
L
W
2r
W
2r
⎡
⎣⎢
⎤
⎦⎥=
−
⎡
⎢
⎢
⎢
⎤
⎥
⎥
⎥
⎦
⎡
⎣⎢
⎤
⎦⎥
r
Figure-7: A front wheel-steered mobile robot.

The equations describing motion for this robot are:
?
?
?
01
θ
⎣⎦
⎣⎦
, (9)
( )
( )
θ
x
y
cos
sin
0
0
θ
ν
ω
⎡
⎢
⎢
⎢
⎤
⎥
⎥
⎥
=
⎡
⎢
⎢
⎢
⎤
⎥
⎥
⎥
⎡
⎣⎢
⎤
⎦⎥
ν
ω
ω
⋅
h
ωϕ
( )
⎡
⎣⎢
⎤
⎦⎥=
⋅
⋅⋅
⎡
⎣⎢
⎤
⎦⎥
r
r Ltan
h
where L is the normal distance between the center of the front
wheel and the line connecting the rear wheels, Th is angular
speed of the rear wheels, and N is the steering angle of the
front wheel (B/2>N>-B/2). The inverse operator is:
ω
φ
tan
In the above cases, it was possible to perfectly invert the
actuation stage. If this is not possible, the pseudo inverse
approach may be used to construct the inversion operator F.
The equation of motion for many nonholonomic mobile
robots may be written as:
?
?
?
θ
⎣
⎢
⎦
⎥
where G is a nonlinear vector function. At a certain (x,y)
point in space, equation-11 may be linearized at the current
operating condition and the motion of the robot described as:
?
?
?
01
θ
⎣
⎢
⎦
⎥
⎣
⎢
⎦
⎥
ν
ω
where A need not necessarily be full rank. In this case the
inverse operator may be constructed as:

u
2
where A+ is the pseudo inverse of A [23] and A is derived
from G.
. (10)
ν
θν
hr
r
⎡
⎣⎢
⎤
⎦⎥=
⋅
⎡
⎣⎢
⎤
⎦⎥
−
L
1(/ ()
∆
(11)
( , , , , )
G x y
θ ν ω
x
y
⎡
⎢
⎢
⎤
⎥
⎥
=
(12)
( )
( )
θ
x
y
cos
sin
0
0
θ
ν
ω
⎡
⎢
⎢
⎤
⎥
⎥
=
⎡
⎢
⎢
⎤
⎥
⎥
⎡
⎣⎢
⎤
⎦⎥
and , (13)
θ
⎡
⎣⎢
⎤
⎦⎥=
⎡
⎣⎢
⎤
⎦⎥
A x y( , , )
u
u
1
2
(14)
u
1
⎡
⎣⎢
⎤
⎦⎥=
⎡
⎣⎢
⎤
⎦⎥
+
A x y( , , )
r
θ
θ
ν
∆

ν
∆
θ
r
⎡
⎣⎢
⎤
⎦⎥
u
u
1
2
⎡
⎣⎢
⎤
⎦⎥
Figure-8: The close loop, HPF-based, nonholonomic system.

Page 4

III. Performance Analysis
In this section two properties of the above controller are
proven. It is shown that the closed loop system in figure-8 is
stable for a general rigid nonholonomic robot. For the specific
cases of the differential drive and front wheel steered robots,
where perfect actuator inversion is possible, convergence to
the target position and orientation is guaranteed. It is also
proven that the trajectory of the robot can be made arbitrarily
close to the trajectory laid by the gradient dynamical system
directly derived from the HPF. This proves that the robot
trajectory also satisfies all the provably-correct properties of
a holonomic HPF path.

The proofs of closed loop stability assume that the pseudo
inverse is used in constructing the inverse operator of the
actuators. Proofs for this case subsumes the proofs for the
cases where actuators inversion is perfect.

Proposition-1:A matrix P constructed as the product of a
matrix A by its pseudo inverse (P=A+A) is positive semi-
definite (A is not full-rank).

Proof: by definition the pseudo inverse of A is:

A lim A AIA
0
→
δ
Let Q=ATA, and Z=*AI. Since Q is symmetric and Z is
positive definite, they may be jointly diagonalizable [24] into:
Q=UT7U and Z=UTU,
where U is an orthonormal matrix and 7 is a diagonal matrix
containing the eigenvalues of Q (8i). Substituting (16) into
(15) in order to compute P we have:
P = lim U UU
0
δ →
=
lim UI
0
δ→
= lim UI
0
δ →
λ
λ
λ
λ
→
⋅⋅
⋅
⎣
Since Q is constructed as the product of a matrix by its
transpose, its eigenvalues are non-negative. Therefore, the
eigenvalues of P lie in the interval [0,1), i.e. they are non-
negative. Therefore P is positive semi-definite.

Proposition-2: The closed loop system constructed by using
the control law in (14) with the system in (12,13) is stable. If
A is full rank, then the robot will converge to the target
position and orientation.

Proof: Consider the Liapunov function candidate:
1
2(
note that the HPF V is a Liapunov function that is positive
everywhere in S except at x=xT and y=yT where it is equal to
zero[16]. The derivative of = with respect to time is:
?
?
y
Manipulating (12), (13) and (14) we obtain:
?
?
?
01
θ
⎣
⎢
⎦
⎥
⎣
⎢
⎦
⎥
where P=AA+. Substituting (20) in (19) we get:
. (15)
limA
0
→
δ
A AI
T1TTT1
+−−
=+⋅=+⋅δδ
()()
(16)
UUU
TT1T
−
+
()
ΛΛ
(17)
UUU
1TT
−−−
+
[( ) ( ) () ]
11
ΛΛ
U
1
−
+
[()]
T
ΛΛ
= .
lim U
0
δ
1
00
0
1
0
00
1
U
1
1
2
2
N
λ
N
λ
+
+
⋅
+
⋅
⋅⋅
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
[]
T
(18)
Ξ∆=+
V(x,y))
θ
2
(19)
?
Ξ
?
∆= ∇
⎡
⎣⎢
⎤
⎦⎥−
⋅
Vx
θ θ
(20)
x
y
cos( )
sin( )
0
0 Pcos ( )
⎣
V
θ
θ
θ
∆
θ
α
⎡
⎢
⎢
⎤
⎥
⎥
=
⎡
⎢
⎢
⎤
⎥
⎥
− ∇
⎡
⎢
⎤
⎥
⎦
. (21)
[]
?
Ξ
( )
( )
θ
()
()
∆
∆
∆
∆
∆
= ∇
⎡
⎣⎢
⎤
⎦⎥
− ∇
⎡
⎢
⎣
⎤
⎥
⎦
−
− ∇
⎡
⎢
⎣
⎤
⎥
⎦
Vcos
sin
0
0Pcos
V
0 Pcos
V
θ
θ
θ
θ
θ
θ
α
α
The gradient of V may be expressed as:
⎡
⎣⎢
sin arg
substituting (22) into (21) we have:

[
Ξ
= − ∇⋅
V cos
(22)
∇= ∇⋅
−∇
−∇
+
+
⎤
⎦⎥
VV
cos arg
(
(
V
V
(
(
)
)
)
)
π
π
(23)
]
[]
?
()
()
()
∆
∆
∆
∆
∆
∆
− ∇
⎡
⎣⎢
⎤
⎦⎥
−
− ∇
⎡
⎣⎢
⎤
⎦⎥
0 Pcos
V
0Pcos
V
θ
θ
θ
θ
θ
θ
α
α
or ,
[]
?
Ξ
()
()
∆∆
∆
∆
= − ∇⋅
− ∇
⎡
⎢
⎣
θ
⎤
⎥
⎦
V cosScos
V
θθ
θ
θ
α
where . (24)
SPcos
⎣
0
10
1
=
⎡
⎢
⎤
⎥
⎦
−α
()
∆
There are two cases one for " even and the other for " odd. If
" is odd, cos"-1()2) is non-negative for any value of )2. Since
P is positive semidefinite, it can be shown by direct
computation of the eigenvalues of S that the eigen values of
S are non-negative provided that P is positive semidefinite. In
other words, S is also positive semidefinite. For the case where
perfect inversion of the motion actuation process is possible
(i.e. P=I), (24) is zero at *LV*=0, )2=0 and )2=B/2.
Equation (4) may be used to compute the minimum invariant
set of the system: *LV*=0, )2=0 to which the robot will
converge. Since it is proven that an HPF is Morse [21]
convergence of *LV* to zero implies convergence of x and y
to xT and yT. Also convergence of )2 to zero implies that the
robot will converge to the orientation encoded by the HPF at
xT and yT. For the case where actuator inversion is not perfect
(i.e. P…I), convergence analysis will depend on the structure
of P. However, from the above it can be easily sown that this
will only cause deadlock. For the case where " is even, the
sign of cos "-1()2) is negative for* )2*>B/2. However from
(4) it can be easily shown that :

limt arg
t→∞
In other words, )26 0 and S is negative semidefinite.

Proposition-3: Let D be the spatial projection of the trajectory
X(t) laid by the gradient dynamical system in (2). Also let Dn
be the spatial projection of the trajectory laid by the suggested
nonholonomic planner (figure-9). Let *(t) be the deviation
between D and Dn at time t. Let *m be the maximum deviation.
If the motion actuation process is fully invertible, then by
setting " high enough, *m may be made arbitrarily small,
lim
m
α →∞

(25)
V
→ −∇θ( )( ).
0
δ→
.
Figure-9: deviation between the HPF trajectory and the
nonholonomic trajectory.

Page 5

Proof: Let e be a unit vector normal to -LV
. (26)
e
sin arg
cos arg
−
V
V
=
−∇
−∇
(
⎡
⎣⎢
⎤
⎦⎥
(( ))
( ))
The rate of change of *(t) may be computed as:
?
?
y
sin
= − − ∇⋅
V sin(
∆θ
The maximum of the above is achieved at:
(27)
?
( )
( )
θ
()
δ
θ
θ
θ
∆
α
=
⎡
⎣⎢
⎤
⎦⎥=
⎡
⎣⎢
⎤
⎦⎥
∆
− ∇
⎡
⎢
⎣
⎤
⎥
⎦
e
x
e
cos0
0
cosV
TT
∆
cos)().
θ
α
‘ (28)
∆θ
α
+
α
=
−
cos
1
1()
Since V is harmonic, *LV*#Cm
where Cm is a finite positive constant [25]. A upper bound on
d*/dt (*dm) may be constructed as:

δ
dC
1
(
As can be seen, the rate of change in * tends to zero as " goes
to infinity. Since the system is stable, we have:
δδ
( ),( )00 lim0
t
→∞
x,y 0 S (29)
(30)
α
α
+
α
α
mm1
≤⋅
+
)
. (31)
δ
?( )
∞ →δ
?( )
,
,
0
lim
t
→∞
0 andt dt0
0
=→
=
∞
∫
Since (27 ) satisfies the Lipschitz conditon and its convergence
to zero is independent of " and depends mainly on (4), an
upper bound on *m may be constructed as follow:
≥
+≥
−≥>
>
⎣
0t
where T is the effective time d*/dt converges to zero. Using
the above the maximum deviation may be bounded as:
*m # *dm@T/2
which also tends to zero as " becomes very large.
, (32)
?
/
/
δ
δ
δ
=
>
⎡
⎢
⎢
⎢
⎢
00t
t
T 2
T
d
d
T 2
T
0
t
m
m
(33)
Proving that the deviation between the trajectory generated
directly from the gradient of the HPF and the nonholonomic
path may be driven to zero. This in turn implies that the
nonholonomic path inherits all the provably-correct properties
of the gradient, this includes obstacle avoidance.

IV. A Suggested Extension: The Kinodynamic Case
The dynamic behavior of the differential drive robot that ties
the torques applied to the right and left wheels (TR, TL) to the
position and orientation of the robot may be described using
two, coupled differential equations. The first one is obtained
by differentiating (6) with respect to time,
??
??
??
01
θ
⎣
⎢
⎦
⎥
⎣
⎢
⎦
⎥
and the second is derived using Lagrange dynamics in the
natural coordinates of the robot,
⎡
⎢
⎢
⎢
⎦
WW
Where M is the mass of the robot. As demonstrated by
simulation (figure-15), using the control scheme for a massless
robot with robots that have mass will cause instability. To
stabilize the system an omni-directional, linear viscous
dampening force applied in the natural coordinates of the robot
is used to generate the control signal:
T
T
arg
L
−∇
⎣
⎣
⎢
(

, (34)
( )
( )
θ
?
?
( )?
( )?
0
x
y
cos
sin
0
0
sin
cos
0
0
0
θ
ν
ω
θ θ
θ θ
ν
ω
⎡
⎢
⎢
⎤
⎥
⎥
=
⎡
⎢
⎢
⎤
⎥
⎥
⎡
⎣⎢
⎤
⎦⎥+
−
⎡
⎢
⎢
⎢
⎣
⎤
⎥
⎥
⎥
⎦
⎡
⎣⎢
⎤
⎦⎥
(35)
?
?
ν
ω
⎡
⎣⎢
⎤
⎦⎥=
− ⋅
4 r
⋅
⎣
⎤
⎥
⎥
⎥
⎡
⎣⎢
⎤
⎦⎥=
⋅⎡
⎣⎢
⎤
⎦⎥
1
M
1
r
1
r
4 r
T
T
B
T
T
33
R
L
R
L
, (36)
BK
Varg
V
V
K
R
Pd
⎡
⎣⎢
⎤
⎦⎥=
⋅⋅
− ∇⋅ −∇
−
−
⎡
⎢
⎤
⎥−
⎦
⋅⎡
⎣⎢
⎤
⎦⎥
⎡
⎢
⎤
⎥
⎥
⎦
−1
cos ()
α
θ
θ
ρ
θ
?
(
)
)
?
where KP and KD are positive constants, B-1 is the inverse of
B, and is the radial speed of the robot,
? ρ
ρ =+
xy
The block diagram of the planner is shown in figure-10.

. (37)
?
??
22
(?,?)
ρ θ
Figure-10: A dynamic, HPF-based planner with linear dampening,
nonholonomic case.
Rate feedback in the natural coordinates of the robot is needed
to stabilize the response and make the system yield to the
guidance signal derived from the HPF. Significant transients
are expected for a small coefficient of rate feedback. Although
increasing this coefficient reduces the transients, it results in
reducing the speed of the robot. One way to cope with this
problem is to sensitize the dampening to the guidance signal
is to notice that changing the speed of the robot is not needed
if the actual speed of the system is equal to the reference
speed. This is realized using the control signal:
T
T
L
⎣
⎡
⎣⎢
(
θ
It was proven in [16] that the gradient dynamical system in (2)
which is constructed from an underlying harmonic potential
guarantees convergence from any point in S to a specified
target point. The proof makes use of the fact that a harmonic
potential is also a Liapunov function candidate. The following
proposition shows that the procedure suggested in (38) makes
it possible for the dynamical system in (2) to steer a
differential drive robot with second order dynamics from any
initial position and orientation in S to the target position and
orientation encoded in the harmonic field V. A variant of
Liapunov method, called the LaSalle invariance principle [26],
is used in the proof.

Propositon-4: The control law in (38) applied to a differential
drive robot with second order dynamics described by the
system equation in (34,35) guarantees global asymptotic
convergence of the robot from any initial position and
orientation in S to the target potion point (xT,yT) and
orientation (arg(-LV (xT,yT)) encoded in the harmonic potential
V provided that Kp>0 and Kd>0.

Proof: consider the following Liapunov function candidate:
1
2K
1
2I
where M is the mass of the robot, I is its inertia , Kp and Kd are
positive constants. Notice that V(x,y) is a valid liapunov
function [16]. It is always positive except at the target point
(xT,yT) where it is equal to zero. As a result = is always
positive except at the target position and orientation when the
robot is at a standstill. The time derivative of = is:

?
?
? ??
? ??
+ ⋅ ⋅+⋅⋅
IM
θ θρ ρ
(38)
BK
[
V cos
⋅
arg
0
V
K
argV
R
P
d
⎡
⎣⎢
⎤
⎦⎥=
⋅⋅
− ∇−∇−
⎡
⎢
⎤
⎥
⎦
−⋅
−−∇−
⎤
⎦⎥
−1
(())
?
?
() )]
α
θ
ρ
θ
(39)
Ξ =⋅⋅+ ⋅ ⋅
I (arg(
−∇−
+⋅+⋅
KM V(x,y)
V(x,y))
1
2M
Pd
2
θ
θ
?
ρ
?
22
(40)
?
x
y
?(
θ
())
Ξ =⋅⋅∇
⎡
⎣⎢
⎤
⎦⎥−
⋅ ⋅ ⋅
I
−∇−
KMV(x,y)KV(x,y)
P
T
d
arg
θ

Page 6

Notice that: (41)
∇∇
−∇
−∇
+
+
⎡
⎣⎢
⎤
⎦⎥
V(x,y) = V(x,y)
V(x,y)
V(x,y)
cos(
sin(
⎤
⎦⎥
(
(
)
)
)
)
arg
arg
π
π
and . (42)
?
?
y
?cos( )
sin( )
x
⎡
⎣⎢
⎤
⎦⎥=
⎡
⎣⎢
ρ
θ
θ
Substituting (35), (38), (41) and (42) in (40) and noticing that
for a differential drive robot B+=B-1, we have:
?
?
cos(
?((
?
− ⋅ ⋅
−⋅
+ ⋅ ⋅ ⋅ −∇
+⋅⋅ ⋅ ∇⋅
KM V(x,y)
P
ρ
Therefore:
?
?
Ξ = − ⋅ ⋅
KI
d
θ
As can be seen the time derivative of the Liapunov function is
negative semi-definite. According to LaSalle principle motion
will converge to a subset of the set of points (E) for which the
time derivative of = is zero:

E 0,
===
{?
?
ρθ
The subset is called the minimum invariant set (S) and may be
computed as the set of point for which the gradient dynamical
system in (2). It was shown in [16] that motion for (2) is
guaranteed to converge to the target point xT, yT , hence the
orientation of the robot will converge to arg(-LV(xT, yT )). In
other words the dynamical differential drive robot will
converge to set:

S 0,0, x = x , y = y ,
T
===
{?
ρθ
provided that Kp and Kd are positive.

V. Simulation Results
The suggested controller is tested for the massless case using
the gradient guidance field in figure-11. This field encodes the
simple behavior of move right and stay at the center.

(43)
())
))
?
?(
θ
())
(
?
cos())
Ξ = −⋅⋅ ⋅ ∇⋅ −∇−
− ⋅ ⋅ ⋅
I
−∇−
−
arg
−
−∇−
KM V(x,y)V(x,y)
KV(x,y)
KI
K
KI V(x,y)
V(x,y)
?
⋅ρ
P
d
d
P
d
ρθ
θ
θ
ρ
θ
θ
θ
arg
arg
M
arg
2
2
. (44)
K
P
22
M
. (45)
0, x, y,}
θ
(46)
V x , y
−∇
(
TTT
=
?
arg( )}
θ
Figure-11: Move right and stay at center gradient guidance field.
The trajectories obtained for different values of " are shown
in figure-12. The simulation is carried out for both the
differential drive robot and the front wheel-steered robot. The
time step )T=.01 second and the total duration of the
stimulation is 6 seconds. The trajectories obtained for both
robots are identical.

Figure-12: Trajectories from the nonholonomic, kinematic, HPF planner.
The control signal for both robots for the case 2(0)=B/2 and
"=9 are shown in figures 13,14 below.



Figure-13: control signal, DDR, "=9.
Figure-14: control signal, FSR, "=9.
The controller designed for the massless case is used with a
differential drive robot with a mass M=1 added for 2(0)=B/2.
As expected direct use of the guidance force as a control signal
will fail and cause instability (figure-15).

Figure-15: Adding mass causes instability.
To stabilize the system an omni-directional, linear viscous
dampening force applied in the natural coordinates of the robot
is used to generate the control signal. The response of the
system is shown in figure-16 for different values of KP and Kd
and an "=0. The two cases are simulated for the same
duration. As can be seen, the use of rate feedback in the
natural coordinates of the robot did stabilize the response and
made the system yield to the guidance signal derived from the
HPF. Significant transients are observed for a small coefficient
of rate feedback. Although increasing this coefficient reduces
the transients, it results, as in the holonomic case, in reducing
the speed of the robot.
In figure-17, the direction sensitive dampening is compared to
the linear dampening case using same coefficients for the
planner. As can be seen sensitizing the dampening to direction

Page 7

significantly reduced the overshoot and settling time without
compromising the speed of the robot.


Figure-16: Response of the planner in (13) for different Kp and Kd.

Figure-17: response of the planner in (15) compared to the one in (13).

In figure-18 the direction sensitive controller in (38) simulated
for two values of "=0,1. As can be seen the case where "=1
has lower overshoot compared to the case where "=0.

Figure-18: response of the planner in (15) compared to the one in (16).

Using a Kp=.001 and a Kd=60, The controller in (38) is tested
in a cluttered environment. Figure-19 shows the harmonic
gradient guidance field that is used to motivate the motion of
the robot and the holonomic, kinematic trajectory such a field
generates. Figure-20 shows the dynamic trajectory the
controller generates and the orientation of the robot as a
function of time. As can be seen, the nonholonomic, dynamic
trajectory is very close in shape to the holonomic, kinematic
trajectory with a satisfactorily smooth orientation profile. The
control torques applied to the right and left wheels of the robot
are shown in figure-21.


Figure-19: Guidance field and trajectory of a kinematic, holonomic, HPF
planner.


Figure-20: Trajectory and curvature using the planner in (16) and the guidance
field in fig. 19.

Page 8

Figure-21: Torque control signals corresponding to fig. 20.
Figure-22: trajectory in the presence of actuator saturation.
In figure-22 the robustness of the proposed controller in the
presence of actuator saturation is tested. The magnitude of the
torques (TR and TL) is restricted not to exceed Tm:
T C max(max(Tn (t)),max(Tn (t)))
m
t
where TnR and TnL are the torques for then non-saturated case,
C is a constant representing the percentage saturation. The
maximum torque for the non-saturated actuators is equal to
.103 N/M. The controller showed remarkable robustness to
saturation. The trajectory was virtually unaffected up to 99.8%
saturation (i.e. C=.002); however, a sudden breakdown in
performance is observed beyond this limit.
(47)
R
t
L
=⋅
VI. Conclusions
In this paper the ability of the HPF approach to accommodate
nonholonomic constraints when planning a trajectory for a
robot is demonstrated. This adds a significant improvement to
the already existing set of constraints the approach can subject
a planning process to in provably-correct manner. It also
shows that the wealth of properties harmonic potential fields
have (e.g. the goal seeking ability utilized in this paper) is a
great asset of the HPF approach and the key to extending the
capabilities of the approach.

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