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206 A. Kammerdiner et al.
ting the respective cointegration rank of the respective 8-dimensional phase series
against time interval values. The top picture in Fig. 10.4 and the bottom picture in
Fig. 10.5 include a legend that specifies which group (or row) of channels is repre-
sented by each line. Four plots are displayed, each containing 200 consecutive time
intervals. The horizontal axis represents time interval indexes, and the vertical axis
denotes the value of cointegration rank. In Fig. 10.4, the top plot depicts cointegra-
tion rank values for time intervals 1–200, and the bottom plot shows cointegration
ranks on intervals 201–400. In Fig. 10.5, the top and the bottom plots correspond to
intervals 401–600 and 601–800, respectively.
As can be seen from Figs. 10.4 and 10.5, most of the time all four groups of
channels exhibit a zero cointegration rank, which implies the absence of general-
ized phase synchronization. On the other hand, at certain times there are sudden
short spikes in cointegration rank that can be detected in all four 8-channel rows.
Interestingly, the patterns of the sharp transient increases in the rank values are com-
plex, characterized by a number of different combinations of rows often displaying
a simultaneous increase in their respective cointegration ranks.
Furthermore, the increased values of cointegration ranks range between 4–8, with
the cointegration rank of 8 being observed the second most often after the prevalent
rank of 0. The patterns in the range of cointegration rank values were further studied
using scatter plots of the experimental stage against the cointegration rank values
during the respective stage. Recall from Sect. 10.3.1 that each successful trial in the
experiment consisted of five distinct stages, and note that the intervals in between
trials were designated as experimental stage 0.
Figures 10.6 and 10.7 display the scatter plots of experimental stage versus the
respective cointegration ranks for each of the four groups of LFP channels. The in-
dexes of experimental stages, ranging from 0 to 5, are represented by the horizontal
axis, and the cointegration rank values found during each stage are displayed on the
vertical axis and range from 0 to 8. Although all the coordinates (the stages and
the ranks) are integers, we added a random jitter to the cointegration rank values
in order to be able to display multiple points having the same coordinates. The four
scatter plots in Figs. 10.6–10.7 correspond to four different groups of channels, with
the top left plot displaying stage indexes versus cointegration ranks for LFP 1–8, the
top right presenting the scatter plot for LFP 9–16, and the bottom left and bottom
right plots illustrating the stage/rank relationship for LFP 17–24 and LFP 25–32
groups, respectively. Figures 10.6 and 10.7 highlight some interesting similarities
among the four different groups of electrode, each of which represents a single row
of the microwire array. In particular, whereas no synchronized activity in terms of
generalized phase synchronization can be detected at stages 2 and 3 (after the center
pad is hit and until it is released) in all four groups with a single instance exception
of stage 3 for LFP 17–24, stages 0, 1, 4, and 5, are all characterized by a short but
strong presence of generalized phase synchronization. Although the rank results for
stage 0 (between trials) are difficult to interpret with certainty, the cointegration of
the instantaneous phases during experimental stage 1 may be attributed to the mo-
tor planning induced by activation of the central light at the beginning of each trial
and/or execution of motor action that leads to central pad hit. Similarly, for experi-
mental stage 4, the positive and large rank values may be associated with execution
10 Integration of Signals in Complex Biophysical Systems 207
Fig. 10.4 Linear plots of cointegration rank values on 200 consecutive time intervals for four
groups of LFP channels. Top: First 200 time intervals (i.e., intervals 1–200). Bottom:Timeinter-
vals 201–400
208 A. Kammerdiner et al.
Fig. 10.5 Linear plots of cointegration rank values on 200 consecutive time intervals for four
groups of LFP channels. Top: Time intervals 401–600. Bottom: Time intervals 601–800
of the movement beginning upon the release of the central pad and ending with the
target being hit. Finally, the evidence of short-term synchronization at stage 5 may
be connected to maintaining the grasp for a certain time until the reward is given.
10 Integration of Signals in Complex Biophysical Systems 209
Fig. 10.6 Scatter plots of
cointegration rank vs.
experimental stage for four
groups of LFP channels.
A random jitter is added to
the cointegration rank values
to display coinciding points.
Top: LFP 1–8. Bottom:
LFP 9–16
10.4 Conclusion
Experimental data recorded during reach and grasp experiments in monkeys were
analyzed using a novel technique of generalized phase synchronization. Our inves-
tigation into the dynamics of neural activation in the motor and premotor cortical
areas engaged in movement planning and execution discovered the presence of short
term cointegration in the instantaneous phases of neural channels located in a com-
mon row of the implanted microwire arrays. Since short-term synchronization of
neuronal discharges is widely believed in neuroscience research community to be
responsible for the dynamical integration of activity of distributed groups of neurons
210 A. Kammerdiner et al.
Fig. 10.7 Scatter plots of
cointegration rank vs.
experimental stage for four
groups of LFP channels.
A random jitter is added to
the cointegration rank values
to display coinciding points.
Top: LFP 17–24. Bottom:
LFP 25–32
that is necessary for execution of cognitive and motor tasks, the strong presence of
cointegration in instantaneous phases may be attributed to transient fusion of neural
signals.
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Chapter 11
An Info-Centric Trajectory Planner
for Unmanned Ground Vehicles
Michael A. Hurni, Pooya Sekhavat,
and I. Michael Ross
Summary We present a pseudospectral (PS) optimal control framework for au-
tonomous trajectory planning and control of an Unmanned Ground Vehicle (UGV)
with real-time information updates. The algorithm is introduced and implemented
on a collection of motion planning scenarios with varying levels of information. The
UGV mission is to traverse from an initial start point and reach the target point in
minimum time, with maximum robustness, while avoiding both static and dynamic
obstacles. This is achieved by computing the control solution that solves the ini-
tial planning problem by minimizing a cost function while satisfying dynamical and
environmental constraints based on the initial global knowledge of the area. To over-
come the problem of incomplete global knowledge and a dynamic environment, the
UGV uses its sensors to map the locally detected changes in the environment and
continuously updates its global map. At each information update, the optimal con-
trol is recomputed and implemented. Simulation results illustrate the performance
of the planner under varying levels of information.
11.1 Introduction
Autonomous trajectory planning of unmanned vehicles has been one of the main
goals in robotics for several years. In recent years, this problem has become partic-
ularly important as a result of rapid growth in its applications to both military and
civilian missions. Various control methods have been proposed and examined for
autonomous guidance and control of unmanned vehicles [4, 13].
There are two approaches to optimal trajectory planning for a dynamic system:
The decoupled approach and the direct approach. The decoupled approach involves
M.A. Hurni (
) · P. Sekhavat · I.M. Ross
Naval Postgraduate School, Monterey, CA 93943, USA
e-mail: mahurni@nps.edu
P. Sekhavat
e-mail: psekhava@nps.edu
I.M. Ross
e-mail: imross@nps.edu
M.J. Hirsch et al. (eds.), Dynamics of Information Systems,
Springer Optimization and Its Applications 40, DOI 10.1007/978-1-4419-5689-7_11,
© Springer Science+Business Media, LLC 2010
213
214 M.A. Hurni et al.
first searching for a path (using a path planner) and then finding a time-optimal con-
trol solution on the path subject to the actuator limits. The direct approach searches
for the optimal control trajectory directly within the system’s state space [4].
Optimal control trajectory planning using numerical optimization, as described
in [4], is a direct approach to the complete motion-planning problem, which deter-
mines the path to the target by searching for the optimal control trajectory within
the vehicle’s state space. The result is the complete state space and control solu-
tion from start to goal. The basic concept of how optimal path planning works fol-
lows from [4, 9]. The planner is given the kinodynamic equations of the vehicle,
the obstacles’ approximate location and geometry (to be coded into smooth path
constraint functions), and the mission’s boundary conditions and cost function. The
kinodynamic equations can also be viewed as constraints (similar to the obstacles),
defining the relationship between the vehicle state and the control input. The actual
obstacles’ geometry need not be smooth, but the constraints used to mathematically
define the obstacles must be made up of one or more smooth functions. The optimal
control technique finds a solution to the state equations that takes the vehicle from
the initial state at time zero to the final state at the final time, while avoiding obsta-
cles, obeying vehicle state and control limits, and minimizing the cost function. The
cost function can be any function of state variables, control variables and time, as
long as it is sufficiently smooth (i.e., continuous and differentiable).
Recent advances in optimal control and numerical optimization tools have made
the method a viable approach for many applications. It has been demonstrated that
using the initial (open-loop) trajectory generated off-line as the bias (guess) can
produce run times of sufficient speed. Subsequently, each real-time solution can be
used as the bias for the next run, thus maintaining solution speed and autonomy. This
approach involves using the optimal control algorithm in a feedback form (closed
loop), thus self-generating the needed bias. References [1–3, 6–8, 11, 12]solvevar-
ious path planning problems using optimal control with numerical optimization in a
feedback control algorithm. In [1, 3, 6–8, 11], the use of optimal control in a feed-
back form using a bias that is obtained autonomously is demonstrated on a Reusable
Launch Vehicle, a tricycle, an Unmanned Ground Vehicle, an Unmanned Air Vehi-
cle, and a spacecraft. In reference to the computational speed (cost), it has been
shown in [1–3, 6–8, 11, 12] that the use of a bias to help steer the solution trajectory
speeds up the run times significantly from seconds to fractions of a second. So the
bias does not just aid in achieving feasible solutions, but also speeds up the solution
process considerably. Also, advancements in sparse linear algebra, development of
new algorithms, and improved computer processor speeds have made solving op-
timization problems relatively easy and fast [3]. Recent applications of real-time
optimal control [1–3, 6–8, 11, 12] have proven to be very promising in facilitating
feedback solutions to complex nonlinear systems [3].
This paper presents the implementation of a pseudospectral (PS) optimal control
framework for autonomous trajectory planning and control of a UGV with real-time
information updates on numerous motion planning scenarios with varying levels of
available information. Simulation results illustrate the effects of varying levels of
information and how more or less information would influence the performance of
the planner.
11 An Info-Centric Trajectory Planner for Unmanned Ground Vehicles 215
11.2 Problem Formulation and Background
The UGV that is used in this paper is modeled as a four-wheeled car with front-
wheel steering [6]. Only the front wheels are capable of turning and the back wheels
must roll without slipping. L is the length of the car between the front and rear axles.
r
t
is the instantaneous turning radius. The state vector is composed of two position
variables, x and y (m), an orientation variable, θ(rad), the car’s velocity, v (m/sec)
and the angle of the front wheels, φ(rad) with respect to the car’s heading, (11.1).
The x–y-position of the car is measured at the center point of the rear axle.
x
∈X :=
⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
x :x
min
≤x(t) ≤x
max
y :y
min
≤y(t) ≤y
max
θ :−2π ≤θ(t)≤2π
v :−1 ≤v(t) ≤1
φ :−1 ≤φ(t) ≤1
⎫
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎭
(11.1)
The control vector consists of the vehicle’s acceleration, a (m/sec
2
) and the rate
of change of the front wheel angle, ω (rad/sec), (11.2).
u
∈U :=
a :−0.5 ≤a(t) ≤0.5
ω :−0.33 ≤ω(t)≤0.33
(11.2)
Using the fact that L = r
t
tan(φ) and v = r
t
˙
θ, assigning the length L = 0.5m
results in the following kinematic equations of motion for the UGV:
˙x
=
⎡
⎢
⎢
⎢
⎢
⎣
˙x
˙y
˙
θ
˙v
˙
φ
⎤
⎥
⎥
⎥
⎥
⎦
=
⎡
⎢
⎢
⎢
⎢
⎣
v cos(θ)
v sin(θ)
2v tan(θ)
a
ω
⎤
⎥
⎥
⎥
⎥
⎦
(11.3)
The p-norm was used to algebraically model the shapes of the obstacles used in
this work. Using the p-norm, one can easily model any square, rectangle, circle, or
ellipse. These shapes are all that is needed in path planning since any obstacle can
be modeled by fitting one of those shapes around it. Modeling an obstacle’s exact
shape and size is more complex and unnecessary. Equation (11.4) shows the general
form of the equation used to model the obstacles.
h
i
x(t),y(t)
=
x(t) −x
c
a
p
+
y(t) −y
c
b
p
−
c
p
=0 (11.4)
Development of the cost function is discussed next. The objective is to minimize
maneuver time while simultaneously maximizing robustness. Robustness is charac-
terized by the vehicle being able to execute the mission in the presence of a certain
amount of uncertainty and not hit any obstacles. This uncertainty can come from