This document presents research on controlling a robotic convoy over rough terrain using potential field methods. The researchers implemented quarter car, half car, and full car models in MATLAB to simulate vehicle dynamics. They also used potential fields to direct multiple robots in a convoy formation while avoiding obstacles. Simulations tested the convoy's response to terrain disturbances and showed errors between vehicle positions.

1. Control of a
high-speed robotic
convoy over rough
terrain
RBE595
Final Project
Prof Padir
April-19-2012
Michael Audi
Aaron Fineman
Yifan Li
Michael Raineri
Annette M Rivera Courtesy of http://simhqcom/forum/ubbthreadsphp/topics/2122888html

2. Introduction:
 Commonly seen in military convoy driving
 Vehicles must:
follow each other while
maintaining a safe and constant
distance
avoid obstacles that the preceding
vehicle failed to detect
continue unimpeded, rather than
become trapped waiting for the
leader to continue
 Inspired by searching for ways to control full-
scale convoys
Courtesy of http://simhqcom/forum/ubbthreadsphp/topics/2122888html
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3. Purpose/Goal:
 To apply various high-level control algorithms, in
particular potential field based methods, for use in
robotic convoys driving at high speed over rough
terrain
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4. Related Work:
 Robotic convoying
 Significant research problem in mobile robotics
Addressed from different points of view, eg, artificial vision, control
theory, artificial intelligence, etc [1]
Most existing mobile robot systems still involve a single robot working
alone, while there is a wide range of potential applications for robots
acting in concert
 Potential Fields
 Assisting a robot to move from one initial configuration to a
desired final configuration without colliding with any obstacles
 Car Models
 Dynamics of a vehicle system
typically modeled as a quarter car, a half car, and a full car
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5. Robotic convoying:
 Following Robot Imitates its lead robot
 Each of the following robots track the angular and linear velocity of
its lead robot.
Courtesy of Belkhouche and Belkhouche (August 2005)
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 Convoy with constant
distance between robots
 Constant Distance Between Robots
with
 Velocity Pursuit
 Deviated Pursuit
 Proportional Navigation
Modeling and Controlling a Robotic Convoy Using Guidance Laws Strategies
Fethi Belkhouche, Member, IEEE, and Boumediene Belkhouche, Member, IEEE
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B:
CYBERNETICS, VOL. 35, NO. 4, AUGUST 2005 (p.813-825)

6. Potential Fields:
 Additive Attractive/Repulsive
Potential Functions
 attracting the robot to the goal position
while repelling the robot from the
obstacles by superposing these two
effects into one resultant force applied on
the robot.
𝑭 𝒒 = −𝛁 𝑼𝒂 𝒒 + 𝑼𝒓 𝒒
Courtesy of Firat Uyanik (n.d.)
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Additive attractive/repulsive
potential function
A study on Artificial Potential Fields
Kadir Firat Uyanik
KOVAN Research Lab. Dept. of Computer Eng.
Middle East Technical Univ. Ankara, Turkey

7. Car Models:
 𝑀 𝐱 + 𝑏 𝐱 + 𝑘 𝐱 = 𝐅
Courtesy of Jazar (2009)
Quarter Car Half Car
Full Car
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Courtesy of Jazar (2009)
Courtesy of Jazar (2009)

8. Procedure:
 A version of the potential fields control method was
implemented in MATLAB This program was then used to
directly control three Khepera III's utilizing a serial over
Bluetooth link
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9. Procedure:
 The potential field models are governed by the following
characteristic equation:
𝐹 𝑞 = −𝛻 𝑈𝑎𝑡𝑡𝑟𝑎𝑡𝑖𝑣𝑒 𝑞 + 𝑈𝑟𝑒𝑝𝑢𝑙𝑠𝑖𝑣𝑒 𝑞
Figure:
Potential field shape
 The characteristic shape used for
our potential fields is
demonstrated in Figure
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10. Robot Following Simulation (Fixed
Offset)
 Leading Robot:
𝑞𝑛 = 𝑞 + −𝐾 ∗ 𝐹 𝑞 𝑞
 Sequential Following
Robots:
𝑞𝑛 𝑟 = 𝑞(𝑟+1) − 𝑜
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11. Robot Curve Smoothing Simulation
 Potential Field:
 𝐹 𝑞 =
𝑟 = 1, −𝛻(𝑈𝑎(𝑞) + 𝑈𝑟 𝑞
𝑟 > 1, −𝛻(𝑞 𝑟+1 + 𝑈𝑟 𝑞
 Position:
 𝑞𝑛(𝑟) = 𝑞𝑟 + −𝐾 ∗ 𝐹 𝑞 𝑞
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12. Hardware Implementation
 Khepera 3 Robots
 Potential Field
Implementation
 Controlled by
Matlab
 Almost Completed
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13. Hardware Implementation
 Khepera 3 Robots
 Potential Field Implementation
 Controlled by Matlab
 Almost Completed
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14. Procedure:
 2D (half) and 3D (full) car models were implemented in
MATLAB showing the forward dynamics of the system,
which takes on the following characteristic form:
𝑀 𝐱 + 𝑏 𝐱 + 𝑘 𝐱 = 𝐅
(m being the mass coefficients, b being the damping coefficients, k being
the spring coefficients)
Figure:
2D (half) car model
Figure
Courtesy of Jazar (2009)
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15. Procedure:
𝑀 𝐱 + 𝑏 𝐱 + 𝑘 𝐱 = 𝐅
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% Mass Matrix
m = [ms 0 ;
0 mu];
% Damper matrix
b = [bs -bs ;
-bs (bs+bu)];
% Spring matrix
k = [ks -ks ;
-ks (ks+ku)];
Figure:
Quarter car model
Courtesy of Jazar (2009)
Quarter Car

16. Figure:
Full car model
Procedure:
𝑀 𝐱 + 𝑏 𝐱 + 𝑘 𝐱 = 𝐅
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% Mass matrix
m = [mu 0 0 0 0 0 0;
0 Ix 0 0 0 0 0;
0 0 Iy 0 0 0 0;
0 0 0 mf 0 0 0;
0 0 0 0 mf 0 0;
0 0 0 0 0 mr 0;
0 0 0 0 0 0 mr];
% Damper matrix
C11 = 2*cf + 2*cr;
C12 = b1*cf - b2*cf - b1*cr + b2*cr;
C13 = 2*a2*cr - 2*a1*cf;
C14 = -cf;
C15 = -cf;
C16 = -cr;
C17 = -cr;
C21 = C12;
C22 = b1^2*cf + b2^2*cf + b1^2*cr + b2^2*cr;
C23 = a1*b2*cf - a1*b1*cf - a2*b1*cr + a2*b2*cr;
C24 = -b1*cf;
C25 = b2*cf;
C26 = b1*cr;
C27 = -b2*cr;
C31 = C13;
C32 = C23;
C33 = 2*cf*a1^2 + 2*cr*a2^2;
C34 = a1*cf;
C35 = a1*cf;
C36 = -a2*cr;
C37 = -a2*cr;
C41 = -cf;
C42 = -b1*cf;
C43 = a1*cf;
C44 = cf;
C45 = 0;
C46 = 0;
C47 = 0;
C51 = -cf;
C52 = b2*cf;
C53 = a1*cf;
C54 = 0;
C55 = cf;
C56 = 0;
C57 = 0;
C61 = -cr;
C62 = b1*cr;
C63 = -a2*cr;
C64 = 0;
C65 = 0;
C66 = cr;
C67 = 0;
C71 = -cr;
C72 = -b2*cr;
C73 = -a2*cr;
C74 = 0;
C75 = 0;
C76 = 0;
C77 = cr;
b = [C11 C12 C13 C14 C15 C16 C17;
C21 C22 C23 C24 C25 C26 C27;
C31 C32 C33 C34 C35 C36 C37;
C41 C42 C43 C44 C45 C46 C47;
C51 C52 C53 C54 C55 C56 C57;
C61 C62 C63 C64 C65 C66 C67;
C71 C72 C73 C74 C75 C76 C77];
Full Car
Courtesy of Jazar (2009)
% Springmatrix
K11 = 2*kf + 2*kr;
K12 = b1*kf - b2*kf - b1*kr + b2*kr;
K13 = 2*a2*kr - 2*a1*kf;
K14 = -kf;
K15 = -kf;
K16 = -kr;
K17 = -kr;
K21 = K12;
K22 = kR + b1^2*kf + b2^2*kf + b1^2*kr + b2^2*kr;
K23 = a1*b2*kf - a1*b1*kf - a2*b1*kr + a2*b2*kr;
K24 = -b1*kf - (1/w)*kR;
K25 = b2*kf + (1/w)*kR;
K26 = b1*kr;
K27 = -b2*kr;
K31 = K13;
K32 = K23;
K33 = 2*kf*a1^2 + 2*kr*a2^2;
K34 = a1*kf;
K35 = a1*kf;
K36 = -a2*kr;
K37 = -a2*kr;
K41 = -kf;
K42 = K24;
K43 = a1*kf;
K44 = kf + ktf + (1/(w^2))*kR;
K45 = -kR/(w^2);
K46 = 0;
K47 = 0;
K51 = -kf;
K52 = K25;
K53 = a1*kf;
K54 = -kR/(w^2);
K55 = kf + ktf + (1/(w^2))*kR;
K56 = 0;
K57 = 0;
K61 = -kr;
K62 = b1*kr;
K63 = -a2*kr;
K64 = 0;
K65 = 0;
K66 = kr + ktr;
K67 = 0;
K71 = -kr;
K72 = -b2*kr;
K73 = -a2*kr;
K74 = 0;
K75 = 0;
K76 = 0;
K77 = kr + ktr;
k = [K11 K12 K13 K14 K15 K16 K17;
K21 K22 K23 K24 K25 K26 K27;
K31 K32 K33 K34 K35 K36 K37;
K41 K42 K43 K44 K45 K46 K47;
K51 K52 K53 K54 K55 K56 K57;
K61 K62 K63 K64 K65 K66 K67;
K71 K72 K73 K74 K75 K76 K77];

17. Procedure:
 The inverse dynamic models were then implemented in
MATLAB These simulate a following vehicle with perfect
virtual sensors measuring the changes in the center of the
leading vehicle so that the road disturbances may be
estimated
𝑀 𝐱 + 𝑏 𝐱 + 𝑘 𝐱 = 𝐅
Courtesy of Jazar (2009)
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18. Procedure:
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 Use the last measurement for the position,
velocity, and input force to calculate the current
acceleration
 The assumption made here is the acceleration for the current time
is approximately equal to the acceleration for the last point in time
 A(t) = A(t-1) = M-1 * (F(t-1) – B*V(t-1) – K*X(t-1))
 Using the estimated acceleration, the input force
matrix was calculated for the current instance in
time
 F(t) = M*A(t) + B*V(t) + K*X(t)
 The force matrix contains the input ground disturbance

19. Simulations:
 Response to a 2 cm bump:
 Position
 Velocity
 Error
Courtesy of Spenko, Iagnemma, Dubowsky (nd)
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20. Quarter
Car Model:
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Quarter Car:
2cm Response
Quarter Car:
2cm error

21. Half
Car Model:
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Half Car:
Position Response
Half Car:
2cm Velocity Response

22. Half
Car Model:
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Half Car:
2cm error

23. Full
Car Model:
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Full Car: 2cm Response

24. Full
Car Model:
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Full Car: Error

25. Full Car Model:
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26. Summary:
 Modeled the car dynamics of the robots as Linear
models.
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Technologies Rating
Bluetooth
MATLAB
Khepera III's
 Potential Fields for
Path and Trajectory
Planning in a convoy.
 Used a control algorithm to
compensate for obstacle
avoidance.