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A Four Legged Walking Robot with Obstacle
Overcoming Capabilities
Antonio Bento Filho
1
, Paulo Faria S Amaral
2
and Benedito G Miglio Pinto
3
1
Mecanical. Eng. Department, Fed. Univ of Espirito Santo, abento@ele.ufes.br
2
Electrical Eng. Dept., Fed. Univ of Espirito Santo, pfsamaral@automatica.com.br
3
Automatica Tecnologia S/A, Serra, ES, Brazil, bmiglio@automatica.com.br
Abstract. This work deals with the four legged walking robot Guará, built with
obstacle overcoming capabilities. It was designed as a research robot platform, to
be used in indoor environment, walking straight and curved paths and detecting
and overcoming known obstacles. Straight paths are done with gait matrix strategy
and curved paths are accomplished by the four legs which have differential strokes
in inclined paths relatively to the robot longitudinal axis. Obstacle overcoming is
done using only information from contact sensors installed on the robot’s feet.
Complex movements and tracking sequences are proposed to be built from a small
group of simple movements sequenced according to the contact keys switching se-
quence. Results include: gait generation for straight and curved paths; integrated
movement sequence to go one step up in the floor; and a description of the devel-
opment, construction and testing of Guará quadruped robot.
Keywords: obstacle, legged robot, mobile robot, robot, anthropomorphic.
1. Introduction
There has been great effort in studying mobile robots that use legs as their lo-
comotion system. However, at the same time, legs pose a number of problems of
their own. Indeed, legged-robot research focuses on everything related to leg mo-
tion and coordination during robot navigation [6].
The difficulty in developing legged vehicles is mainly to coordinate the large
number of degrees of freedom that should be triggered on a simple motion of the
robot from a point A to point B. The precise coordination of walking movements,
so that the resulting dynamic forces during navigation keep the robot in balance, is
a big challenge. These restrictions can be overcome by modeling statically, the
dynamic balance of forces resulting from the movement of the robot.
2
Collisions with obstacles or irregularities in the floor can destroy the robot bal-
ance and it can fall due to lack of additional points of support or by collision of
one leg with some elevation. Robot navigation requires obstacles detection and
avoidance, while guiding the robot to its goal. It must consider known and un-
known, moving or fixed obstacles in the robot’s path and the needing for detection
and avoidance strategies.
To operate safely in public environments, many successful systems use sensor
based modules for the collisions prevention, to control the mobile robot. The main
control structure is feeling - react: the reading of the sensor is analyzed conti-
nuously to determine a free collision movement.
Wheeled and legged mobile robots path planning has received great attention
and several works have been developed on the detection and overcoming of ob-
stacles and in the development of skills that allow robots to walk in irregular sur-
faces with obstacles [2, 5, 7, 8]. In addition to the navigation skills, the robot’s
balance strategies are essential so that it can perform the job overcoming or avoid-
ing the obstacles in its path.
2. Problem Definition
The effort to develop legged mobile robots have been conducted on several
fronts including: the studies of locomotion and biomechanics of animals and in-
sects with legs, the construction of models to represent the movement of animals
and insects with legs; the construction of mathematical models of equilibrium for
legged robots in order to simulate the models built for the animals and insects with
legs; the design and build of robots, legs and devices for special applications, re-
search in animal locomotion and validation of locomotion models; the design and
construction of energy efficient actuators and mobile robot legs, capable of dis-
playing both reactivity and adaptability to their operating environment.
This work concerns with a strategy for overcome or avoid obstacles using only
information from sensors installed on the feet to perceive the obstacle and manage
an autonomous sequence of complex integrated movements to move across the ob-
stacle or deviate from it, preserving the robot’s balance and stability.
Concerning with the design and build of robots, legs and devices, it is presented
a mechatronic solutions for the construction of an anthropomorphic like robot leg
[2]; the design of the quadruped robot Guará for statically stable walking; the im-
plementation of the algorithms for obstacle detection and surpassing, using only
information obtained from sensors on the robot feet; the implementation of the ki-
nematic and equilibrium models; the implementation of an interface for the con-
struction of static gaits; and the implementation of all these features in a software
platform with data structures, interfaces and models, built within object-oriented
programming (OOP).
3
The obstacle overcoming problems addressed are transposition or bypass in
regular and firm ground, using only information obtained from sensors installed on
the robot’s feet. The obstacles considered are the step up, step down, step close,
and channel slope. The approach used can be expanded to other complex auto-
nomous movements like stair climbing and other features with a legged.
2.1. The Guará Robot
The four legged walking robot Guará, which is showed in Fig. 1, was designed
as a research robot platform, to be used in indoor environment, walking straight
and curved paths and detecting and overcoming known obstacles [4].
Fig. 1 Legged robot Guará turning left while walking.
2.2. Leg Design and Reference System
The normal human step describes six movements [10]. These movements in-
crease the length of the step and flatten the circular path of the hip by increasing
the effective length of the leg in support; smooth the transition between the two
phases of leg support; and transfer the body's weight on the supporting leg during
foot transition.
The Guará’s movements are inspired accordingly to the normal human step. It’s
legs have two revolute joints connecting the femur to the hip, performing the pitch
and roll of the femur, one revolute joint for the knee, connecting the femur to the
tibia and performing the pitch of the knee, and one revolute joint for the ankle,
connecting the tibia to the foot and performing the pitch of the foot, according to
Fig. 2. The sixteen leg joints are driven with toothed belt drive by DC motor and
reducers installed in the robot body. With this joint distribution we can allow the
lateral movement of the hip which improves the robot stability margin by moving
its gravity center towards the supporting polygon.
4
Fig. 2:
Leg revolute joints and joint reference systems
2.3.
Leg’s Inverse Kinematics
A typical method for obtaining the inverse kinematics for a manipulator with
six degrees of freedom is to get the
position and orientation of the
the first three joints followed by
With four degrees of freedom, the leg is a
matics and this
method will not work because
trix for the end
effector. In this case, however, it is possible to solve the inverse
kinematics
geometrically, directly from a joint, since the last three joints are
rallel and perpendicular to the
to obtain the inverse kinematics of
used in MVACS (Mars Volatiles and Climate Surveyor) launched by NASA in
1999.
Referring to Fig.
2
1, 2 e 3, we have:
2
3
2
3
3
2
cos yx
∗
++
=θ
Geometrically we can find
=
−
2
3
1
2
tanθ
x
z
Leg revolute joints and joint reference systems
, according to Denavit-
Hartenberg
Leg’s Inverse Kinematics
A typical method for obtaining the inverse kinematics for a manipulator with
six degrees of freedom is to get the
reference
position of the wrist from the desired
position and orientation of the
end
effector, and then use geometric methods for
the first three joints followed by
the three wrist joints.
With four degrees of freedom, the leg is a
n
unusual problem of inverse kin
method will not work because
there is not a c
omplete rotation m
effector. In this case, however, it is possible to solve the inverse
geometrically, directly from a joint, since the last three joints are
rallel and perpendicular to the
first joint. This scheme was used in
Bonitz (1997),
to obtain the inverse kinematics of
a
manipulator with four degrees of freedom,
used in MVACS (Mars Volatiles and Climate Surveyor) launched by NASA in
2
, applying the cosines law to the triangle
formed by the joints
( )
32
2
3
2
2
2
3
aa
aaz
∗
∗
+−
Geometrically we can find
θ
2
from Fig. 2 as follows:
∗+
∗
−
+
−
332
33
1
2
3
3
cos
sin
tan θ
θ
aa
a
z
z
Hartenberg
:
A typical method for obtaining the inverse kinematics for a manipulator with
position of the wrist from the desired
effector, and then use geometric methods for
unusual problem of inverse kin
e-
omplete rotation m
a-
effector. In this case, however, it is possible to solve the inverse
geometrically, directly from a joint, since the last three joints are
all pa-
Bonitz (1997),
manipulator with four degrees of freedom,
used in MVACS (Mars Volatiles and Climate Surveyor) launched by NASA in
formed by the joints
(1)
(2)
5
From the coordinates of the end of the shank (x3, y3, z3) in Fig. 2, the rotation
θ
1
is taken as follows:
−
=
3
31
tan
1y
x
θ
(3)
The foot moves with
θ
4
always known, which gives the coordinate of the joint 3
in relation to the supporting point coordinates (x4, y4, z4) as follows:
∗−
∗∗−
∗∗−
=
444
1444
1444
3
3
3
sin
sinsin
cossin
θ
θθ
θθ
az
ay
ax
z
y
x
(4)
The mapping of the foot coordinates from the operational space into joint space
in done by equations (1) to (4). The matrix of the gait being executed synchronizes
each foot trajectory and body lateral movements in the leg’s operational space and
leg’s inverse kinematics maps them in joint space.
2.4. Foot Stoke and Flight Trajectories
The foot stroke trajectory is a straight line, as it’s desired that the robot body
would be maintained in a horizontal plane. Other configurations can be con-
structed to walk inclined paths, maintaining the robot body horizontal, and walk a
horizontal path, maintaining the robot body inclined.
Any continuous foot flight trajectory could be implemented regarding one foot
do not strike the other and the ground during flight to a new stroke position.
The ideal foot flight trajectory has boundary conditions of zero speed at the foot
takeoff and landing and the ideal curve for the foot flight would be a cycloid.
However, due to the flexibility of the legs, the joint mechanical slash and balanc-
ing of the robot, at the beginning of the foot flight to a new stroke position, the
foot stayed on the ground a little later after starting the foot flight movement. At
the end of the flight, a little before the landing desired position, the foot always
touches the ground early before the time predicted by the cycloid model. During
these periods, the foot is being moved in the opposite direction to the movement,
causing reverse thrust in takeoff and landing, which causes undesirable balancing
and instability to the robot.
To avoid this, a polygonal trajectory, was implemented, as shown in Fig. 3,
where leg’s thrust is from right to left, the vertical distances are measured from
leg’s coordinate system origin and the dimensions are in millimeters.
At taking off point, associated with the left inclined segment in the generated
path (in blue), the foot is moved upward with horizontal speed equal to the robot
speed, and a more high vertical speed, until it leaves completely the ground. Then
it’s accelerated to achieve a higher horizontal speed and describes the superior
6
three segments trajectory and at the end of the third superior segments, it begins
the landing movement in the right inclined segment in the generated path. During
landing, the horizontal speed is set to the robot’s ongoing speed and the vertical
speed is lowered, until the feet touch the ground. The received line (in red) is the
foot path built by direct kinematics of joint coordinates received from the robot. It
deviates from the sent values because the joint mechanical slashes, but it could be
noted that the foot take off begins when predicted and the reverse stroke in landing
and taking off of the feet.
Fig. 3: Leg 3 joint 3: generated and received paths in operational space, saggital planeRobot De-
sign and Reference Systems
Fig. 4 shows robot starting walking with leg 3 flying to a new stroke position.
Robot legs are numbered as: 0, the fore-left; 1, the rear-left; 2, the fore-right and 3,
the rear-right leg. Principal dimensions are: L: length; W: width; G: geometric cen-
ter; and G
xy
: geometric center projection on the ground.
The global reference system (
0
x
0
,
0
y
0
,
0
z
0
) and the robot reference system (
r
x,
r
y,
r
z) are parallel when the robot starts walking. Regarding the robot reference sys-
tem, each leg has its own reference system, as shown (
0
x,
0
y,
0
z) for the leg 0 in
Fig. 4.
2.5. Robot Kinematic Model
As shown in Fig. 5 (a), (b), (c) and (d), there are four possible combinations of
three feet supporting patterns, each one defining two foothold position vectors
r
p
ij
where the right superscript r is the onboard robot reference frame and the left sub-
script ij is the leg’s numbers [2].
-290
-280
-270
-260
-250
-240
-100 -80 - 60 -40 -20 0 20 40 60
Generated
Received
For each supporting pattern it is computed their
uct which
allow the defini
points
on board reference system [3]
ing updated and the final rotation from the initial position is computed every step
and could be checked and correcte
other robot’s stereo orientation reference.
(a)
r
P
10
r
P
1 0
3 2
2.6.
Statically Stable
In statically stable walking a legged robot must have, at least, tree feet on the
ground. Fig. 6
shows a
the rate
between the
The leg phase angle
φ
Gait matrix
define
columns
are associated with leg state
columns with a one
umns order
s containing the first
the total of columns.
To allow
finding the ideal
construction library. The gait matrix is entered directly
Fig. 4: Joints and robot reference systems.
For each supporting pattern it is computed their
unit normal vector
cross pro
allow the defini
tion of the rotation matrix between
two consecutive
on board reference system [3]
.
As the robot walks, the rotation matrix is b
ing updated and the final rotation from the initial position is computed every step
and could be checked and correcte
d with on board compass and inclinometers or
other robot’s stereo orientation reference.
(b) (c)
r
P
10
P
20
r
P
31
r
P
20
r
P
32
r
P
31
1 0
3 2
1 0
3 2
1
3
Fig. 5: Legs supporting patterns.
Statically Stable
Gaits.
In statically stable walking a legged robot must have, at least, tree feet on the
shows a
periodic regular symmetric gait. The gait loa
d factor
between the
stroke time and total gait time, and is the same for all legs.
φ
is the rate between its landing time and the
total gait time
define
s the gait pattern; its lines are
associated with the legs and
are associated with leg state
. In the gait matrix,
β
is the rate
between the
enter and the total of columns and
φ
is
the rate between
s containing the first
one enter after a zero
entry, from left to right
finding the ideal
β
and
φ,
the robot’s software also has
a gait matrix
construction library. The gait matrix is entered directly
in the gait building library
7
cross pro
d-
two consecutive
set
As the robot walks, the rotation matrix is b
e-
ing updated and the final rotation from the initial position is computed every step
d with on board compass and inclinometers or
(d)
r
P
32
0
2
In statically stable walking a legged robot must have, at least, tree feet on the
d factor
β
is
stroke time and total gait time, and is the same for all legs.
total gait time
.
associated with the legs and
its
between the
the rate between
col-
entry, from left to right
, and
a gait matrix
in the gait building library
8
which defines the gait load factor
β
and the leg’s phase angle
φ
. The total number
of columns defines the gait cycle, the total of zero entries defines the flight portion
and the one entry defines the stroke portion of the gait cycle. The gait matrix and
all its properties can be tested using in the robot graphic interface and/or in robot
walking. In the 32 segment gait of Fig. 6, there are 27 stroking and 5 flying gait
segments, resulting in a load factor
β
= 0,84375.
1/2
1/4
3/4
1(t/T)
0
1
2
3
Foot
20
22
24
16
14
12
10
8
6
4
2
0
18
28
30
26
32
Segment
Fig. 6: Regular symmetric gait.
In Fig. 6, the gait cycle begins in segment 0, with the four legs in stroke moving
the robot body forward and to the left, relatively to the feet; at segment 3 the rear
right leg 3 flights to a new stroke position; at segment 8 the four legs are in stroke
again; at segment 11 the front leg 2, on the same side, flights to a new stroke posi-
tion; and at segment 16 the four legs are in stroke, moving the robot forward and
to the right and then repeat the movement sequence for the rear let 1 and the front
left 0 legs.
Legs cooperative movements that move the robot are synchronized through the
gait matrix and an addressing algorithm that moves the robot body laterally during
straight or curved walking trajectories.
2.7. Curved Paths
Fig. 7 shows the curved path stroke. The algorithm implemented is a drivabili-
ty strategy that allows the robot turning while walking, without losing equilibrium
and stability. Curved paths are accomplished by the four legs which incremental
differential strokes tangent to the robot curve and inclined to its longitudinal axis.
The stroke path is done by small straight line segments in the foot operational
space, as shown in Eq. (1), where:
θ
is the curve angle; R
o
and R
i
are the foot out-
side and inside curve radius;
λ
o
e and
λ
i
are the stroke pitch for the legs outside
and inside the curve; and L and W are length and width between legs reference
systems.
9
2/1
22
2/1
22
22
*
22
*
+
−==
+
+==
LW
RR
LW
RR
oii
ooo
θθλ
θθλ
(1)
As the straight path is also implemented through small straight line increments,
to turn the robot in a curved path the only additional computational effort is to sum
four 3x1 matrix with the small rectilinear segments that form the foot curved tra-
jectory in foot’s operational space to find the next foot position. The inverse kine-
matics then maps foot cartesian coordinates to leg joint coordinates.
Fig. 7: Curved path stroke.
2.8. Obstacles Detection and Overcoming
The environmental information around the robot is used for planning the ro-
bot’s path, and, to detect low and tight, and air obstacles in robot’s path, it’s com-
mon the use of special recognition systems.
An approach to the locomotion problem for of the four legged robot"JROB-1”,
with Scara like legs, in complex environments, is presented in [8] and a scheme to
detect obstacles in the robot legs path using proximity sensors, to improve the lo-
comotion skill in complex environments, is presented in [7].
Inside
stroke li
Outside
stroke le
External
radis Re
Internal
radius Ri
Radius Rc
Center
W/2W/2
L/2
L/2
q
q
q
3i+1
3i
4i
2i
2i+1
1i+1
1i
r i
x
r i+1
x
r i
yr i+1
y
GiGi+1
4i+1
10
This work presents a strategy for detecting and overcoming obstacles placed in
the robot’s foot trajectory using contact keys type collision sensors installed in the
robot’s feet. Fig. 8 shows the contact keys type collision sensors installed on a
Guara’s feet. The contact keys lay out is to feel contact of the foot with horizontal
and vertical surfaces.
There are two double keys parallel connected horizontally installed on the front
and on the back foot and two single keys vertically installed on the front and on
the back foot. In a collision, the system identifies the obstacle from the switch pat-
tern and starts an integrated movement sequence to move the robot to transpose, or
to turn aside the obstacle.
Figure 9 shows the feet contact keys switching patterns. The keys can detect
contact with: horizontal surfaces, Figure 9 (b) pattern [1011], and (c) pattern
[1101]; vertical surfaces, Figure 9 (a) pattern [0111] and (d) pattern [1110]; and
no contact pattern [1111], Figure 9 (e).
Fig. 8: Contact keys on the feet.
Figure 9: Feet touch contact keys.
Contact switching occurs also in foot’s taking off and landing events before and
after its flight to a new stroke position, or when it’s being moved during obstacle
overcoming. The movement sequences studied were for the step up, step down,
channel, step close, ramp and divert obstacles.
Fig. 10 shows the main obstacle overcoming control cycle. Each obstacle has
its own unique recognition key pattern and other obstacles patterns can be imple-
mented. Fig. 11 shows the step up obstacle overcoming control cycle, which re-
sults are presented.
(b) (c)
(a) (d) (e)
11
Fig. 10: Obstacle overcoming control cycle.
Fig. 11: Step up obstacle overcoming control cycle.
2.9. Main Control Structure
The robot’s system control architecture is showed in Fig. 12 The three layers
control system structure comprises a supervisor and a coordinator level connected
by a CAN bus and an actuator level. Processes associated with walk and overcom-
ing obstacles always reach states that start other processes which still can start oth-
er processes, up to 3 levels. All the main control cycle tasks run in the supervisor
level.
The robot’s main control strategy is based in “case” and “if then else” patterns
which sense and react to contact keys state changes; executes the first level task
&
&
(Landed)
(Pattern[1111])
&(Timeout Channel)
&
&
&(Pattern[1101])
&(Pattern[1011])
(Stroke)
(Pattern[1001])
&
&
(On step up)
(Pattern[1111])
&
&
(Stroke)
(Pattern[1111])
&
&
(Landed)
(Pattern[1101])
&
&
(Landed)
(Pattern[1111])
(Pattern[1110])&
&
&
(Landed)
(Pattern[1011])
!(Timeout)
!(Timeout)
!(Timeout)
Step down
Step up
Ramp up
Channel
Ramp down
Colision
process
unit
Leg stroke
Step close
Leg flight
!(Landed)
Moving back
Lowering
Equalizing
Moving robot
platform Stepping
up Finished
Step up
Move back
the feet to the
last taking
off point
Move down
the feet to the
last taking off
point
Equalizing
legs positions
Move robot
structure to
left
kinematic
limit
Move up
and land fore
right feet on
the step
up
Move robot
structure to
right
kinematic
limit
Land fore
left foot on
the step
up
!(Position OK)
(Position OK)
!(Landed)
(Landed)
!(Equalized)
(Equalized)
!(Position OK)
(Position OK)
!(Position OK)
(Position OK)
!(Position OK)
(Position OK)
12
which generates direct kinematics of legs according to the kinematic model, co
straints and kinematic limits of the robot; process the inverse kinematics; and
communicates with robot on board controlle
run processes which manage groups of movements to perform a task in hand.
The main con
trol cycle is showed in
Collision Processing Unit
Processing Unit
which manages the walking tasks; and a
which manages the control system and robot’s hardware communication.
In Fig. 13, the
Main
processing level, checking all process branches and ends each control cycle ru
ning the communication unit processes.
The
Gait Processing Unit
tion of the robot straight and curved paths, ac
set up.
The
Collision Process Unit
sequences and is fired by the main unit if the switching pattern switches to an
known obstacle previously programmed. When a collision occ
control cycle starts the collision unit and then all processes until the robot returns
to its walking state is done within the collision unit.
Process associated with robot communication and man machine interface are
performed each contr
implements the joint controllers and the CAN network bus with the supervisor
level. The actuator level performs direct joint control and contact keys data acqu
sition [9].
which generates direct kinematics of legs according to the kinematic model, co
straints and kinematic limits of the robot; process the inverse kinematics; and
communicates with robot on board controlle
r.
The second and third state levels
run processes which manage groups of movements to perform a task in hand.
Fig. 12: Control system architecture
trol cycle is showed in
Fig. 13.
The internal processes are: the
Collision Processing Unit
which manages the obstacle overcoming tasks; the
which manages the walking tasks; and a
Communication Unit
which manages the control system and robot’s hardware communication.
Main
unit runs a fixed sequence polling process
in the first
processing level, checking all process branches and ends each control cycle ru
ning the communication unit processes.
Gait Processing Unit
process walking tasks for the incremental constru
tion of the robot straight and curved paths, ac
cording to previous gait parameters
Collision Process Unit
concerns with the overcome obstacles movement
sequences and is fired by the main unit if the switching pattern switches to an
known obstacle previously programmed. When a collision occ
urs, the main unit
control cycle starts the collision unit and then all processes until the robot returns
to its walking state is done within the collision unit.
Process associated with robot communication and man machine interface are
performed each contr
ol cycle in the Communication Unit
. The coordinator level
implements the joint controllers and the CAN network bus with the supervisor
level. The actuator level performs direct joint control and contact keys data acqu
which generates direct kinematics of legs according to the kinematic model, co
n-
straints and kinematic limits of the robot; process the inverse kinematics; and
The second and third state levels
run processes which manage groups of movements to perform a task in hand.
The internal processes are: the
which manages the obstacle overcoming tasks; the
Gait
Communication Unit
in the first
processing level, checking all process branches and ends each control cycle ru
n-
process walking tasks for the incremental constru
c-
cording to previous gait parameters
concerns with the overcome obstacles movement
sequences and is fired by the main unit if the switching pattern switches to an
urs, the main unit
control cycle starts the collision unit and then all processes until the robot returns
Process associated with robot communication and man machine interface are
. The coordinator level
implements the joint controllers and the CAN network bus with the supervisor
level. The actuator level performs direct joint control and contact keys data acqu
i-
13
Fig. 13: Guará’s main control cycle.
3. Results
3.1. Guará robot
Fig. 1 shows a picture of the Guará robot. Its walking system comprises four
anthropomorphic like legs electrically actuated by DC motors and gear reducers
and the navigation and balance scheme is based on human’s walking. Its obstacles
overcoming capabilities use only the information obtained from contact sensors
installed in the robot feet. This information is used to manage integrated move-
ment sequences to transpose or to turn aside the obstacles.
3.2. Robot Control System Software
The Guará software, developed in OOP, is an integrated environment for robot
driving, gait development, parameters adjustment and robot diagnosis. The soft-
ware environment also allows the development of new integrated movement se-
quences to transpose new obstacles, by adding new obstacle code libraries in C++.
The robot start walking begins by loading the selected gait matrix file, which
defines the load factor
β
, the phase angle
φ
and the remaining stroke before each
leg reach its kinematic limit. Then, the number of points to build the foot trajecto-
ry in the operational space, the maximum stroke to be used, which is less or equal
the minimum available leg’s stroke, and the height of the foot flying movement
are entered.
Connect
button
Collision
processing
unit
Gait
processing
unit
Send
data to
robot
Read
data from
robot
Refresh
control
variables
Stop
Walking
Processing
Idle
Step
loop state
Key change
Main
Each set
point
No key
change
Read contact
keys
Speed c ontrol
slidi ng bar
Stering spin
buttom
Obstacl e overcomi ng
Feet take off or landi ng
Connect
buttom
Start
Communicati on unit
14
3.3. Operational and joint coordinate in straight walking
Fig. 14 shows operational coordinates (x
3
, y
3
, z
3
) generated and executed for
leg 0, joint 3, in robot’s reference frame according to Denavit-Hatenberg conven-
tion, using 108 stroke points and 20 flight points to build the foot trajectory.
As it can be seen from Fig. 6, that it were used 4 points for each segment in the
basic gait pattern shown. Referring to Fig. 6, foot 3, which flight starts after 3
segments, will fly in 12 step counts. The robot starts walking moving his body to
left kinematic limit, as shown by x
3
in Fig. 14, for the flight of foot 3 and 2. Then,
referring to Fig. 6, at segment 16, which corresponds to 64 step counts, the robot is
being moved forward and to its right kinematic limit for the flight of foot 1 and 0
which is the last foot to fly to a new stroke position. Coordinate z
3
begins in stroke
and then moves to a new position with y
3
showing the flight of the foot at the same
step count interval.
Fig. 14: Operational coordinates generated and executed for leg 0, joint 3, robot’s reference
frame.
It could be noted the very short time the foot have to flight to a new position
represented by y
3
coordinate which is synchronized with the z
3
transverse path.
From the x
3
coordinate display it can be seen that foot 0 only flies at the end of the
x
3
lateral movements and after foot 2 flight.
-0,30
-0,25
-0,20
-0,15
-0,10
-0,05
0,00
0,05
0,10
1
101
201
301
401
501
[m]
x
3
y
3
z
3
S
troke
F
light
F
light
Fig. 15: Joint
coordinates ge
Fig. 15shows
joint coordinates
joint 3,
in robot’s reference frame
build the foot trajector
The angle
θ
0
is the
body laterally to the left kinematic limit when robot starts walking.
shows flat segments, where the legs on the same side fly to new position, and i
clined
segments corresponding to the lateral excursion of the robot body from one
side to another.
The robot body stays in horizontal plane as the roll movement
counter clockwise direction for all the legs. Then a clockwise rolling movement is
applied to a
ll legs to move the robot body to the right kinematic limit.
the end of each flat
segment corresponds
quired
greater excursion of the joint
during flight.
The angle
θ
1
is the
backward the thigh
. The
interval, where a greater excursion of the joint angle is required. It can be seen
a small r
ipple synchronized with
position from left to right kinematic limit which requires stretching of the leg
length.
The angle
θ
2
is the
backward the shank
and it can be seen the same ripples described for
occurs synchronized with the flight interval of the foot.
The angle
θ
3
is the ankle joint
and it can be seen the same ripples described f
above,
also occurs synchronized with the flight interval of the foot.
θ
0
coordinates ge
nerated and executed for leg 0, joint 3, robot’s
reference frame
joint coordinates
(
θ
0
,
θ
1
,
θ
2
,
θ
3
)
generated and executed for leg
in robot’s reference frame
, using 108 stroke points and 20
flight points to
build the foot trajector
y.
is the
hip joint 0
roll degree of freedom which moves the robot
body laterally to the left kinematic limit when robot starts walking.
The
shows flat segments, where the legs on the same side fly to new position, and i
segments corresponding to the lateral excursion of the robot body from one
The robot body stays in horizontal plane as the roll movement
counter clockwise direction for all the legs. Then a clockwise rolling movement is
ll legs to move the robot body to the right kinematic limit.
The ripple at
segment corresponds
to the foot flight
interval, where
greater excursion of the joint
0
to keep the vertical movement of the foot
is the
hip joint 0
pitch degree of freedom which moves for and
. The
discontinuities in this curve also occur in the foot flight
interval, where a greater excursion of the joint angle is required. It can be seen
ipple synchronized with
the interval which
θ
0
is changing
the robot body
position from left to right kinematic limit which requires stretching of the leg
is the
knee joint 1
pitch degree of freedom which moves for and
and it can be seen the same ripples described for
θ
1
which
occurs synchronized with the flight interval of the foot.
is the ankle joint
2
pitch degree of freedom which rotates the foot
and it can be seen the same ripples described f
or
θ
1
which
for the same reasons as
also occurs synchronized with the flight interval of the foot.
[Step counts]
θ
1
θ
2
θ
3
15
reference frame
.
generated and executed for leg
0,
flight points to
roll degree of freedom which moves the robot
The
θ
0
line
shows flat segments, where the legs on the same side fly to new position, and i
n-
segments corresponding to the lateral excursion of the robot body from one
The robot body stays in horizontal plane as the roll movement
is in
counter clockwise direction for all the legs. Then a clockwise rolling movement is
The ripple at
interval, where
it’s re-
to keep the vertical movement of the foot
pitch degree of freedom which moves for and
discontinuities in this curve also occur in the foot flight
interval, where a greater excursion of the joint angle is required. It can be seen
also
the robot body
position from left to right kinematic limit which requires stretching of the leg
pitch degree of freedom which moves for and
which
, also
pitch degree of freedom which rotates the foot
for the same reasons as
16
3.4. Operational and joint coordinate in curved walking
Fig.
16
shows the stroke and flight paths of front left leg 0, and front right leg 2,
in leg’s operational coordinate system, with the robot performing a curve with a
radius of approximately 1.4 [m] and center of curvature to the right. It can be ob-
served that leg 0 stroke pitch is greater than that of the leg 2. As shown in Fig. 7
and according to the proposed model, foot 0 has a greater radius as it is in the out-
er portion of the curve and has to be moved a greater extension in each step count.
Fig. 17 shows the roll angle
θ
0
for front left leg 0 and front right leg 2. Accord-
ing to the curve model, it can be seen the reversed slope of the lines in the period
of thrust, corresponding to lateral displacement, in contrast to the flat lines in the
thrust straight path, shown in Fig. 14. These lines correspond to the feet 0 and 2
lateral reversed movements that will turn right the robot body.
Fig. 16: Front left foot 0, and front right foot 2stroke and flight paths in leg’s reference frame.
The discontinuities that occur in joint roll angle
θ
0
curves in leg 0 and leg 2 are
due to the additional excursion needed for the foot being lifted vertically to go to a
new stroke position, while the robot body is being moved laterally.
Fig. 18 shows Guará robot turning right in a 90
0
corner with a radius of approx-
imately 1.4 [m]. In Fig. 18(a) the front right foot 2 is flying to a new stroke posi-
tion. In Fig. 18(b) the foot is landed, and the right rear foot 3 is flying, and it can
be seen that the foot excursions a little to the center of the curve. In Fig. 18(c) the
robot is moving forward and is being turned right as the front feet are being strok-
ing from the center and the rear feet are being stroking out from the center of the
curve. In Fig. 18(c), (d) and (e) the outside legs 0 and 1 are then moved to a new
stroke position.
-0,08
-0,06
-0,04
-0,02
0
0,02
0,04
0,06
0,08
1 96 191 286 381 476 571 666
[m]
[Step counts]
Leg 0
Leg 2
17
Fig. 17 Joint coordinates generated and executed for leg 0, joint 3 in leg’s reference frame.
Fig. 18: Guará robot turning right in a 90
0
corner.
3.5. Obstacle overcoming implementation
To overcome obstacles the proposed approach uses only the feet contact keys
information and perform a sequence of movements to move the robot to overcome
the obstacle, or deviate from it, preserving its balance and stability. To start over-
coming any obstacle, before any leg aerial movement is done, the robot is moved
to a know position that ensures the highest stability margin with the other legs that
will remain in support.
-0,20
-0,15
-0,10
-0,05
0,00
0,05
0,10
0,15
0,20
1
100
199
298
397
496
595
694
[Step counts]
[rd] θ
0
, leg 0 flight θ
0
,
l
eg 2
flight
(a)
(b)
(c)
(d)
(e)
(f)
θ
0
, leg 0
stroke
θ
0
, leg 2
stroke
18
3.5.1. Overcoming a step up obstacle
The Guará quadruped robot is presented in Fig. 19,
Fig.
20
and Fig. 21 going up a step up in front of its walking path. The step is
indefinitely placed and will only be perceived by the robot at the touch of one
front feet key, which causes a state change in the feet key pattern.
In Fig. 19(a), the robot is walking, with right rear foot 3 flying to a new stroke
position, and in Fig. 19(b), the front right foot collides with the step up. In Fig. 19
(c), it’s started a movement sequence that moves back and raises the right front
feet 2 to its flight start position and moves the robot body left and back to its ki-
nematic limits, before moving the foot 2 to the step up position.
In
Fig.
20
(a) the front right foot 2 is being moved to the step up position. In
Fig.
20
(b) the front right foot 2 is already in the step up position and starts mov-
ing the robot body to the right kinematic limit before moving the front left foot 0
to the step up position. In
Fig.
20
(c) the front left foot 0 is being moved up to the step up surface and in
Fig.
20
(d) the front left foot 0 is landed in the step up surface and the robot
starts walking with the front feet 0 and 2 on the step upper surface and the back
feet 1 and 3 in the step lower surface.
In Fig. 21(a) the robot is walking and leg 3 collides with the step and the same
sequence done for the front 0 and 2 feet is repeated for the rear 1 and 3 feet. In Fig.
21(b) the right rear foot 3 is moved back and raised to its flight start position. the
robot is being moved forward and to the left kinematic limit before the lifting of
the rear right foot 3 to the step up. In Fig. 21(c) the rear right foot 3 is landed in
the step up, and the robot is being moved forward and to the right kinematic limit.
(a)
(b)
(c)
Fig. 19: Gait showing the flight collision of the foot 2 on the first run
In Fig. 21(d) the robot is still being moved forward and laterally to the right ki-
nematic limit, before the lifting of the rear left foot 1 to the step up position. In
Fig. 21(e) the robots is lifting up the rear left foot 1, which is landed in step up po-
sition in Fig. 21(d). The task of lifting the entire robot to the step up surface is
completed with the four foot supporting in the upper step.
19
Fig. 20: Front feet going up a step.
(a) (b) (c)
(e) (f)(d)
Fig. 21: Positioning the rear legs on the step up to restart walking.
4. CONCLUSIONS AND FUTURE WORK
The method used to address the problem of obstacle overcoming was to divide
a complex movement task in a group of simple movements implemented in a
known sequence that ensures robot stability, rather than develop a complex model
to search a generic solution for the problem.
The main characteristic of this approach is that it allows the solution space di-
mension reduction and that it has no limits, being able to implement complex tasks
such as going up in a stairs. The limits are the robot’s leg’s kinematic limits, which
are very closed in Guará. Robots with stronger actuators and wider kinematic lim-
(a)
(b)
(c) (d)
20
its could support the implementation of libraries of obstacles, associated with the
work environment of low speed robots. It could be seen that that, according to ob-
stacle shape and dimensions acquired from contact keys and to the robot kinematic
limits, sets of simple movements are combined to overcome the obstacle giving
navigation skill and autonomy to the legged robot in an environment with known
obstacles.
Although it’s not known the time of implementation and enforcement of more
sophisticated models to perform the same tasks, the approach probably makes the
implementation slower, but allows complex movements sequences implementa-
tions while ensuring the robot stability.
A
CKNOWLEDGMENT
To Automatica Tecnologia S/A, the resources and manpower to build the robot
Guara and for the experiments that were performed.
R
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