KINEMATICS, DYNAMICS AND TRAJECTORY GENERATION
OF A THREE-LEGGED CLIMBING ROBOT
Tarun Kumar Hazra and Nirmal Baran Hui
Department of Mechanical Engineering, National Institute of Technology, Durgapur, West Bengal, 713209, India
Keywords: Climbing Robots, Tripod Robot, Kinematics analysis, Dynamic Analysis, Trajectory Generation.
Abstract: In the present paper, an attempt has been made to design a three-legged climbing robot. Each leg of the
robot has been considered to have two revolute joints controlled separately by two differential drive motors.
Both forward and inverse kinematics analysis have been conducted. The problem of trajectory generation of
each joint (both for swing phase and support) has been solved to suit the basic motion laws of Newton's.
Dynamic analysis of each link of all the legs has been derived analytically using Lagrange-Euler
formulation. Both kinematic and dynamic analysis models of the robot have been tested through computer
simulations while the robot is following a straight line path. It is important to mention that the direction of
movement of the robot has been considered in the opposite direction of the gravitational acceleration.
1 INTRODUCTION
It is extremely difficult to develop a robot which can
manoeuvre freely in rough terrain, specifically in
stiff surfaces. There are robots specifically designed
to perform pre-defined task and move in a particular
terrain. For example, wheel robots are good for flat
surface movement over legged robots because of
higher speed and less hazard like gate planning. On
the other hand, legged robots are preferred in uneven
surfaces and for staircase ascending and descending
purposes because of relatively better dynamic
stability (Song and Waldron, 1989). Therefore, there
is a huge demand of a robot, which is capable of
manoeuvring all type of landscapes while carrying
some pay-load. If it is so, then we can use them for
the purpose of surveillance, military operation,
exploration etc.
Quite a large number of researchers developed
and/or analysed biped (Vukobratovic et al., 1990,
Goswami, 1999), quadruped (Koo and Yoon, 1999)
and hexapod robots (Barreto et al., 1998, Erden and
Leblebicioglu, 2007). There are many advantages of
legged robots over the wheeled robots and some
disadvantages too. The main disadvantage is that a
legged robot needs to plan both its path as well as
gait (the sequence of leg movement) simultaneously
during locomotion. However, it is extremely
difficult job and complexity increases as the number
of legs increases. As a result, stable gait generation
of a hexapod robot is more critical than a quadruped
robot. On the other hand, hexapod robot is more
statically as well as dynamically stable than the
quadruped or the biped one. It is because of the fact
that for maintaining stability of a multi-legged robot,
its projected center of gravity (CG) should lie within
its support region, which is a convex hull passing
through its supporting feet. As the number of legs
reduces, number of supporting feet reduces and the
convex hull becomes smaller. Therefore, it is a
fertile area of research and many unsolved research
problems still exist.
It is also important to mention that research with
the robot having odd number of legs is limited. Bretl
et al. (2003) have presented a framework for
planning the motion of three-legged climbing robots.
They have given stress mostly on the development
of motion planning strategy. On the other hand, it is
necessary to analyze kinematics and dynamics of
any robot before assessing its stability or controlling
the robot. There exist less number of published
article dealing the issues of gait planning and
dynamic stability of three-legged robot. It is
nevertheless to mention that the work of Bretl et al.
(2003) is inspiring in this context. During
locomotion, at least one leg must be in swing phase
(i.e., ground reaction forces in that leg would be
zero) and it results in instability of the robot. This
problem becomes highly complex, if it is planned to
move in the uneven surface.
161
Kumar Hazra T. and Baran Hui N..
KINEMATICS, DYNAMICS AND TRAJECTORY GENERATION OF A THREE-LEGGED CLIMBING ROBOT.
DOI: 10.5220/0003540701610166
In Proceedings of the 8th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2011), pages 161-166
ISBN: 978-989-8425-74-4
Copyright
c
 2011 SCITEPRESS (Science and Technology Publications, Lda.)

Rest of the paper is structured in the following
manner. In Section 2, both forward as well as the
inverse kinematics of the said robot has been
discussed. The foot trajectory planning of the robot
has been explained in Section 3 and formulation of
the dynamics model has been presented in Section 4.
Results are presented and discussed in Section 5.
Finally, some concluding remarks have been made
and scope for future work has been indicated in
Section 6.
2 KINEMATIC MODEL
In this paper, an attempt has been made to develop a
suitable model of a planar three legged robot as
shown in Figure 1.
Figure 1: A planar three-legged robot.
Following assumptions are considered during the
kinematic and dynamic analysis of the robot.
• Links of the robots are made of rigid bodies
and their physical properties are considered
to be constant,
• The Center of Gravity (CG) of the robot
body is assumed to be coincided with the
geometrical center of the body,
• During locomotion, trunk of the robot is
considered to be parallel to the plane on
which the robot will be moving. Also,
height between the trunk and maneuvering
plane has been considered to be constant
equal to
h .
• The direction of gravitational acceleration
has been considered along the -ve Y-
direction of the body attached coordinate
frame of the robot.
A possible kinematic posture of the robot model is
shown in Figure 2. The robot consists of a trunk of
triangular cross section with each side is equal to a
and three legs, which are symmetrically distributed
around the three sides of the triangular trunk body.
Each leg has two links connected each other and
with the trunk by two rotary joints. It is also
important to mention that each joint will be
controlled separately using differential drive DC
servo motors. The Denavit-Hartenberg (D-H)
notations (Denavit and Hartenberg, 1955) have been
followed in kinematic modeling of each leg. The
base frame (Σ
0
) is placed at the centroid of the robot.
The other frames (Σ
1
, Σ
2
, Σ
3
, Σ
4
etc.) are defined as
body frames and are placed at the different joints of
the robot. The ‘XY’ plane has been considered to be
parallel to the robot body and the ‘Z-axis’ of all the
joints is made vertical to the robot body. Table 1
shows four D-H parameters of a leg (say, i), namely
link length (a
j-1
), link twist (α
j-1
), joint distance (d
j
)
and joint angle (θ
j
) by following the concept
described in Craig (Craig 1986).
Figure 2: A 2D schematic sketch showing the frames
assigned to the first leg of the robot.
Table 1: D-H parameter table for leg-i.
Joint No. (j) α
j-1
a
j-1
d
j
j
θ
CG
0 0 0
1
θ
i
1
0
/2 3a
0
2
θ
i
2
0 L
1
0
3
θ
i
Tip point
0 L
2
-h 0
It is important to mention that for simplicity, link
lengths of all the legs are made same. Therefore, the
first link of a leg is denoted by L
1
and second link is
represented by L
2
. From the above relationship,
differences between the coordinates of the foot tip
point (x
end
i
, y
end
i
, z
end
i
) and CG (x
c
, y
c
, z
c
) of i-th leg
can be determined for the supplied joint variables
(θ
2
i
and θ
3
i
) as follows.
ICINCO 2011 - 8th International Conference on Informatics in Control, Automation and Robotics
162

i
1 1 12 2 123
iii
1 1 12 2 123
i
(-) ,
23
a
(-) s+Ls+Ls,
23
(-)
iiii
xendc
ii
yendc
i
zendc
a
pxx cLcLc
pyy
pzzh
⎫
==++
⎪
⎪
⎪
==
⎬
⎪
⎪
==−
⎪
⎭
(1)
Solving equation (1) algebraically, the joint
angles θ
2
i
and θ
3
i
can be calculated. There will be
two solutions for each posture of the robot and two
values of θ
3
i
can be determined as
(
)
(
)
{}
31 33 32 33
i
3
22
22
1112
3
12
atan2 , , and atan2 - ,
where c can be found out from the following equation
---
23 23
2
iiii ii
ii ii
xy
i
sc sc
aa
pcpsLL
c
LL
θθ ==
⎡⎤
⎧⎫⎧⎫
++
⎢⎥
⎨⎬⎨⎬
⎩⎭⎩⎭
⎢⎥
⎣⎦
=
(2)
It is important to note that the value of
3
c
i
must
lie between -1 and 1 and knowing the values of θ
3
i
,
two values of θ
2
i
can be obtained as
() ()
()
ii i
21 y 1 x 1 31
ii i
21 y 1 x 1 32
iiiii
31 1 1 2 1 31 1 1 2 1 31
ii
32 1 1 2
aa
θ = atan2 p - s , p - c -φ
23 23
aa
θ = atan2 p - s , p - c -φ
23 23
where φ =atan2 L s + L sin θ +θ ,L c + L cos θ +θ
φ =atan2 L s + L si
iii
iii
ii
⎡⎤
⎧⎫⎧⎫
⎨⎬⎨⎬
⎢⎥
⎩⎭⎩⎭
⎣⎦
⎡⎤
⎧⎫⎧⎫
⎨⎬⎨⎬
⎢⎥
⎩⎭⎩⎭
⎣⎦
() ()
(
)
iii
132 11 2 132
n θ +θ ,L c + L cos θ +θ
ii
(3)
One attempt has been made to find the reachable
workspace of the robot. It is graphically obtained in
the following manner. Firstly, movement of any two
legs was fixed and two links of the other leg is
rotated one after another manually and tested how
much they can rotate. Angular movement up to
which the joints could move before losing its
stability provided the reachable workspace. This is
important to mention that this test was carried out
manually and not optimal in any sense. In the
present study solutions belonging to the reachable
workspace and lying in non-overlapping zone, have
been considered.
3 PLANNING OF JOINT FOOT
TRAJECTORY
Motion planning of the robot can be done in three
stages (Jamhour and Andre, 1996). During this
study, following things are to be satisfied.
(i) Trajectory should be planned in such a way so
that the motion could be maintained smoothly
and uninterruptedly (Mi et al. 2011, Mohri et
al. 2001),
(ii) Joint angle values must satisfy the reachable
workspace of the robot.
3.1 Foot Trajectory Generation
In this paper, the joint trajectory is interpolated as a
linear function with parabolic blend at the beginning
and at the end of the trajectory to consider
continuous position and velocity (Craig, 1986).
Let us consider,
oij
θ
and
θ
fij
are the initial and
final joint displacements of the j-th link of leg ‘i’, t
fij
denotes the time interval of the j-th link of leg ‘i’
and
θ

cij
represents the constant acceleration during
the blended parabolic trajectory.
For the joint velocities to be continuous, the joint
velocity at the end of the first blend must be equal to
the beginning of linear segment i. e, at the point (t
bij
,
θ
ij
), where t
bij
denotes the time where first blend
will occur. Therefore, it must satisfy the following
equation.
2
0
4( )
11
()
22
cij
ij
fij fij
bij fij
cij
t
tt
θθθ
θ
×−× −
=× −


and
fij 0 fij 0
22
fij fij
44
tt
ij ij
cij
K
θ
θθθ
θ
×− ×−
≥=

(4)
The value of
K
must be greater than unity and
in the present study it has been considered to be
equal to 1.05. Finally, joint angle expression is
presented in equation (5) and joint speed and
accelerations are derived by differentiating equation
(5) with respect to time.
Joint Angle:
2
0
2
0
2
1
( ) , for 0
2
1
() , for ( )
2
1
( ) , for ( )
2
ij cij fij bij
ij ij cij bij cij bij bij fij bij
f
ij cij fij fij bij fij
ttt
ttttttt
tt tt tt
θθ
θθ θ θ
θθ
⎧
+→≤≤
⎪
⎪
⎪
=− + →<≤−
⎨
⎪
⎪
−
−→−<≤
⎪
⎩

 

(5)
3.2 Gait Planning Strategy
The tripod robot model is shown in Figure 1. During
locomotion, a one-step movement is normally
followed by human being and Bretl et al. (2003)
have mentioned that one step movement can also be
used for planning gaits of three-legged robots. In the
present work the gaits of the robot have also been
planned in the similar manner. It has been assumed
that during movement at a time only one leg will be
in swing phase and other two will be in support
KINEMATICS, DYNAMICS AND TRAJECTORY GENERATION OF A THREE-LEGGED CLIMBING ROBOT
163

phase. At first say leg-2 and leg-3 are in support
phase and leg-1 is in the swing phase moving along
a specified path. When leg-1 will reach to its goal
configuration, leg-1 and leg-3 will switch to support
phase, and leg-2 will be in swinging phase.
Thereafter, leg-3 will be in swing phase and the
other two will be in support phase. In this way the
tip of legs of the tripod reach to the new position
with different configurations and completes one
locomotion cycle.
Let us say, initial position of the geometric
center of the robot is (x
c
, y
c
, z
c
). The CG of the robot
is moving in a straight line path along ‘Y’ axis with
a constant speed. The time for a full locomotion
cycle is considered to be equal to
tΔ . After t
Δ
time
interval the new position of the CG is (x
c
, y
c
+
y
Δ
,
z
c
). The above movement is achieved in three stages.
For each stage, coordinates of the CG of the robot,
foot tip point of three legs at different instant of time
have been presented in Table 2.
Table 2: Positions of different parts of the robot at four
instant of time of a locomotion cycle.
Time
(
ttk=Δ
)
CG
Foot tip point
1
st
leg 2
nd
leg 3
rd
leg
k=0 x
c
,y
c
,z
c
x
1
, y
1
,
z
1
x
2
, y
2
,
z
2
x
3
, y
3
,
z
3
k=1/3
x
c
,(y
c
+
/3yΔ ), z
c
x
1
, (y
1
+
y
Δ ),z
1
x
2
, y
2,
z
2
x
3
, y
3,
z
3
k=2/3
x
c
, (y
c
+
2/3yΔ ),
z
c
x
1
, (y
1
+
y
Δ ),z
1
x
2
, (y
2
+
y
Δ )
,
z
2
x
3
, y
3,
z
3
k=1
x
c
, (y
c
+
y
Δ
), z
c
x
1
, (y
1
+
y
Δ
),z
1
x
2
, (y
2
+
y
Δ
)
,
z
2
x
3
,
(y
3
+
y
Δ
)
,
z
3
4 DYNAMICS OF THE ROBOT
Dynamics of different kind of robot have been
explained in (Mi et al. 2011, Mohri et al. 2001). In
the present paper, Lagrangian Euler-based
formulation has been used. Torque expression for
first joint of i-leg can be derived as follows.
()
()
22 2
111212221232
12112
2
22 212 3 3
22 123
212 33 2 3 1 1 12
2 2 123 2 1 12
13 13
2
12 12
13
12
1
2
2
1
2
ii
i
i
ii
c
i
ii i i i
iii
F
mL mL mL mLLc
mmLc
mL mLLc y
mLc
mLLs mgLc
mgLc mgLc M
τθ
θ
θθθ
⎡⎤
=+++
⎢⎥
⎣⎦
⎡⎤
+
⎡⎤
++ +
⎢⎥
⎢⎥
⎣⎦
+
⎢⎥
⎣⎦
−++
+++




(6)
Similarly, for second joint of each leg torques can be
calculated using the expression
22
22221232223
2
22 123 212 32
22123
13 13
12 12
+
1
2
ii i
i
iii
c
ii
F
mL mLLc mL
mLc y mLLs
mgLc M
τ
θθ
θ
⎡
⎤⎡ ⎤
=+ +
⎢
⎥⎢ ⎥
⎣
⎦⎣ ⎦
⎡⎤
⎡⎤
++
⎣⎦
⎣⎦
⎡⎤
+
⎢⎥
⎣⎦
 


(7)
Here,
12
and mm
denote mass of links 1 and 2,
respectively. Acceleration due to gravity is
represented by
g , speed of the CG of the robot is
denoted by
c
y

and
i
F
M
represents the torque due to
foot reaction forces at i-th Leg and it is zero for the
leg which is in swing phase. It is important to
mention that all the joint torque expressions have
been derived with respect to the coordinated frame
attached to the CG of the robot. The value of
g is
considered to be equal to
[
]
2
09.810/ms−
.
5 SIMULATION RESULTS
Developed mathematical models have been tested
through computer simulations. In the present case,
the leg stroke of the one step movement (Δy), body
height (
h ), side length of the triangular-shaped cart
(
a ) and time step are assumed to be equal to 0.03m,
0.05m, 0.12m and 6 seconds, respectively. During
analysis, following data have been considered:
L
1
=0.04m, L
2
=0.06m, m
1
=0.002Kg, m
2
=0.012Kg,
coordinates of CG (0,0,0.05)m, foot tip points during
starting for the first leg (0.09,0.05,0)m, for the
second leg (-0.09,0.05,0)m and third leg at (0.01,-
0.11,0)m.
It is important to mention that forward
kinematics always leads to a single pose matrix for
any robot. However, several robot configurations
(i.e., joint angle values) may result in the same foot
tip point corresponding to a fixed location of the
CG. In the present study, two solutions are obtained
that will generate the same foot tip point of the
robot. It provides freedom in the trajectory planning.
In the present study, only those combinations of
solutions have been preferred, which are falling
within the reachable workspace of the robot.
Maximum joint angle speed and acceleration values
at different instant of time for legs 1, 2 and 3 are
calculated and it has been observed that those values
are higher during swing phase than the support
phase. It may be due to the absence of support
reaction forces during swing phase.
ICINCO 2011 - 8th International Conference on Informatics in Control, Automation and Robotics
164

From equations (6) and (7), it is clear that joint
torques is comprised of three components: inertial
(M Comp), centrifugal and/or Coriolis (H Comp)
and Gravity (G Comp) (refer to Figure 3).
-3.E-05
-2.E-05
-2.E-05
-1.E-05
-5.E-06
0.E+00
5.E-06
1.E-05
2.E-05
2.E-05
0123456
Time (s)
M, H Comp of Torques (N-m
)
6.E-03
6.E-03
6.E-03
6.E-03
6.E-03
6.E-03
6.E-03
6.E-03
6.E-03
6.E-03
6.E-03
G Comp of Torques (N-m
)
M Comp
H Comp
G Comp
(a) Joint 1 of Leg-1.
-2.E-05
-1.E-05
0.E+00
1.E-05
2.E-05
3.E-05
4.E-05
5.E-05
6.E-05
0123456
Time (s)
M, G Comp of torque (N-m
)
0.E+00
5.E-07
1.E-06
2.E-06
2.E-06
3.E-06
3.E-06
4.E-06
4.E-06
5.E-06
H Comp of torque (N-m
)
M Comp
G Comp
H Comp
(b) Joint 2 of Leg-1.
-2.E-05
-2.E-05
-1.E-05
-5.E-06
0.E+00
5.E-06
1.E-05
2.E-05
2.E-05
0123456
Time (s)
M, H-Comp of torque (N-m
)
-3.7E-03
-3.7E-03
-3.7E-03
-3.7E-03
-3.6E-03
-3.6E-03
-3.6E-03
-3.6E-03
-3.6E-03
-3.5E-03
-3.5E-03
G-Comp of torque (N-m
)
M Comp
H Comp
G Comp
(c) Joint 1 of Leg-2.
-1E-04
-1E-04
-1E-04
-8E-05
-6E-05
-4E-05
-2E-05
0E+00
2E-05
4E-05
0123456
Time (s)
M, G Comp of torque (N-m
)
0E+00
1E-08
2E-08
3E-08
4E-08
5E-08
6E-08
7E-08
8E-08
H Comp of torque (N-m
)
M Comp
G Comp
H Comp
(d) Joint 2 of Leg-2.
-6.0E-05
-4.0E-05
-2.0E-05
0.0E+00
2.0E-05
4.0E-05
6.0E-05
8.0E-05
0123456
Time (s)
M, H Comp of torque (N-m
)
-4.E-03
-3.E-03
-3.E-03
-2.E-03
-2.E-03
-1.E-03
-5.E-04
0.E+00
G Comp of torque (N-m
)
M Comp
H Comp
G Comp
(e) Joint 1 of Leg-3.
-6.E-05
-4.E-05
-2.E-05
0.E+00
2.E-05
4.E-05
6.E-05
8.E-05
0123456
Time (s)
M, G-comp torque (N-m)
-1.E-07
9.E-07
2.E-06
3.E-06
4.E-06
5.E-06
H-comp torque (N-m)
M Comp
G Comp
H Comp
(f) Joint 2 of Leg-3.
Figure 3: Contribution of M-comp, H-comp and G-comp
of on the joint torques over the entire locomotion cycle.
During this study, following common
observations are made.
(i) During the first 1/4-th and last 1/4-th time
period of every stages of locomotion cycle,
inertial component of torque has been found to
be approximately constant and in-between it is
observed to be zero. It is because of the
acceleration distributions considered in the
study. That is why, it is almost constant for
second joint, whereas it is varying little bit for
the first joint. These small variations might
have occurred due to the contributions from the
other joints.
(ii) For the First Joint: Contribution of gravity
component in all the legs has been found to be
more compared to the other two components.
Gravity component has been observed to be
varying in the positive side only for Leg 1. On
the other hand it is varying both in positive as
well as in negative side for the other two legs.
It clearly indicates that the requirement of
torque for the first leg is more in compared to
the other legs.
(iii)For the Second Joint: Contribution of
centrifugal and/or Coriolis component in all the
legs is observed to be very low. It might be due
to the fact that first joint does not have any
contribution on this component of torque.
Total torque requirement for the first joint of
KINEMATICS, DYNAMICS AND TRAJECTORY GENERATION OF A THREE-LEGGED CLIMBING ROBOT
165

every leg has been found to be more in
compared to the second joint. Moreover, for
controlling of first joint of Leg 1, torque
requirement is observed to be considerably
higher than the other legs. It is also to be noted
that torque requirement for each joint during
swing phase is less in compared to the support
phase. This is because of the presence of
support reaction forces.
6 CONCLUDING REMARKS
This paper presented kinematics, dynamics and
trajectory planning of a three-legged robot. Direct
and inverse kinematics has been analyzed, while the
robot is following a straight line path. Movement of
the robot is ensured considering that at any instant of
time only one leg can be in swing phase while the
other two will provide necessary support. Joint
torques has been computed continuously for the full
locomotion cycle and compared between the legs.
All the developed mathematical models have been
tested through computer simulations on a P-IV PC.
Computational complexity of the code developed for
solving the mathematical expressions is found to be
very low, making it suitable for on-line implementa-
tions. More torque requirement has been observed
for the first joint of each leg and for every joint
during support phase than in swing phase. For the
first joint of each leg, torques varies between
(-0.004N-m to 0.00585N-m) and those for the
second joint vary between (-0.00003N-m to
0.00007N-m). This is a very low torque requirement
and low power servo motors will be sufficient to
control them.
The present study can be extended in a number
of ways, such as, static and dynamic stability
analysis, optimization of joint torques of the robot
while it is following a curvilinear path. Moreover,
presently the performance of the robot has been
tested through computer simulations. Real
experiments will be more interesting in this regard.
The authors are working with some of these issues
presently.
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