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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