Optimum stable gait planning for a 4DOF biped robot using … · · 2012-12-07Position of Center of Gravity (COG) and Zero Moment Point (ZMP) ... for getting the optimum stable

Int j simul model 10 (2011) 4, 177-190

ISSN 1726-4529 Professional paper

DOI:10.2507/IJSIMM10(4)2.186 177

OPTIMUM STABLE GAIT PLANNING FOR AN 8 LINK

BIPED ROBOT USING SIMULATED ANNEALING

Sudheer, A. P.*; Vijayakumar, R.

* & Mohandas, K. P.

**

* Department of Mechanical Engineering, NIT Calicut, India

** Department of Electrical Engineering, NIT Calicut, India

E-Mail: [email protected]

Abstract

Biped or humanoid robots should have a higher level of autonomy to achieve different tasks

in service and manufacturing sectors. Biped locomotion is a complex dynamical task because

of the intermittent interaction with its environment and the ground through its feet. Most of

the previous gait planning efforts were made by complex and time consuming algorithms with

maximum one limb in the upper body for controlling the gaits. This work presents the

kinematic and dynamic modelling and simulation of 8 link biped robot walking with two

limbs in the upper body. Optimum cycloidal gait trajectory is simulated and compared

between the static and dynamic walking cases. Easy and comparatively less time consuming

simulated annealing algorithm is used for the optimization. Zero Moment Point criteria is

used for the stability analysis. (Received in December 2010, accepted in August 2011. This paper was with the authors 2 months for 1 revision.)

Key Words: Gait Planning, Sagittal Plane, Zero Moment Point, Simulated Annealing,

Degrees of Freedom

1. INTRODUCTION

One of the main objectives of humanoid robotics is to create autonomous systems for

imitating humans at a wide variety of dynamic, complex and novel environments. Experts

from various fields of science, engineering and linguistics have tried to combine their efforts

to create a robot as human-like as possible. Although this field has witnessed tremendous

progress in recent years, realistic, periodic and rhythmic gait generation is not yet achieved

due to the limitations in kinematics, dynamics and control techniques.

Biped robot walking consists of two stages, namely single support phase in which one

foot is in contact and double support phase in which both feet are in contact with the ground.

In a complete walking cycle, stability has to be analysed in both the above phases during

static and dynamic conditions. Position of Center of Gravity (COG) and Zero Moment Point

(ZMP) are two important criteria for analyzing the static and dynamic stability of legged

robots. Qiang Huang et al. [1] interpreted ZMP, as that point on the ground about which the

sum of all moments of the active forces is equal to zero. This concept adopts Vukobratovic et

al.’s [2] claim that the ZMP has been vaguely related to the ground surface, regardless of

referring to the support foot polygon of the legged robot. Researchers have been working on

gait planning and executing gaits for humanoid or biped robots based on genetic algorithms

(GA), artificial neural networks (ANN) and fuzzy logic control (FLC) and other conventional

algorithms [3-12] considering stability mostly only in the sagittal plane. Goswami Dip et al.

[3] optimized the walking gaits using genetic algorithm (GA) by considering the Zero

Moment Point (ZMP) and walking speed. The optimal walking gaits are also experimentally

realized on the biped robot also. Maitray Shrivastava et al. [4] used a trajectory generation

method using GA for a 8 degrees of freedom (DOF) robot that can walk on flat terrain and

mailto:[email protected]

Sudheer, Vijayakumar, Mohandas: Optimum Stable Gait Planning for an 8 Link Biped …

178

climb stairs with deformation at the sole. This method incorporates the dynamics of an actual

8 DOF robot to find the most energy optimal gait.

Jih-Gau Juang [5] developed a technique based on artificial intelligence for a 5 DOF biped

robot for the trajectory control. A three layered ANN is used as a controller. It provides the

control signal in each stage of walking gait. Chenglong Fu and Ken Chen [6] proposed

optimum walking control for humanoid stair climbing, which consists of a stair climbing gait

and a sensory control strategy. The climbing gait is considered as feed forward control and is

modeled to satisfy the environmental, kinematic and ZMP constraints. J. B. Goncalves and D.

E. Zampieri [7] used a recurrent neural network to determine the trunk motion for a biped

walking by considering the ZMP criterion. Sathoshi Ito et al. [8] introduced a ZMP feedback

control algorithm for 4 DOF biped balance in the lateral stepping motion. Stability of the

stepping motion is examined analytically and through computer simulation. Effectiveness of

this method is demonstrated by biped robot experiment during the lateral stepping motion. N.

Bigdeh et al. [9] developed a method for dynamic modelling of a five link robot based on

neural network. This method considerably reduces the complexities in solving the dynamic

model equations. Jong Hyeon Park and Moosung Choi [10] proposed a method that

minimizes the energy consumption in the locomotion of a 7 DOF biped robot. A real coded

GA is employed in order to search for the locomotion pattern and optimum center of mass

locations of links of the biped robot. Rahul Kumar Jha et al. [11] modelled the gait generation

problem for the step climbing using fuzzy logic controller and its rule base is optimized

offline using GA in the simulation of the gait trajectory. Panda Ranga Vundavilli and Dilip

Kumar Pratihar [12] proposed a method for the design of a neural network based gait planner

for a two legged robot negotiating uneven terrains. Genetic algorithm is used to provide

training off-line to the gait planner and the trajectory is simulated. These authors have

extended their work for biped motion up and down the sloping surfaces and ditch crossing by

combining soft computing techniques like GA, FLC and ANN.

In this paper, Simulated Annealing (SA), an intelligent random-search technique is used

for getting the optimum stable gait trajectory in both static and dynamic conditions of biped

robot by carefully controlling the rate of control parameter. An 8 link biped robot with

frictionless joints is used in this work for the stability analysis and simulation. Static and

dynamic stability cases are considered separately, for the optimum gait planning and analysis.

Peak error in ZMP is minimized for getting the static and dynamic stability when biped gait is

generating one walking step. Simulated annealing algorithm gives the solution within a short

time for this problem. To the knowledge of the authors, no work based on simulated

annealing has been reported in the area of humanoid or biped robots. This paper is structured

as follows. In Section 2, both kinematic and dynamic modelling are described. The

fundamental theory of the center of mass (COM) and ZMP, in single and double support

phases are given in Section 3. Formulation of the problem and simulated annealing algorithm

are shown in Section 4 and 5 respectively. Section 6 deals with the optimal gait trajectory for

stepping motion along with the computer simulation results. Finally Section 7 presents the

concluding remarks with outlook.

2. MODELLING OF BIPED ROBOT

An important part of robot simulation is the treatment of robot kinematics and dynamics. The

advances in humanoid robotics and the increased usage of biped or humanoid robots have

raised the need for computer simulation of robots among the aims of which are the design of

new robots, task planning of existing robots, performance evaluation and cycle time

estimation. Modelling of robots includes both kinematic and dynamic modelling.


179

2.1 Kinematic modelling

There are different methods available for the kinematic modelling and analysis. In general, the

inverse kinematic problem can be solved by various methods such as screw algebra (Kohli,

1975), dual matrices (Denavit, 1956), iterative (Uicker, 1964), geometric approach (Lee,

1984) and decoupling of position and orientation (Pieper, 1968). Geometric approach is used

for deriving the inverse kinematic equations in this work. Biped robot model with mass of

each link, assumed to be concentrated at different position for the gait synthesis, is as shown

in Fig. 1. The coordinates of hip joint J3 and ankle joint J6 of the swing leg are (xh, yh) and (xe,

ye) respectively. Forward and inverse kinematic equations are formed for the motion and

stability analysis, based on the geometry of the biped. Inverse kinematic equations derived

based on the hip position for the single support phase (SSP) and double support phase (DSP)

are listed below. Joint positions θ2 & θ3 are same for both phases and θ1 & θ8 are calculated

from the simple geometry.

222

23

22

22

2

2arccosarctan

2hh

hh

h

h

yxl

llyx

x

y (1)

223

22

23

22

3

2arccosarctan

hh

hh

h

h

yxl

llyx

y

x (2)

226

27

26

22

6

)(2

)(arccosarctan

hhe

hhe

h

he

yxxl

llyxx

y

xx (3)

227

26

27

22

7

)(2

)(arccosarctan

2hhe

hhe

he

h

yxxl

llyxx

xx

y (4)

Figure 1: Biped robot model.


180

Inverse Kinematic Eqs. (1) to (6) gives the instantaneous angular positions of various

links during the stepping motion. The motion of a biped robot comprises of time-functions of

angular positions and velocities of the joints and links of the robot. The straight forward

approach is to generate the joint time trajectories by solving inverse kinematics, to maintain

physical stability of the biped. Eqs. (5) and (6) are used for finding joint parameters of the

sixth and seventh links in SSP.

226

27

26

22

6

)()(2

)()(arccosarctan

lyyxxl

lllyyxx

lyy

xx

ehhe

ehhe

eh

he (5)

227

26

27

22

7

)()(2

)()(arccosarctan

2 lyyxxl

lllyyxx

xx

lyy

ehhe

ehhe

he

eh (6)

Joint angles θ4 and θ5 are obtained directly from the simulation by using the simulated

annealing algorithm in both static and dynamic cases. Table I shows the parameters of various

links of the biped robot. The moment of inertia values listed in the table are used for the

dynamic stability analysis.

Table I: Parameters of biped robot links.

Link (i) Mass (mi)

[kg]

Length (li)

[m]

Center of mass

from the joint (ri)

[m]

Moment of

inertia (Ii)

×10-2

[kgm2]

1 0.06 0.05 & a = 0.05 0.03 0.0188

2 1.5 0.30 0.24 2.3400

3 2.5 0.35 0.28 5.3084

4 8 0.25 0.2 8.6666

5 2 0.15 0.12 0.7800

6 2.5 0.35 0.28 5.3084

7 1.5 0.30 0.24 2.3400

8 0.06 0.05 & a = 0.05 0.03 0.0188

2.2 Dynamic modelling

Dynamic model of a robot is useful for the computation of torque and forces required for

execution of a typical work cycle, which is vital information for the design of links, joints

drives and actuators. A general dynamic model for biped walking related to the joint

coordinates vector and joint torque vector without considering the friction and other external

disturbances is given in Eq. (7).

DGCM )(),()( (7)

where, M is the 8×8 inertia matrix, C = )()( hB is the 8×8 Coriolis and centrifugal matrix

and G = Cg(θ) is the 8×1gravity vector, D is the 8×8 coefficient matrix of joint torques.

M(θ) = [qij cos(θi – θj)]; B(θ) = [qij sin(θi – θj)]; C = – [diag(hi)];

θ = [θ1 θ2 θ3 θ4 θ5 θ6 θ7 θ8]T; Th 2

82

72

62

52

42

32

22

1)( ;

g(θ) = [sin θ1 sin θ2 sin θ3 sin θ4 sin θ5 sin θ6 sin θ7 sin θ8]T;


181

τ = [τ1 τ2 τ3 τ4 τ5 τ6 τ7 τ8]T and

1 0 0 0 0 0 0 0

0 1 -1 0 0 0 0 0

0 0 1 -1 0 0 0 0

0 0 0 1 -1 0 0 0

0 0 0 0 1 0 0 0

0 0 0 0 0 1 -1 0

0 0 0 0 0 0 1 -1

0 0 0 0 0 0 0 1

τi is the torque at the ith

joint θi and

.

i are the angular position and velocities of link i. The

parameters qij and hi are constants derived by Langrangian equations of motion.

3. STABILITY CRITERIA

In static gait planning problems, biped robot is stable if the projection of COM falls within the

convex hull of the foot support polygon. In dynamic locomotion, link acceleration, inertial

forces, and ground reaction force are also to be considered and the ZMP should be within the

convex hull of the foot polygon for satisfying the stability criterion. The dynamic locomotion

is highly nonlinear and difficult to analyze in real environment. The condition in the static and

dynamic stability of the biped along the sagittal plane during the single support phase is that

the location of the ZMP (Xzmp, 0, 0) must be inside the convex hull of the supporting foot.

Therefore, Xmin < Xzmp < Xmax, based on the biped foot dimension. In double support phase

ZMP or projection of COM should lie within the convex hull of support polygon formed by

left and right foot. In the case of static stability analysis Xzmp equation contains only mass and

the displacement terms. The Xzmp can be expressed as:

Xzmp =

8

1

8

1

8

1

8

1

)(

)(

iii

i iiiiiii

iii

gym

wIyxmxgym

, Yzmp = 0 and Zzmp = 0 (8)

4. PROBLEM FORMULATION

Mathematical interpolation is one of the simplest methods for providing suitable gait

trajectory in accordance to the given boundary conditions. Generally the trajectory of the

swing foot of human is a cycloidal profile in normal walking [5]. Cycloidal profile reduces

the effects of sudden acceleration at the beginning and deceleration at the end during the gait

generation. In this simulation, swing foot trajectory is taken as cycloid and the hip joint

trajectory is taken as a modified cycloid. The modified Cartesian cycloid equation used for the

hip joint motion is given as:

x = rx (θ – sin θ) and y = ry (1 – cos θ) (9)

The objective of this work is to generate an optimal cycloidal gait trajectory by

minimizing peak error in ZMP by changing the hip and upper body link positions of the biped

robot. If the ZMP variations are minimum we can select a minimum foot size so that biped

can move easily without any external and self collisions. One walking step begins with the

heel of the right foot leaving the ground. Single support phase is the predominant portion

D =


182

because its duration is longer than double support and transition phases. In this paper, a

complete cycloidal gait trajectory is optimized based on the ZMP value when the hip joint is

moving through number of modified cycloidal paths and also by changing upper body

positions. The objective function is:

Minimize f = Minimize (Max(XzmpError)) = Minimize (Max[|(Xzmp – Xzmp – reference)|]) (10)

where: ,,,,(),,,,,,,( 5544 iiiififihihiyx yyxxfyxrrfXzmpError iw );

with constraints: 0 ≤ rx ≤ 30; 0 ≤ ry ≤ 30; -30 ≤ xhi ≤ 0; 40 ≤ yhi ≤ 70; 0 ≤ θ4i ≤ π/4;

0 ≤ θ4f ≤ π/4; 0 ≤ θ5i ≤ π/4; 0 ≤ θ5f ≤ π/4.

Cartesian and joint positional constraints used in both static and dynamic walking

simulation are in centimeters and in degrees respectively. The variables which define the

modified cycloidal path of hip are rx, ry, xhi, and yhi ; where rx and ry are the span and height

and (xhi ,yhi) is the starting coordinates of cycloid. Other variables that determine the optimal

gait trajectories are θ4i, θ4f, θ5i and θ5f , the initial and final upper body joint variables. These

eight variables determine the optimal gait trajectory of the swing foot with minimum stance

foot size. Origin of the inertial reference is taken as the ZMP reference in this work.

Following assumptions are made for the optimal gait planning.

Swing foot in the first half of the stepping time moves with uniform acceleration and in

the second half moves with uniform deceleration.

During single support phase, stance foot is assumed to remain in flat contact on the

ground.

The coefficient of friction for rotation at joints is zero and the links are perfectly rigid.

The contact between the floor and the foot is smooth and floor for walking is rigid.

5. SIMULATED ANNEALING

Simulated annealing (SA), a random-search technique, was developed by Kirkpatrick et al.

[13]. This is a powerful optimization method inspired by metallurgical annealing process

which exploits an analogy between the way in which a metal cools and freezes into a

minimum energy crystalline structure. SA's major advantage over other methods is an ability

to avoid becoming trapped in local minima. It has been proved that by carefully controlling

the rate of cooling of the temperature, SA can find the global optimum in a reasonable amount

of time. SA algorithm does not need to calculate the gradient descent that is required for most

traditional search algorithms. Various steps involved in the simulated annealing based

optimization are depicted below.

Step 1: Choose a set of initial variables or point Xt, say X1 = (rx, ry, xhi, yhi, θ4i, θ4f, θ5i, θ5f) and

set a starting temperature Ts a sufficiently high value, cooling rate Cr, and set final

temperature Te. The algorithm starts from a randomly chosen point X1 of which the fitness

value of the objective function, E(Xt) is computed.

Step 2: A neighbouring point Xt+1, say X2 gets chosen using Gaussian distribution and E(Xt+1)

is computed.

Step 3: Calculate ∆E = [E(Xt+1) − E(Xt)], if ∆E ≤ 0, i.e., the cost of the neighbouring point is

better than the cost of the point, the new point is accepted. Otherwise the new point is

accepted by a probability p = exp (−∆E / Ts) if the random number r generated in the range

(0, 1) is less than p. Else go to step 2. This completes one iteration of the simulated annealing

procedure.


183

Step 4: Calculate the new temperature, Tnew= Ts ×Cr and then check the terminating condition.

The algorithm is terminated when Tnew ≤ Te or a small enough change in fitness value is

obtained. Flow chart of this particular work with the selection of various parameters is given

in Fig. 2.

Figure 2: Flow chart for simulated annealing algorithm.

6. SIMULATION RESULTS AND DISCUSSION

Statically and dynamically stable optimal walking trajectory is of 8 links biped robot is

determined by using simulated annealing algorithm. Present paper gives an idea about the

influence of upper links and hip positions in the biped stable walking. Initialization module

includes the selection of control parameter Ts = 200, cooling rate Cr = 0.95 and the final

control parameter Te = 30 as shown in Fig. 2. These selected values are found to be good for

this problem because the solution is arrived at within a reasonable time. The possibility of

being trapped in a local minimum was overcome with more iterations of simulation within the

0 < r < 1


184

required value of fitness. Statically stable walking is analyzed by shifting only the masses of

various limbs of the robot without considering the parameters like acceleration, inertial forces

and other disturbances. But in dynamic stable walking these parameters are also considered in

the simulation. Simulation is carried out on a Core 2 Duo Intel processor using MATLAB.

Nine intermediate points in between starting and ending positions in the cycloidal

trajectory of the swing foot are considered for analyzing the gait. Initially the maximum value

of absolute ZMP error is determined out of all intermediate position ZMP errors in each

cycloidal motion based on modified cycloidal hip trajectory and upper body positions. Then

minimized the peak absolute error in ZMP in the solution space by using the simulated

annealing algorithm, which will give the idea of the required the foot size for an optimum gait

of the biped.

6.1 Static stable biped walking

Statically stable walking pattern is simulated and the kinematic diagram is given in Fig.3. The

required optimum value of peak error in ZMP is fixed in this case, that is when (XzmpError, 0, 0)

≤ (0.3, 0, 0) the simulation is stopped. The period required for one walking step is assumed as

5 s with a step length 20 cm. The simulation is carried out ten times for checking the number

of iterations and time taken for each fitness value. It is found that the time taken in all the

simulations is reasonable. Table II gives an idea of the time taken, peak error and the values

of variables of simulation of biped optimum gait planning when the robot is walking in

statically stable condition. Fig. 3 depicts the optimal gait trajectory of the swing foot of the

biped which satisfies the required static stability criteria. This stick diagram gives the idea of

the influence of the upper body and the hip motions for generating an optimum gait. Angular

positional variations of fifth, sixth and seventh links are more compared to other links as

shown in Fig. 4. That is the angular positional variations of these links help more to bring the

biped in statically stable condition. ZMP error variations and fitness variations are plotted in

Figs. 5 and 6 respectively. From the fitness plot it is clear that simulated annealing algorithm

helps to escape from the local minimum because of the probabilistic hill climbing

characteristics of SA and independency of initial conditions. This characteristics of SA helps

to find global minimum. Table II shows the comparison of parameters in ten set of static

stable simulations. Hip positions in 6th

, 7th

and 8th

simulation are same and reasonably good

with the same value of peak error. In 4th

and 5th

simulation peak error obtained is minimum

with the same variations of upper body and hip positions.

Figure 3: Static walking.


185

-40

-30

-20

-10

0

10

20

30

40

50

60

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time (S)

Jo

int

An

gle

s(D

eg

ree)

Theta 2

Theta 3

Theta 4

Theta 5

Theta 6

Theta 7

Figure 4: Variations of joint angles. Figure 5: Variation of ZMP.

Table II: Comparison of parameters in statically stable simulation.

Simulation

No.

Minimum

peak error

(< 0.3) [cm]

Time

taken

[s]

No. of

iterations

Variables

hs = [rx, ry, xhi, yhi, θ4i, θ4f, θ5i, θ5f ]

1 0.2851 115.55 439 [ 0.2483 3.0819 -7.7969 62.3894

0.2860 0.2321 0.7569 0.0594 ]

2 0.2851 114.97 436 [ 0.2483 3.0819 -7.7969 62.3894

0.2860 0.2321 0.7569 0.0594 ]

3 0.2479 119.96 459 [ 2.8911 4.9687 -6.7284 60.1678

0.2492 0.3673 0.1114 0.0522 ]

4 0.1931 133.83 524 [ 2.3783 3.1394 -6.3684 60.3379

0.2198 0.2832 0.0305 0.0037 ]

5 0.1931 132.70 536 [ 2.3783 3.1394 -6.3684 60.3379

0.2198 0.2832 0.0305 0.0037 ]

6 0.2969 131.17 511 [ 3.9685 5.0167 -3.3416 62.177

0.0221 0.1370 0.0065 0.4182 ]

7 0.2969 130.42 504 [ 3.9685 5.0167 -3.3416 62.177

0.0221 0.1370 0.0065 0.418 2]

8 0.2969 131.47 515 [ 3.9685 5.0167 -3.3416 62.177

0.0221 0.1370 0.0065 0.4182 ]

9 0.2926 139.08 521 [ 2.5563 3.1811 -4.0167 62.0584

0.0086 0.0309 0.5515 0.4858 ]

10 0.2926 140.09 534 [ 2.5563 3.1811 -4.0167 62.0584

0.0086 0.0309 0.5515 0.4858 ]


186

Figure 6: Variation of fitness function.

6.2 Dynamic stable biped walking

Dynamic walking analysis is important when robot is running, jumping or performing other

types of sudden or fast movement. Here, record as time for one walking step is 1 s and step

length is 20 cm. Kinematic diagram of dynamic walking is given in Fig. 7. When the required

peak error in ZMP value is very small, usually simulation time will increase or at times it may

not give optimum solution. During attempts with increased number of iterations it was found

that simulation takes more time and peak error value obtained is higher than in the static case.

Hence a reasonable limiting value of Peak error in ZMP is selected, (XzmpError, 0, 0) ≤ (1, 0, 0).

Dynamic stability is achieved by changing the position of the hip and upper body of the

biped. Stability of the biped is influenced more compared to other links because the angular

positional variations of fifth, sixth and seventh links as shown in Fig. 8. Various plots given

below are for the first set of simulation from the dynamically stable simulation results given

in Table III. It is clear from the table that the time taken for all simulations is more or less the

same. Hip positions are the same and reasonably good in the 2nd

, 3rd

and 4th

simulation but

minimum peak error is obtained in 5th

and 6th

simulation.

Angular velocity of the fifth link at the upper body is more as given in Fig. 9.This depicts

the influence of the fifth link in stable walking. Angular acceleration, linear velocity and

linear accelerations of lower leg of the swing foot as shown in Figs. 10, 11 and 12 are of high

value because the right foot has to move in a cycloidal path for the step length of 20 cm in 1 s

and also this lower leg has greater significance in the minimization of peak error in ZMP. It is

clear from the Fig. 13 that the fitness value is much higher up to around 400 iterations and

then reached the required minimum peak error value of 0.9996 after the 641 iterations. The

simulation took a reasonable time of 268.56 seconds. Up and down in the fitness plot

symbolizes the ability of simulated annealing algorithm to achieve global optimum by

escaping from the local minimum because it is independent of gradient descent and initial

conditions. Peak error in ZMP is plotted in Fig. 14. ZMP plot gives the idea of the foot size

variation in X direction for the optimum stable gait generation.


187

Figure 7: Dynamic walking.

Figure 8: Variations of joint angles.

-250

-200

-150

-100

-50

0

50

100

150

200

250

300

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time (s)

An

gu

lar

Accele

rati

on

(ra

d/s

2)

Ang acc of stance

leg-link 2

Ang acc of stance

leg-link 3

Ang acc of upper

body-link 4

Ang acc of upper

body- link 5

Ang acc of swing

leg-link 6

Ang acc of swing

leg-link 7

Figure 9: Variations of angular velocity. Figure 10: Variations of angular acceleration.

Join

t A

ng

les

(Deg

rees

)


188

Figure 11: Variations of linear velocity. Figure 12: Variations of linear acceleration.

Figure 13: Variations of fitness function. Figure 14: Variations of ZMP.

Table III: Comparison of parameters in dynamically stable simulation.

Simulation

No.

Minimum peak

error (< 1) [cm]

Time

taken [s]

No. of

iterations

Variables

hs = [rx, ry, xhi, yhi, θ4i, θ4f, θ5i, θ5f ]

1 0.9996 268.56 641 [ 1.2849 2.5492 4.2741 62.5425

0.4807 0.2091 0.2016 0.6929 ]

2 0.9998 260.41 638 [ 0.3459 1.9946 -3.5998 62.2865

0.2492 0.1094 0.6027 0.1156 ]

3 0.9998 266.39 644 [ 0.3459 1.9946 -3.5998 62.2865

0.2492 0.1094 0.6027 0.1156 ]

4 0.9998 262.47 638 [ 0.3459 1.9946 -3.5998 62.2865

0.2492 0.1094 0.6027 0.1156 ]

5 0.9735 246.98 574 [ 0.3778 2.1674 -4.2163 62.0183

0.2599 0.0408 0.6642 0.5454 ]

6 0.9735 241.47 568 [ 0.3778 2.1674 -4.2163 62.0183

0.2599 0.0408 0.6642 0.5454 ]


189

7. CONCLUSIONS AND OUTLOOK

Stable optimum gait generation of 8 links biped robots is presented in this work. Simulation

clearly illustrates the influence of hip and upper body positions for stable walking with the

minimum foot size. From the literature it is noted that, in most of the planar robot walking

cases, upper body is not considered and this is the only work carried out with two links in the

upper body. This paper demonstrated that the proposed gait planning method can achieve

statically and dynamically stable biped gait with small values of ZMP. When the foot size of

the biped is large, it is very difficult to move easily because of external and self collision. In

application point of view, design and control of smaller foot is better even though the stable

locomotion is difficult. Currently, researchers are trying to minimize the foot size for getting

easy, fast and stable walking. This work helps to decide the foot dimensions. During walking,

various motion parameters such as joint angles, velocity, accelerations, linear velocities and

accelerations are plotted and analyzed. Proposed SA algorithm gives better solution in a

reasonable time. The variation of fitness function shown in Figs. 6 and 14 indicate the

significance of SA in the optimization problems. The use of simulated annealing optimization

algorithm is not reported by other researchers specifically in the area of humanoid or biped

robotics. These type of work helps to design joints and to select the proper actuators by

realizing the utility of various joints in the optimum stable walking. It is seen that this

algorithm can be used for the optimization of various gaits for stair ascending and descending,

stepping over obstacle, ditch crossing and obstacle avoidance etc. The analysis given above is

restricted to the motion in sagittal plane. However it is possible to extend this methodology

for motion in the frontal plane also.

REFERENCES

[1] Huang, Q.; Yokoi, K.; Kajita, S.; Kaneko, K. (2001). Planning Walking Patterns for a Biped

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