978-1-4799-6743-8/14/$31.00©2014 IEEE

Effect of Motion Parameters on Time Optimal

Path of Biped Robot Using PSO Algorithm

Nasim Zafari,

School of Mechanical Engineering,

Shahid Chamran University of Technology

Ahvaz, Iran

Mohammad Hasan Ghesemi

School of Mechanical Engineering

Babol Noshirvani University of Technology

Babol, Iran

Abstract-Designing of stable time optimal path for bipeds is one of the main issues in motion of bipeds. Motion parameters such as participation of motion phases and the distance between hip joint and ankle joint of support leg at the beginning and the end of the single support phase, have important effect in designing of time optimal path. In this paper the effect of these two motion parameters on time optimal path is discussed. At first, time optimal path is designed by particle swarm optimization algorithm considering definite participation of double support phase and actuators torque limitation. To ensure Stability, motion of a biped needs to follow specific pattern to comply with certain stability criterion such as zero moment point. Then the effect of double support phase on time optimal path and torque limitation on participation of the double support phase which satisfies time optimal path are studied. Finally effect of distance between hip joint and ankle joint of support leg at the beginning and the end of the single support phase on time optimal path is investigated.

Keywords—biped robot, motion parameters, time optimal path, PSO algorithm

I. INTRODUCTION

Researches on biped robot due to its structural and ability to cross rough surfaces compared to wheeled robots have increased. Study and research on bipedal robots began from the late 60th century with the works of Frank, Vukobratovic, Hemami and McGheer. Kato in Waseda University conducted numerous studies in this field and the first prototype of biped robots was designed by him. at first stability was the main problem of bipedal robots and many research have been done in this field. Vukobratovic made a huge change in the dynamic stability of biped robots with offering the zero moment point stability criteria, which still is the most widely used criteria [1]. Kajita Park, Erbatur and Handharu used Linear Inverted Pendulum model for the design of biped robots path. [2-5] Huang et al. presented a new method for path planning of biped robots. At first they designed the path of ankle joint with respect to kinematic constraint and Interpolation of a cubic polynomial. Furthermore the horizontal path of hip joint was designed parametrically by Interpolation of a cubic polynomial. Finally, by using the iterative method and the optimization, the hip joint parameters were calculated in such a way that maximum stability was achieved [6]. The speed of

biped robots is always the most important topic discussed. Sedigh and Mansuri examined effects of time and length parameters in the speed of walking robots movement [7].

In this paper the effect of two motion parameters,

participation of motion phases and the distance

between hip joint and ankle joint of support leg at

the beginning and the end of the single support

phase, on time optimal path is discussed. At first,

time optimal path is designed by PSO algorithm

considering determined participation of double

support phase and actuators torque limitation. To

ensure Stability, motion of a biped needs to follow specific pattern to comply with certain stability criterion such as zero

moment point. Then the effects of double support phase on time optimal

path and torque limitation on participation of the double support phase which satisfies time optimal path are studied. Finally, effects of 𝑥𝑒𝑑 and 𝑥𝑠𝑑on time optimal path are investigated.

II. DYNAMIC MODEL OF BIPED ROBOT

For determining the time optimal path with actuators torque limitation, at first angular motion of joints with considering kinematic constraint are designed by using Interpolation of polynomial [7], then the torques applied to the joints of the robot is obtained by the equations of the robot. The desired biped robot has seven links and seven degrees of freedom, which moves in a plane. The robot is composed of two legs, upper body and each of the legs consist of three members: foot, leg and thigh. Each complete walking cycle consists of two half-cycles and each half cycle consists of two single support phase (SSP) and double support phase (DSP). The walking model of robot include the time of each phase and also how to move the legs during walking. Model which has been considered for the study of human movement and the motion parameters has been shown. (fig.1-2).

The dynamic model of biped robot is shown in figure 3. The generalized coordinates is considered the angle of each of the links with vertical axis.

The motion equation of robot represents the relationship

between the generalized equations of motion and torque of actuators. In each dynamical system, number of independent equation is equal to degrees of freedom in the system. Here, the Lagrange method is used to obtain the equations of motion of the biped robot. Lagrange equation is as follows:

𝑑

𝑑𝑡 ∂L

∂𝑞 −

∂L

∂𝑞𝑟= 𝑄𝑟 (1)

In this equation, 𝑞𝑟 is the 𝑟 component of generalized coordinate vector 𝑞 and 𝐿 is the Lagrange of the model and equals:

𝐿 = 𝐾 − 𝑃 (2)

𝐾 =1

2 𝑣𝐺𝑖

𝑇 𝑚𝑖𝑣𝐺𝑖𝑛𝑖=1 +

1

2 𝜔𝑏𝑖

𝑇 𝐼 𝑖𝜔𝑏𝑖𝑛𝑖=1 (3)

𝑃 = 𝑚𝑖𝑔 𝑧𝐺𝑖𝑛𝑖=1 (4)

𝐾 Is kinematic energy, 𝑃 is potential energy, 𝑚𝑖 is the

mass of 𝑖-th link , 𝐼 𝑖 Mass moment inertia tensor of 𝑖-th link respect to the coordinate system which is connected to its mass centered, 𝜔𝑏𝑖 is the absolute angular velocity vector of the 𝑖-th link in the coordinate system which is connected to the link. 𝑣𝐺𝑖 is also the velocity vector of mass center of 𝑖-th link in the inertial system. In Lagrange equation Qr is the 𝑟-th component of the generalized force vector corresponding to 𝑞𝑟 and equals:

Qr = ∂𝑣𝑖

∂𝑞 𝑖.𝑅𝑖

𝑛𝑖=1 +

∂ω0𝑖

∂𝑞 𝑟𝑇𝑖

𝑛𝑖=1 (5)

𝑅𝑖 is the effective force of the robot and 𝑇𝑖 is the

effective torque, 𝑣𝑖is the absolute linear velocity of the

robot, and the ω0𝑖 is the absolute angular velocity. By

substituting the above equations in the Lagrange

conjunction and the simplification, the dynamic equations in

unconstraint mode of the robot can be obtained as follows:

𝐌 𝐪 𝐪 + 𝐇 𝐪,𝐪 = 𝐁𝛕 (6)

In this equation, 𝒒 is the generalized coordinate vector, 𝐌 𝐪 is symmetric inertia matrix, 𝐇 𝐪,𝐪 is the summation of coriolis, gravity and centrifugal forces, 𝝉 is the torque vector of joints.

The two-legged robot structure is such that has different degrees of freedom in each phase of motion. During single support phase, the support foot is in contact with the ground and the swing leg will move forward. In this phase, the robot has six degrees of freedom. During double support phase, one of foots is always on the ground completely and the other one is only in contact with the ground in a point. In this phase, the robot has five degrees of freedom.

Hence the equations of motion for two-legged robots vary depending on the phase. Then the equations of motion in each phase are extracted. In this robot 𝑞4 = 0 is considered.

III. BIPED ROBOT STABILITY CRITERIA

In order to maintain stability of the robot during motion zero moment point stability criteria is used. Based on these criteria for robot balancing, the location of zero moment point must place within the supporting polygon. The location of the zero moment point will be determined by following equations.

𝑥𝑍𝑀𝑃=

𝑥𝐺𝑖𝑚 𝑖(𝑧 𝐺𝑖𝑖

+𝑔)𝑛𝑖=1 − 𝑧𝐺𝑖

𝑚 𝑖𝑥 𝐺𝑖− 𝐼 𝑦𝑖

𝑞 𝑦𝑖𝑛𝑖=1

𝑛𝑖=1

𝑚 𝑖(𝑧 𝐺𝑖+𝑔)𝑛

𝑖=1 (7)

𝑞1

𝑞2

𝑞3

𝑞4

𝑞5

𝑞6

𝑞7

X

+

Fig3.dynamic model and generalized coordinates

Lan

𝑑4

𝑙3

𝑙2 𝑑2

𝑑3

Lab Laf

X

Z

Fig2. Parameters of robot

Hmin

Left

𝜃𝑓 𝜃𝑏

xsd

Hmax

xed

0 Td1 Tm Td

Hao

SSP

Lao Ds

Tc

DSP

Fig1.walking cycle of robot

Z

𝑦𝑍𝑀𝑃=

𝑦𝐺𝑖𝑚𝑖 𝑧 𝐺𝑖 𝑖 + 𝑔 𝑛𝑖=1 − 𝑧𝐺𝑖

𝑚𝑖𝑦 𝐺𝑖 − 𝐼 𝑥𝑖𝑞 𝑥𝑖𝑛𝑖=1

𝑛𝑖=1

𝑚𝑖 𝑧 𝐺𝑖 + 𝑔 𝑛𝑖=1

(8)

(𝑥𝐺𝑖,𝑦𝐺𝑖

, 𝑧𝐺𝑖) are the center mass position of 𝑖-th link

relative to the inertia coordinates system, 𝑚𝑖 is the mass of 𝑖-th link, 𝑛 is the number links and 𝑞𝑥𝑖 and 𝑞𝑦𝑖 are the

absolute angular position of 𝑖-th link in the positive direction of x and y axis.

Proper stability region of the model and movement patterns in biped robots is shown In figure (4). In this figure, bold lines indicate the boundaries of the area and dotted lines indicate the ideal zero moment point trajectory.

IV. PARTICLE SWARM OPTIMIZATION (PSO)

ALGORITHM

In 1995 for the first time, this algorithm was emerged by Eberhart, Kennedy and Shi as a non-deterministic search method for the optimization of a function. The algorithm inspired from the movement of a group of birds looking for food. The group of birds is randomly searching for food in space. There is only one piece of food in space. None of the birds would know the place of food. One of the best strategies would be to trace the bird which has the least distance to the Food.

Each solution, which is called as a bird, in the particle swarm algorithm is equivalent to a particle. Each particle has a fitness value which is calculated by an optimization function. The more a particle in the search space is close to the target-food in the birds’ movement model-the more has fitness value. Also each particle has a velocity, which is responsible for direction of its movement.

Implementation of the algorithm is as follows: at first, the initial populations of particles form from a certain number of particles (N particles) and for each particle two parameters: position and velocity in N-dimensional space is described which sequentially are modeled with a position vector and velocity vector. Position vector of particles is initialized randomly at arbitrary intervals (usually

proportional to the problem). The number of 𝑁 is arbitrary

and the number of 𝑛 is equal to design parameters.

Particles move to the n-dimensional space since with calculating function value, search the new options which are possible. A memory will be assigned to the best position of

each particle (𝑥𝑝𝑏𝑒𝑠𝑡 ) and a memory will be assigned to the

best position among all particles (𝑥𝑔𝑏𝑒𝑠𝑡 ).

In each iteration, the velocity and position of each particle is updated based on 𝑥𝑝𝑏𝑒𝑠𝑡 and 𝑥𝑔𝑏𝑒𝑠𝑡 according to

the following equation:

𝑉 = 𝑊 𝑉 + 𝑐1𝑅1 𝑋 𝑝𝑏𝑒𝑠𝑡 −𝑋 + 𝑐2𝑅2 X 𝑔𝑏𝑒𝑠𝑡 −𝑋

(9)

𝑋 = 𝑋 + 𝑉 (10)

𝑐1 , 𝑐2 = 2.05

R1 , R2 ∈ rand[0,1]w = 0.721

(11)

If 𝑓 𝑥 < 𝑓(𝑥𝑝𝑏𝑒𝑠𝑡 ) in each iteration then 𝑥𝑝𝑏𝑒𝑠𝑡 will be

𝑥. 𝑥𝑔𝑏𝑒𝑠𝑡 is also updated in each iteration. Iterations

continued until the value of the function will be small

enough (max iteration).To achieve optimum final value, the algorithm is repeated several times and the average value of the function is selected as the optimal value of the objective function.

V. SIMULATION AND RESULTS

A. Time Optimal Path Using PSO Algorithm

In this part, time optimal path is designed by PSO algorithm considering definite participation of double support phase and actuators torque limitation. Objective function in optimization algorithm for designing time optimal path is as follows:

𝑚𝑖𝑛 𝑑𝑡𝑇𝑐

0 (12)

Subject to

𝛿+ = max 𝑥𝑧𝑚𝑝 − 𝑈 ≤ 0

𝑎𝑛𝑑𝛿− = max 𝐿 − 𝑥𝑧𝑚𝑝 ≤ 0

𝑀 𝑞 𝑞 + 𝐻 𝑞, 𝑞 = 𝐵𝜏𝜏𝑚𝑖𝑛 ≤ 𝜏 ≤ 𝜏𝑚𝑎𝑥

δ+ and δ−are the deviation 𝑥zmp of upper bound (U) and

lower bound (L) of stable region , respectively. Pso algorithm calculate the value of design parameter (𝑇c) such that the objective function is minimized with these situation.

Physical characteristics and motion parameters, and the parameters used in the PSO algorithm are given respectively, in Tables (1)-(3). Time of double support phase is defined by parameter α as follows:

𝑇 𝑑1 𝑇 𝑑

𝑇 𝑐

2𝑇 𝑐

−𝑙𝑎𝑏

𝑙𝑎𝑓

𝐷𝑠 + 𝑙𝑎𝑓

𝐷𝑠 − 𝑙𝑎𝑏

Fig4.stability region and ideal zero moment point trajectory

Td1=

α

2 Tc

Td = αTc

Tm =0.9231 α+1

2Tc

(13)

Tm is when the ankle joint is in the maximum height from the ground.

TABLE I. Physical Characteristics

𝑚4 𝑚3 𝑚2 𝑚1 Mass

(Kg) 30 10 5.7 3.3

𝐼𝑦4 𝐼𝑦3

𝐼𝑦2 𝐼𝑦1

Inertia

(kg.m2) 1.7 0.08 0.02 0.01

𝑑3 𝑑2 𝐿3 𝐿2

Length

(m)

0.15 0.15 0.3 0.3

𝐿𝑎𝑛 𝐿𝑎𝑏 𝐿𝑎𝑓 𝑑4

0.1 0.1 0.13 0.25

𝐻𝑎𝑜 𝐷𝑠

0.2 0.35

𝜏𝑕𝑖𝑝 ≤ 70 𝜏𝑎𝑛𝑘𝑙𝑒 ≤ 70 Nominal

Torques

𝜏𝑘𝑛𝑒𝑒 ≤ 100

TABLE II. Motion Parameters

𝑊𝜃𝑓 𝑊𝜃𝑏 𝑥𝑠𝑑 𝑥𝑒𝑑

constant -0.4 0.4 0.06 0.04

Hmax Hmin α

0.596 0.586 0.3

TABLE III. Parameters Used in PSO Algorithm

20 Population size

1 Dimension of particles

100 iteration

[0.5 10] Initial population

For the conditions governing the robot's path, time optimal path is occurred in 𝑇c = 1.17 s . Angular path of joints and angular torque of left leg joints are shown in figure (5) and (6).

B. Effect of Double Support Phase in Time Optimal Path

In this part, for different value of double support phase, time optimal path without limitations on torques of robot’s joint is determined by PSO algorithm. Motion parameters is shown in tables 1 and 2, except that the amount of α is different and the path has no limitation on torques of robot’s joint. Time of each phase is determined by equation (13). Minimum time for different value of double support phase is shown in figure (7).

According to figure (7), with increasing phase participation α, at first minimum time of robot’s path decreases and then increase. The value of the double support phase which time optimal path occurs is marked in Fig (7). This behavior is because of this reason that there is stability constraint for the biped robot's path. As mentioned previously for path stability, the zero moment point of robot should place in stability margin.

The more the difference between real zero moment path of robot and ideal zero moment path lessen, the more robot’s movement is stable. Thus, the area between the real zero moment path and ideal zero moment path which is given by equation (14), represents the stability of the robot’s path.

0 0.5 1 1.5 2-1.5

-1

-0.5

0

0.5

1

1.5

time(sec)

q (

rad)

q1

q2

q3

0 0.5 1 1.5 2 2.34

-100

-50

0

50

100

time(sec)

(

N.m

)

2

3

1

TcTd Tc+Td

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.5

1

1.5

2

2.5

3

3.5

4

4.5

5

X: 0.42

Y: 0.9006

min

tim

e (

sec)

Fig 5.angualar path of joints

Fig.6.Torque of joints of robot

Fig 7. Minimum time for different value of double

support phase

𝐽 = 𝑥𝑧𝑚𝑝 − 𝑥𝑧𝑚𝑝∗

𝑇c

0𝑑𝑡 (14)

For this purpose, the area between the real zero moment path and ideal zero moment path of time optimal path for different value of double support phase is shown in figure(8).

Fig 8. The area between the real zero moment path and ideal zero moment

path of time optimal path for different α

The results from figure (8) are because of physical characteristic of robot, and it can be the reason of why figure (7) is like this.

C. Effect of Torque Actuators Limitation on Time Optimal

Path

To study the effect of actuators’ torque limitation on determining minimum time of robot’s path, the minimum time for different values of double phase support and different values of the joint’s torque limitation is shown in figure (9). Furthermore, a zoom of figure (9) around the optimal α is shown in figure (10).

Fig 9.minimum time of robot’s path for different torque limitation and

different α

Fig 10. Zoom of figure (9) around the optimal α

According to figure (9) with smaller torque limitation, the value of the double support phase which the minimum time occurs will be increase.

D. Effect of 𝑥𝑠𝑑 and 𝑥𝑒𝑑 on Time Optimal Path

In this part, the effect of the distance between hip joint

and ankle joint of support leg at the beginning (𝑥𝑠𝑑 ) and the

end of the single support phase (𝑥𝑒𝑑 ) on time optimal path is investigated. Therefore, for each 𝑥𝑒𝑑 and 𝑥𝑠𝑑 time optimal path without any limitation on is shown in Figure (11).

Fig 11.minimum time of robot’s path for different 𝑥𝑠𝑑 and 𝑥𝑒𝑑

According to figure (11) with increasing 𝑥𝑠𝑑 for a definite 𝑥𝑒𝑑 , minimum time of robot’s path will be decrease. Furthermore with increasing 𝑥𝑒𝑑 for a definite 𝑥𝑠𝑑 , minimum time of robot’s path will be decrease.

VI. CONCLUSION

1- with increasing participation of double support phase α, at first minimum time of robot’s path decreases and then increase. Furthermore the value of the double support phase which time optimal path occurs can be determined.

2- With smaller torque limitation, the value of double support phase’ participation which time optimal path occurs will be increased.

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

X: 0.36

Y: 0.0702

J

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

min

tim

e (

sec)

|1,|<105 , |2|<10

5 , |3|<10

5

|1|<60 , |2|<100 , |3|<60

|1|<57 , |2|<95 , |3|<57

0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0.48 0.50.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

X: 0.44

Y: 1.006

min

tim

e (

sec)

X: 0.44

Y: 0.9705

X: 0.42

Y: 0.9006

|1,|<105 , |2|<10

5 , |3|<10

5

|1|<60 , |2|<100 , |3|<60

|1|<57 , |2|<95 , |3|<57

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

Xsd (m)

min

tim

e (

sec)

Xed=0.04

Xed=0.06

Xed=0.08

3- With smaller torque limitation, the value of minimum time for each double support phase’ participation increase.

4- With increasing 𝑥𝑠𝑑 for a definite 𝑥𝑒𝑑 , minimum time of robot’s path will be decrease. Furthermore with increasing 𝑥𝑒𝑑 for a definite 𝑥𝑠𝑑 , minimum time of robot’s path will be decrease.

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[3] Park J. H and Kim. K “Biped Robot Walking Using Gravity-

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[4] Erbatur. K and Kurt. O “Natural ZMP Trajectories for Biped Robot Reference Generation”, IEEE Transaction on Industrial Electronics, vol. 56, no. 3, 2009.

[5] Handharu. N, Yoon. J and Kim. G “Gait Pattern Generation with

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