[JIRS-2008] a Novel Method of Gait Synthesis for Bipedal Fast Locomotion

8/14/2019 [JIRS-2008] a Novel Method of Gait Synthesis for Bipedal Fast Locomotion

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A Novel Method of Gait Synthesis for Bipedal Fast

Locomotion

A. Meghdari & S. Sohrabpour & D. Naderi &

S. H. Tamaddoni & F. Jafari & H. Salarieh

Received: 20 May 2007 / Accepted: 5 March 2008 /

Published online: 29 April 2008# Springer Science + Business Media B.V. 2008

Abstract Common methods of gait generation of bipedal locomotion based on

experimental results, can successfully synthesize biped joints profiles for a simple

walking. However, most of these methods lack sufficient physical backgrounds which can

cause major problems for bipeds when performing fast locomotion such as running and

jumping. In order to develop a more accurate gait generation method, a thorough study of

human running and jumping seems to be necessary. Most biomechanics researchers

observed that human dynamics, during fast locomotion, can be modeled by a simple springloaded inverted pendulum system. Considering this observation, a simple approach for

bipedal gait generation in fast locomotion is introduced in this paper. This approach applies

a nonlinear control method to synchronize the biped link-segmental dynamics with the

spring-mass dynamics. This is done such that while the biped center of mass follows the

trajectory of the mass-spring model, the whole biped performs the desired running/jumping

process. A computer simulation is done on a three-link under-actuated biped model in order

to obtain the robot joints profiles which ensure repeatable hopping. The initial results are

found to be satisfactory, and improvements are currently underway to explore and enhance

the capabilities of the proposed method.

Keywords Biped . Locomotion . Gait generation . Mass-spring . SLIP .

Synchronization control

1 Introduction

Motion planning is a crucial step in development of biped robots. Much research has been

done to obtain systematic methods for biped gait generation [14]. The method of

formulating objective functions used in conjunction with controllers to regulate the motionof a planar link-segmental biped robot has been very popular among robotics researchers

J Intell Robot Syst (2008) 53:101118

DOI 10.1007/s10846-008-9233-6

A. Meghdari (*) : S. Sohrabpour: D. Naderi : S. H. Tamaddoni : F. Jafari : H. SalariehCenter of Excellence in Design, Robotics and Automation (CEDRA), School of Mechanical

Engineering, Sharif University of Technology, Tehran, Iran

e-mail: [email protected]


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since 1980s. Through this method, biped locomotion is designed in terms of step length,

progression speed, maximum step height and the stance knee bias angle. The joint angular

displacement profiles will be uniquely determined according to objective functions and

based on the initial angles of each step. To obtain a continuous and repeatable gait, special

constraints are applied in the selection of initial joint angles, objective functions, and theirassociated gait parameters all of which can be extremely challenging.

To overcome these problems, approximation of the biped joint angle profiles to the

desired trajectories was proposed in the literature, namely: time polynomial functions [5],

and periodic spline interpolations [6]. These methods can be used to find satisfactory or

optimal biped joint profiles, and they have the advantage of easily satisfying some desired

motion conditions, such as repeatability, gait optimization, etc. On the other hand, there are

core disadvantages of a high computing load for large bipedal systems and undesirable

features for the joint angle profiles (that may be imposed due to selection of the

polynomials with improper orders [5]), as well as, their malfunction or undesirable

performance in synthesizing the joint profile for a stable fast locomotion [7].

During the past two decades, a tendency has risen among the biomechanics researchers

to model fast human locomotion by a simple spring-mass system which is currently referred

to as spring loaded inverted pendulum or SLIP [8, 9]. This model was based on

observations that revealed the energy level remains approximately constant during running,

hopping and jumping. Furthermore, based on this model and its assumed functionality, the

leg stiffness is defined [10]. Experimental results show that leg stiffness remains

essentially constant during such movements. Thus, the SLIP model seemed to provide

acceptable insight to fast human locomotion and was applied by many researchers [11, 12].

Further surveys showed that in spite of the SLIP model's simplicity, it is still capable of predicting many characteristics of running and jumping. The generally observed force

pattern, shown in Fig. 1, is an example of the SLIP model results.

Stability of the results obtained from the SLIP model is also an interesting subject.

Numerous studies have shown that by properly adjusting the parameters of the model

including leg stiffness and angle of attack, the solution to the spring-mass model becomes

self-stabilized for a minimum running speed [13, 14]. Furthermore, it is concluded that

symmetric stance phases with respect to the vertical axis might result in cyclic movement

Fig. 1 LeftSchematic drawing showing the planar spring-mass model for running. RightThe observed force

pattern compared with the SLIP force response [8]

102 J Intell Robot Syst (2008) 53:101118


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trajectories [15]. A vast area of literature in the field of biomechanics is devoted to these

results; however, this model is relatively recent and there remains a lot to be done in the

future.

Although the SLIP model is of great interest among biomechanics researchers, it has

rarely been used for trajectory planning and control of bipeds which are expected to behave like a human in many ways [16]. So far, robotics research have mostly

concentrated on the biped control as an abstract field, disregarding the fact that biped robots

were invented to be mechanical devices as similar to humans as possible. This approach to

the problem of biped control caused the proposed methods regarding fast locomotion to be

essentially deficient.

In this article, we introduce a novel method of trajectory planning and control of biped

robots in hopping based on a combination of the two aforementioned approaches. Basically,

this method is inspired by the idea that: for a biped to become capable of fast locomotion, it

must be able to follow a human trajectory which is considered as an ideal case. Since the

model used for human motion is a simple spring-mass model, the task is straightforward

and is obtained by synchronization of the two models.

This type of biped control has been recently focused on, and links two fields of

locomotion research. While other studies concentrate on controller design enhancement [17,

18]; the core interest in this article is to provide a physical bone for trajectory planning such

that even a simple PD controller can be applied in an efficient and robust way. Using this

approach, one can provide a physical and biomechanical basis for their analyses. There are,

however, some problems regarding this method; for instance, the torques that can be

exerted by the biped actuators are always major obstacles in practical performance. In other

words, the required input control torques may be larger than the capacity of the bipedactuators.

The other feature of the SLIP model which makes it appropriate for the synchronization

control is that one does not need to consider an input torque exerted at the foothold. The

spring-mass system is self-sufficient in order to provide the necessary momentum for

hopping. Furthermore, it is nearly impractical for a biped robot to exert torque at its

foothold. Thus, when synchronized with a SLIP model, a biped model with one-degree-of

under-actuation is expected to exhibit more natural behavior than when controlled in any

other way.

A link-segmental model of the biped with three degrees of freedom and one degree of

underactuation in the ankle joint is considered for the synchronization method. Using theproposed algorithm, this model is made to follow the corresponding SLIP model from the

same initial conditions. It is discussed later that the initial conditions are set in a way that

periodic motion is achieved. The possibility of such selection of initial conditions is proved

by introducing a mapping between the initial conditions and the final ones. A simulation

has been done and results are presented. Finally, it is verified that by employing the

synchronization method, the model is able to perform a periodic hopping locomotion.

2 Three-Link Biped Model of Hopping: A Case Study

While in walking, three consecutive phases are distinguished: single support phase, single/

double impact, and double support phase [19]. Normal and fast locomotion differs in two

ways: (1) no instance of double support phase occurs in fast locomotion, and (2) the

touchdown in fast locomotion is very different from the impact in walking, because the

running touchdown is regarded to be conservative.

J Intell Robot Syst (2008) 53:101118 103


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The locomotion of the biped hopping on a flat horizontal surface is constrained in the

sagittal plane. One complete gait cycle of hopping in the forward direction, which is

considered for modeling in this study [20], includes two stages: (1) legs are in contact withthe walking surface supporting the whole body and moving the biped center of mass in a

forward hopping direction. If the biped center of mass reaches a sufficient upward velocity,

the feet lose contact with the ground, and (2) the whole body takes-off the ground until the

feet again come into sudden contact with the ground surface while the center of mass still

has considerable velocity downwards and forwards.

Figure 2 shows the link-segmented model of a biped robot in a case study of hopping

locomotion in the sagittal plane. The biped model in this study has three links, each of

which has length, mass, and moment of inertia. In addition, there is a significant constraint

on the biped control system which is the one-degree-of under-actuation in ankle joint of the

biped, so that the robot cannot apply any torque at its foothold.The differential equations of motion may be readily derived using the Lagrangian

formulation. If the joints/links angles are measured with respect to the vertical line, the

potential energy Pof the system in Fig. 2 may be written as:

PX3i1

migyci X3i1

migXi1j1

ljcosqj di cosqi

1

where li,di,mi are the link length, the distance between the link center of mass and its

proximal joint, and the mass of the i-th link. Considering (xci,yci) as the coordinates of thecenter of mass and Ii as the moment of inertia of link i, the kinetic energy of the system may

be expressed as:

KX3i1

1

2mi x

:2ci y

:2ci

1

2Ii

:2

i 2

3

2

1

Fig. 2 The three-link model of a

biped robot



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where for each link, the kinetic energy can be obtained from:

Ki 1

2Ii mid

2i

:2i

12miXi1j1

lj:jcos j 2

12miXi1j1

lj:jsin j 2

midi :iXi1j1

lj:jcos i j

2;

i 1; 2; 3

3

Substituting Eqs. 1, 2, and 3 into the Lagranges equation of motion will provide us with

the desired biped model. The dynamic model describing the motion of the biped in the

contact phase can be written as the following vector equation;

D ::

H;

:

:

G T 4

where D() is the 33 positive definite and symmetric inertia matrix, H ; :

is the 33

matrix related to centrifugal and Coriolis terms, and G() is the (31) matrix of gravity

terms. Also, ; :; ::

, and T are the (31) vectors of generalized coordinates, velocities,

accelerations and torques, respectively. For i,j=1,2,3,

Dij mjdjljP3kj1mk

lilj

cos ij

Hij ;

: mjdjlj

P3kj1mk

lilj

sin ij

:j

Gi mjdjgP3ki1mk

lig

sin i

8>>>>>>>>>>>>>>>>>:

5

The underactuation constraint is imposed on the biped model by nullifying the torque of

the first link. This constraint reduces the biped control inputs to torques exerted in the knee

and torso.

During the flight phase, the biped center of mass undergoes a ballistic trajectory which is

independent of the joints torques. One should consider takeoff as the initial condition for

the flight phase. In addition we have further assumed that the leg of the robot reaches itsmaximum elongation at takeoff. During the flight, the biped system has five degrees-of-

freedom, namely, three joint angles and two degrees locating the biped in xy coordinates.

The latter two variables are selected to be the coordinates of the biped ankle.

Among all types of motion, the periodic gait is of great interest. In order to synthesize a

periodic gait profile for biped hopping, the state variables at each touchdown must satisfy

certain conditions which are described as:

ni n1i ; i 1; 2; 3

:ni :n1i ; i 1; 2; 3 6

There are several methods of trajectory planning that result in a periodic motion. One

can, for instance, use an optimization method to minimize or even nullify the error defined

by the difference between the states at two consequent touch-downs. Here, another method

is applied by passively planning the joint space trajectory.



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At first, we notice that there are two input torques which may be used to predetermine

any two of the joint angles. Considering the influence of the joint space trajectory on the

whole body dynamics, these two angles were chosen to be related to the shank and thigh

links of the biped robot. Then, an interpolation by a polynomial of third degree may be used

to describe the profile of these angles between a takeoff and the consequent touchdown.The conditions imposed on the polynomial at both ends are simply selected such that this

part of the motion becomes periodic.

i tTO;n

TO;ni

:

i tTO;n

:TO;n

i

i tTD; n TD;n1i

:

i tTD; n :TD;n1

i

8>>>>>>>>>:

; i 1; 2 7

where the indices TO and TD correspond to takeoff and touchdown, respectively.As before, the Lagranges equations are applied to derive the equations of motion, which

could be written as a matrix equation similar to Eq. 4, except that the matrices are either

(5 5) or (5 1). Having calculated 1 and 2, there remain three other states, namely, 3, xf,

and yf that are to be determined. The dynamics of the body in flight phase dictates the

latter coordinates. In order to separate them from the two previously determined

variables, the equations of motion are rewritten as;

D11 D12

D21 D22 !X::

1

X::

2 !N1

N2 ! C2

C2 C3C3

00

2

66664

3

77775 8

where X1 1 2 T

and X2 3 xf yf T

, and the matrices are properly partitioned.

Eliminating X::

2 from Eq. 8 yields;

D11 D12D122 D21

X::

1 N1 D12D122 N2

C

C2

C3

!9

where

C1 0

1 1 ! D

12D

1

22

0 1

0 0

0 024 35 10

Thus, the input torques could be calculated in terms of the known variables and substituted

back into Eq. 8 to yield the differential equation governing X2. One could easily integrate

these equations, with the initial conditions set to the values at the take-off, to obtain X2.

Another issue to be properly addressed here is the periodicity of the motion. The

solution of the above system of differential equations may not be related to a periodic

motion, under the following conditions a periodic motion can be achieved;

3 tTD;n TD;n13:

3 tTD;n

:TD;n1

3

yf tTD;n

0

y:

f tTD;n

0

x:f tTD;n

0

11



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At this point, it is argued that the solution of a system of differential equations is

continuously dependent on its initial conditions. Therefore, there exists a continuous

mapping between the initial conditions and the value of the solution at a specific time. One

type of this mapping is referred to as the Poincare map in the literature. A fixed point of a

Poincare

map corresponds to a periodic solution, under certain circumstances. Here,Eq. 8 describes only a part of the motion. Thus, we use the concept of the above mentioned

mapping and search for those initial conditions which give rise to a periodic solution for

these conditions (Eq. 11).

A similar analysis for obtaining periodic solutions of a single SLIP model from an

approximate analytical mapping can be found in [11]. A numerical analysis for the same

purpose is presented in [10], and the analytical solutions are obtained. As there is no

analytical solution to the system of differential Eq. 8, one has to numerically integrate the

equations. The authors performed a trial and error process to find the fixed points of the

mapping. A fixed point, here, refers to any solution satisfying proper conditions (Eq. 11). It

is evident that the parameter xfhas no effect on the periodicity of the motion.

Finally, because of the last two conditions in Eq. 11, there is no need to consider impact

in this model. As the extensive study of a biped model shows [21], impact occurs only if

the tip of the trailing leg has nonzero velocity just before reaching the ground. Also, the

amount of influence of impact on the system, i.e. the impact forces at joints and velocity

changes, is linearly dependent on the velocity change of the trailing limb tip. Since the last

two conditions in Eq. 11 require this velocity change to be zero, impact will not occur and

the velocity profiles remain continuous.

3 Dynamic Model of SLIP

Figure 3 illustrates the parameterization of the mass-spring model as a schematic

representation for the contact phase of hopping or running with at most one foot on the

ground at any time. This model incorporates a rigid body of mass m, possessing a massless

sprung leg attached at the total center of mass (CoM). Figure 3 depicts the angle = formed

between the line joining foothold O to the CoM and the vertical, or gravity axis.

Hopping locomotion of a spring-mass system is divided into a contact phase with

foothold fixed, the leg under compression, and the body swinging forward, i.e., = is

increasing; and a flight phase in which the body describes a ballistic trajectory under thesole influence of gravity. The contact phase ends when the spring unloads; the flight phase

immediately begins afterward, and continues until touchdown occurs on the landing with

Fig. 3 The mass-spring model



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the spring uncompressed and set at a predetermined angle. This defines a hybrid system in

which touchdown and takeoff conditions mark transitions between two dynamical regimes.

Once again, to derive the equations of motion, the Lagrangian formulation is utilized.

The kinetic and potential energies of the body are

K1

2m

22 = 2

12

P mgx cosyUspring 13

where Uspring denotes the spring potential. The equations of motion for the stance phase of

spring-mass system are obtained as below;

::

=: 2

gcos =

U

m 14

=::

2:=:

gsin = 15

where Ux x @Uspring

@x.

Note that neglecting the gravity in stance yields an integrable system. A detailed analysis

of the validity of this approximation for different spring potentials was performed in [22]

using Hamiltonian instead of Lagrangian formulation. Another approximation which

enables an analytical solution to the above equations is by considering = and the maximum

shortening of the spring to be small [11]. Since neither of these approximations isapplicable to biped hopping model, the motion equations of mass-spring system must be

solved numerically.

It should be noted that some of the characteristics of hopping may not be accurately

predicted by the mass-spring model. For instance, it turns out that the vertical component of

the ground reaction force usually exhibits a passive peak just after touchdown, which is

followed by an active peak at nearly the middle of the contact time interval. The mass-

spring model with one sprung mass, though predicting the active peak closely, is incapable

of predicting the passive peak.

In order to build a model for the passive peak, one must consider a more sophisticated

model with two or more sprung masses along with dampers. By adjusting the coefficients

of springs and dampers, the desired model is achieved. Different models with linear and

nonlinear components have been introduced in previous literatures [8]. In this study, for

simplicity, the ordinary spring loaded inverted pendulum, SLIP, model with one linearly

sprung mass is considered.

4 Synchronization Control of SLIP Model and Biped Dynamics

A master-slave synchronization control system is constructed having two systems capable

of modeling the biped locomotion individually. The biped dynamics is considered as the

slave system while the spring-mass dynamics is the master system. Moreover, the joint

torques are taken into account as the control inputs. To synchronize the motion of the robot

with the master dynamics of the SLIP model, two new state variables, namely; the distance

of the biped center of mass from the foothold, , and the angle of the line passing through

the biped center of mass and the foothold with respect to the vertical axis, +, are considered.



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The expressions for these parameters may be derived by considering the coordinates of the

biped center of mass, where xci,yci are the coordinates of the center of mass for each link.

xci Xi1

j1

ljsin qj di sin qi 16

yci Xi1j1

ljcosqj di cosqi 17

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX3i1

mixci

2

X3i1

miyci

2

vuut ,X3i1

mi

18

tan1X3i1

mixci

,X3i1

miyci

19

Double differentiation of these relations yields

::

f ; :

g u+::

f+ ; :

g+ u&

20

In these formulae, f ; :

and f+ ; :

are scalars, while g and g+() are two rowvectors so that their products with the input column vectoru will be a scalar.

Synchronizing control design will be accomplished by defining an error function,

defined as;

e + =

& '21

such that it asymptotically tends to go to zero.From Eqs. 18 and 19, we have;

e::

::

::

+::

=::

& 'f ;

: g u

::

f+ ; :

g+ u =::

& '22

Based on the feedback linearization method of nonlinear control, the control inputs

should be defined such that the error function becomes asymptotically stabilized around

zero. Therefore, considering a simple PD controller with coefficients Kp and Kd, the biped

joint torques will be extracted from the following system of equations;

g g+

!u Kpe Kde

:

=: 2 gcos =

U m

f ; :

2

:=:

gsin =

f+ ; :

( )23



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Consequently, the differential equation of the error function takes the following form:

e::

Kde:

KPe 0 24

The PD controller gains are chosen appropriately such that stabilization of the error

function around zero is guaranteed.Functions appearing in Eq. 20 may be explicitly computed from the aforementioned

biped dynamics. Where could be either of and +, differentiating with respect to time

yields:

k

@k

@qq

25

and,

.

::

:T @2.

@2

:

@.

@

::

26

where :

:

1 :

2 :

3 T

denotes the time derivative of the joint space column vector, and @k@q

is regarded as a row vector and is obtained by differentiating the function with respect to

the states of the joint space. In a similar manner, the symmetric (33) Hessian matrix @2k

@q2

consists of elements which are the mixed second partial derivatives of the function with

respect to the states of the joint space.

By solving Eq. 4 for::

and substituting the result into Eq. 26, we will have:

f. ;

:

:T @2.

@2

:

@.

@ D1

N 27

g. @.

@D

1 28

where N H:

G is defined from Eq. 4.Now, turning our attention back to Eq. 23, we notice that this is, in fact, a system of two

linear equations with three unknowns. There is a certain amount of flexibility in the solution

of such system of equations. Thus, an optimization method may be applied to obtain adesirable solution. However, as it was mentioned before, the system is considered to have

one-degree-of under-actuation in the biped ankle joint, and this constraint is satisfied by

nullifying the torque of the ankle joint. Thus, the vector of generalized torques T is written

in the following form:

T C2

C2 C3C3

0@

1A Mt C2

C3

!29

where C2 and C3 are input torques applied in the knee and hip joints, respectively, and Mt is

a 32 matrix defined by:

Mt 1 0

1 10 1

24

35 30



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Therefore, the system of equations in Eq. 23, is transformed into a system of two

equations with two unknowns, as:

g g

+ !Mt C2C

3 ! Kpe Kde

:

::

f ; :

=

::

f+ ;

:

& ' 31

or

C2

C3

!g g+

!Mt

1KPe Kde

:

::

f ; :

=::

f+ ; :

& ' 32

which always has a unique solution.

The uniqueness of the solution is verified by considering the matrix of coefficients of the

unknowns. The relations 18 and 19 suggest thatg() and g+() are linearly independent. If

the columns of the matrixg

g+ " # are denoted by v1, v2, and v3, it can be easily shown that

the right product of this matrix with Mt is singular, if and only if there exists a real number

, such that:

v2 1 v1 v3 33

As it may be seen, satisfaction of Eq. 33 is not impossible, but it is expected not to occur

in calculations and during simulation, owing to the form ofg() and g+().

5 Simulation and Results

A computer simulation is done to examine the performance of the proposed method in

control of a hopping biped. The objective is to obtain a repeatable hopping cycle for a biped

robot using the method proposed in this article.

The simulation is carried out for the link-segmental model in Fig. 2 whose parameters

are presented in Table 1. Moreover, as mentioned before the spring of the SLIP model is

considered linear. The ratio of the spring stiffness to mass in the mass-spring system is also

included in Table 1.

The initial conditions are set for the link-segmental model and those of the SLIP model

are calculated from relations 16 through 19. The initial conditions and the selected PDcontroller coefficients are presented in Table 2.

Plots showing the simulation results are as follows: Figure 4a through c show the joint

space states, while Fig. 5 show those of the corresponding SLIP model during the contact

phase. Please note that a complete hopping cycle took 0.14 seconds.

Table 1 Parameters of the simulated model

Link Mass (kg) Moment of inertia (kg.m2)

Length (m) CoM distance fromproximal joint (m)

Shank 0.2 Negligible! 0.1 0.05

Thigh 0.2 Negligible! 0.1 0.05

Torso 0.5 0.001 0.1 0.07

Spring stiffness/mass ratio of SLIP model (N/kg. m) 3,000



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Figure 6 reveals the animation of the biped robot hopping based on the control method

of dynamics synchronization of the link-segmental model and the SLIP system.

The control inputs which are the torques applied in knee and torso joints of the biped

robot are plotted in Fig. 7.

The resultant ground reaction forces are compared for the two models in Fig. 8a and b.

Up to now, a new method for generating joint profiles in biped hopping is simulated. Togain a general view of the applicability and accuracy of this method, another simulation is

performed based on the method proposed by Kajita et al. [23]. Although slight

modifications were made to apply Kajitas method to our model, the generality of the

problem is untouched. One may notice that both of these methods utilize the CoM

trajectory to design the joints profiles, yet, different results are achieved. Figure 9 shows the

X and Y components of the bipeds CoM trajectory based on Kajitas method.

It is obvious from the X-component plot that the biped step length is more than 12 cm in

the approach proposed by Kajita. Joint angle profiles are illustrated in Fig. 10 and may be

compared to those of Fig. 4. The periodicity requirement is the core reason for similarities.

Table 2 Initial conditions of the simulated model

Link Initial position (rad) Initial velocity (rad/s)

Shankp10 16.7

Thigh

p

6

13.7Torso

p8 0

Kp 8

Kd 6

ANKLE JONIT

-0.6

-0.4

-0.2

0

0.2

0.4

0 20 40 60 80 100

0 20 40 60 80 100

Percentage of Cycle (%)

Angle(rad)

Angle(rad)

KNEE JOINT

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

0 20 40 60 80 100


TORSO JOINT

0

0.1

0.2

0.3

0.4

0.5


Ang

le(rad)

a b

c

Fig. 4 Biped joint space states; angles of a shank, b thigh, and c torso



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Spring Length in SLIP model

0.17

0.175

0.18

0.185

0.19

0 20 40 60 80 100


Length(m)

Angle in SLIP Model

-0.3

-0.25

-0.2

-0.15-0.1

-0.05

0

0.05

0 20 40 60 80 100 120


Angle(ra

d)

Fig. 5 SLIP states; left spring length, right spring angle

Fig. 6 Synchronization of the

biped model with the

corresponding SLIP model

(the dots track the SLIP

trajectory)

Input Control Torques

-66

-56

-46

-36

-26

-16

-6

4

14

24

34

44

0 20 40 60 80 100


Torque(N.m

)

KNEE JOINT

TORSO JOINT

Fig. 7 Input control torques



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X- Component of Ground Reaction Force

-2.5

-2

-1.5

-1

-0.5

0

0.5

0 20 40 60 80 100


0 20 40 60 80 100


Force

(N)

LINK-SEGMENTAL MODEL

SLIP MODEL

a

Y- Component of Ground Reaction Force

0

5

10

15

20

Force(N)

LINK-SEGMENTAL

SLIP MODEL

b

Fig. 8 Ground reaction forces; a

X-component, b Y-component

Y- Component of CoM Trajectory

0.16

0.165

0.17

0.175

0.18

0.185

0.19


Posit

ion(m)

b

X- Component of CoM Trajectory

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0 20 40 60 80 100 120

0 20 40 60 80 100 120


Position(m)

a

F ig . 9 Trajectory of Bipeds

CoM; a X-component, b Y-com-

ponent



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Also, torques at joints during stance are calculated and plotted in Fig. 11. It is to be

noted that torque of the ankle joint is not necessarily zero in the Kajitas method.

6 Discussions and Conclusions

Recently, most biped robotics researchers apply the two traditional methods to generate

their desired joint profile trajectory for their robots. Although time polynomial function and

periodic spline interpolation result in satisfactory or optimal profiles in biped locomotion,

they can not properly create delicate joint profiles. This is due to the fact that these methods

are based on pure mathematics rather than physics and biomechanics. This deficiency is

much more obvious when these methods are applied to robots performing fast locomotion.

For instance, it is rare that recent biped robots are able to run.

In this article, a novel method is introduced for joint trajectory planning and control of biped robots during fast locomotion. This method, contrary to the traditional methods, is

mostly based on the physical and biological concept of human fast locomotion.

ANKLE JOINT

-0.3

-0.2

-0.10

0.1

0.2

0.3

0.4

0.5

0 20 40 60 80 100 120


Ang

le(rad)

KNEE JOINT

-0.6

-0.5

-0.4-0.3

-0.2

-0.1

0

0.1

0 20 40 60 80 100 120


Ang

le(rad)

a

b

TORSO JOINT

00.05

0.10.150.2

0.250.3

0.350.4

0.45

0 20 40 60 80 100 120


Angle(rad)

c

Fig. 10 Biped joint space states; angles of a shank, b thigh, and c torso

Input Control Torques

-1

-0.5

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8

Percent of Cycle (%)

Torque(Nm

)

ANKLE JOINT

KNEE JOINT

TORSO JOINT

Fig. 11 Input control torques



16/18

Observations show that the complex, nonlinear dynamics of a runner or jumper can be

substituted by a simple mass-spring model, namely spring loaded inverted pendulum or

SLIP, to predict the resultant ground reaction force.

The presented method has taken these observations into account by synchronizing the

dynamics of link-segmental model with the SLIP representation of the biped. Additionally,the mathematical complexities involved in the controller design which is an important

issue in the traditional methods of gait generation and control of bipeds are overcome and

a simple, but efficient, method for biped control is proposed. In fact, this simplification is

the principal contribution of the current work.

As a case study, periodic hopping locomotion is considered in this paper. Therefore, a

simulation is done on trajectory planning and control of a three-link biped robot with one-

degree-of under-actuation in the ankle joint during hopping. The SLIP model was chosen

for this purpose to satisfy the condition of one-degree-of under-actuation, which was

imposed on the system by assuming the torque exerted at the ankle of the biped to be zero.

The synchronization control was accomplished by the feedback linearization method.

During the flight phase, trajectory of two of the states was predetermined and the others

were calculated from the differential equations of motion. All of the states were finally

made to exhibit periodic behavior.

Obtained results indicate that the biped is in stance phase in about two thirds of the

hopping locomotion cycle, while the rest of the cycle consists of takeoff, flight, and

landing. Total behavior of the biped seems satisfactory, and the biped is capable of

performing a complete periodic hop. The SLIP model behavior is depicted in Fig. 5 and

shows a smooth V-shape in length and an increase in the angle in the stance followed by a

decrease in the flight phase.As shown in animation of the whole simulated cycle in Fig. 6, in the stance phase, the

ankle joint angle is increasingly trying to roll the biped center of mass to the forward

direction of the hop. In the mean time, flexion and afterward extension of the ankle joint

help the biped bounce back from the previous flight phase. The torso joint angle is

dominated by the SLIP model behavior and is continuously decreasing.

The main strategy of joint profile generation in the flight phase is to set the biped links in

the landing stance at the same state (including position and velocity terms) as the initial

ones, so that a repeatable hopping cycle is obtained. The overshoots in the ankle and torso

joint angles are due to the extension and flexion of knee joint in the flight phase trying to

regain its initial state.The dynamical results included in the previous section consist of computed biped joint

torque and the resultant ground reaction force. As expected regarding the previous research,

Error in X- Components of Ground Reaction Force

-0.05

0

0.05

0.1

0.15

0.2

0 20 40 60 80 100


Error(N)

Error in Y- Component of Ground Reaction Force

-0.4

-0.3

-0.2

-0.10

0.1

0.2

0.3

0.4

0 20 40 60 80 100


Error(N)

Fig. 12 Error in Ground reaction forces computed using two models; leftX-component, rightY-component



17/18

a peak in both X and Y components of ground reactions are observed in Fig. 8. During

takeoff and touchdown, the ground reaction force becomes zero since there is no connection

with the ground. The error in the prediction of two models of biped, link-segmental and

SLIP, are plotted in Fig. 12.

As shown, the errors are small with regards to the ground reaction forces. The error in X-component forces occurred when the shank is held vertical and the knee joint angular

velocity is changing its sign. However, this error gradually decreased afterwards; the error

between Y-components of ground reaction forces is more noticeable than their X-

components errors. It is believed that the one-degree-of under-actuation of the biped model

causes this deficiency in the Y-direction, where the input torques must compensate for the

gravity.

The torques applied in the biped joints are plotted in Fig. 7; the trend is dominated mostly

by the angle extensions and flexions, and the sudden jump is because of the phase change.

The only weak point is the high joint torque required to perform the desired hopping.

An additional simulation for the same model with the same initial conditions was

performed based on an approach proposed by Kajita et al. [23]. Comparing these two

methods reveals that although both of them use the CoM trajectory as the key to the joint

profile generation, they lead to relatively different outcomes. The main difference between

the results can be seen in the vertical ground force reaction; the vertical ground force is

continuous in our method whereas it is discontinuous in [23].

In addition to the continuity of ground reaction force in our method, some other

advantages can be also emphasized over Kajitas approach: First, one can use our approach to

design gait patterns for either jumping or hopping processes, while Kajitas method has to be

modified to be applicable for the same purposes. Secondly, the transfer phase between the reststate and the steady running state cannot be directly designed by Kajitas method. Lastly,

elimination of undesired impact with the ground and a simple feedback linearization control

law are good features of a biped locomotion control algorithm while considerable impacts and

a complicated five-component controller can be found in Kajitas method. However, the most

important disadvantage of Kajitas proposed method is that it requires solving differential

equations with boundary conditions, which may greatly increase the processing load.

On the other hand, there are, some advantages in Kajitas method over the one presented

here. For instance, larger step length, increased velocity, reasonable input control torques,

and including more trajectory design parameters are some of the advantages. The one

degree of under-actuation in our biped model was the main cause of high joints torque.One may conclude that the presented method is applicable to any biped model for any fast

locomotion such as jumping, hopping or running by using optimization methods. The core

advantage of this method is that the proposed simulation depends only on initial conditions.

Hence, no pre-determined or offline trajectory is required, which reduces the time needed in

real-time robot gait generation. One may apply this algorithm to control a biped while running.

Periodic solutions of the SLIP model have been found, which can be used for this purpose.

However, when considering a running biped, one should select at least five degrees-of-freedom

for the biped model. Hence, the number of input control torques increase to four provided that

the model is to account for one-degree-of under-actuation. This calls for other constraints to beimposed on the trajectory planning, such as minimizing the input power.

Acknowledgement This work was supported by Iran National Science Foundation (INSF) under contract

number 84084/8 to whom the authors would like to give their appreciation. Furthermore, we would like to

appreciate the support of Center of Excellence in Design, Robotics and Automation (CEDRA), Sharif

University of Technology, Iran.



18/18

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