Dynamical Modeling of Humanoid with Nonlinear Floor FrictionXiang Li1, Hiroki Imanishi1, Mamoru Minami1, Takayuki Matsuno1 and Akira Yanou1

1Okayama University, Japan(Tel: 81-86-251-8233, Fax: 81-86-251-8233)

1pzkm87r2@s.okayama-u.ac.jp

Abstract: Biped locomotion created by a controller based on Zero-Moment Point [ZMP] known as reliable control method looksdifferent from human’s walking on the view point that ZMP-based walking does not include falling state, and it’s like monkeywalking because of knee-bended walking profiles. However, the walking control that does not depend on ZMP is vulnerable toturnover. Therefore, keeping the event-driven walking of dynamical motion stable is important issue for realization of human-like natural walking. In this research, a walking model of humanoid robot including slipping, bumping, surface-contacting andpoint-contacting of foot is discussed, and its dynamical equation is derived by the Extended NE method. And we considering thesiliping of the foot of humanoid robot when the friction coefficient is small to create this walking model. In this paper we willintroduce the new model which including the slipping supporting foot of humanoid robot and verify the model.

Keywords: Humanoid, siliping, friction, Bipedal, Dynamical, Extended Newton-Euler Method

1 INTRODUCTION

Human beings have acquired an ability of stable bipedalwalking in evolving repetitions so far. From a view point ofmaking a stable controller for the bipedal walking based onknowledge of control theory, it looks not easy because of thedynamics with high nonlinearity and coupled interactions be-tween state variables with high dimensions. Therefore howto simplify the complicated walking dynamics to help con-struct stable walking controller has been studied intensively.

Avoiding complications in dealing directly with true dy-namics without approximation, inverted pendulum has beenused frequently for making a stable controller[1] [2] [3], sim-plifying the calculations to determine input torque. Further,linear approximation having the humanoid being representedby simple inverted pendulum enables researchers to realizestable gait through well-known control strategy[4] [5] [6].

Our research has begun from a view point of [7] [8] asaiming to describing gait’s dynamics as correctly as possi-ble, including point-contacting state of foot and toe, slip-ping of the foot and bumping. We discuss the dynamics ofwhole-body humanoid that contains head, waist and arms.And that what we think more important is that the dimensionof dynamical equation will change depending on the walk-ing gait’s varieties, which has been discussed by [9] aboutconcerning one legged hopping robot. In fact, this kind ofdynamics with the dimension number of state variables vary-ing by the result of its dynamical time transitions that are outof the arena of control theory that discusses how to controla system with fixed states’ number. Further the tipping overmotion has been called as non-holonomic dynamics that in-cludes a joint without inputting torque, i.e., free joint.

Meanwhile, landing of the heel or the toe of lifting leg in

the air to the ground makes a geometrical contact. Based on[10]. We derive the dynamics of humanoid which is simu-lated as a serial-link manipulator including constraint motionand slipping motion by using the Extended Newton-EulerMethod [11].

The conventional method of the NE could be applied toa robot having an open loop serial linkage structure, but themotion of hand was limited to motions without contactingexternal world. The NE method has not been formulated al-though it was very important for a robot that works under apremise that it must be contacted with the environment whenthe robot was doing some grinding work or assembling work.For this point, the extended NE method was proposed in [11]that is as same as the research of [12], in terms of that theconstraints are strictly satisfied. Meanwhile, the constraintforce which can be included in the iterative calculation of theNE method by calculating the constraint force by a substitu-tion method [13].

In this research that based on [14], a walking modelof humanoid robot including slipping, bumping, surface-contacting and point-contacting of foot is discussed, and itsdynamical equation is derived by the Extended NE method.And we considering the siliping of the foot of humanoidrobot when the friction coefficient is small (on the ice, snow,or wet road) to create this walking model.

In this paper we will introduce the new model which in-cluding the slipping supporting foot of humanoid robot andverify the model.

2 DYNAMICAL WALKING MODELWe discuss a biped robot whose definition is depicted in

Fig. 1. Table 1 indicates length li [m], mass mi [kg] of

q1

q2

q3

q15q9

q5

q6

q11

q10

q7

q8

q4

q13

q14

q12q16

q17

ÜW

zy

x

1

2

3

4

5

7

8

6

9

11

12

10

2

3

4

6

7

1

8

5

13

15

14

17

16

9

10

11

16

12

13

14

15

17

ÜW

z

y

x

_x

Head

Upper body

Upper arm

Lower body

Hand

Lower arm

Waist

Upper leg

Lower leg

Middle body

Foot

: Number of link : Number of joint

Fig. 1. Definition of humanoid’s link, joint and whole body

Table 1. Physical parameters

Link li mi di

Head 0.24 4.5 0.5

Upper body 0.41 21.5 10.0

Middle body 0.1 2.0 10.0

Lower body 0.1 2.0 10.0

Upper arm 0.31 2.3 0.03

Lower arm 0.24 1.4 1.0

Hand 0.18 0.4 2.0

Waist 0.27 2.0 10.0

Upper leg 0.38 7.3 10.0

Lower leg 0.40 3.4 10.0

Foot 0.07 1.3 10.0

Total 1.7 64.2

links and joints’ coefficient of viscous friction di [N·m·s/rad],which are decided based on [15]. This model is simulatedas a serial-link manipulator having ramifications and repre-sents rigid whole body—feet including toe, torso, arms andso on—by 17 degree-of-freedom. Though motion of legs isrestricted in sagittal plane, it generates varieties of walkinggait sequences since the robot has flat-sole feet and kickingtorque. In this paper, one foot including link-1 is definedas “supporting-foot” and another foot including link-7 is de-fined as “floating-foot” or “contacting-foot” according to thewalking state.

In this paper, we derive the equation of motion followingby NE formulation [?]-[?]. So we must consider the struc-ture of the supporting-foot with two situations. When thesupporting-foot is constituted by rotating joint. We first haveto calculate relations of positions, velocities and accelera-tions between links as forward kinematics procedures frombottom link to top link. Serial link’s angular velocity iωi,angular acceleration iωi, acceleration of the origin ipi andacceleration of the center of mass isi based on Σi fixed ati-th link are obtained as follows.

iωi = i−1RTi

i−1ωi−1 + eziqi (1)

iωi = i−1RTi

i−1ωi−1 + ez qi + iωi × (ezi qi) (2)

ipi = i−1RTi

{i−1pi−1 + i−1ωi−1 × i−1pi

+ i−1ωi−1 × (i−1ωi−1 × i−1pi)}

(3)

isi = ipi + iωi × isi + iωi × (iωi × isi) (4)

Then if the supporting-foot is constituted by prismaticjoint. We will switch the equations as the following.

iωi = i−1RTi

i−1ωi−1 (5)iωi = i−1RT

ii−1ωi−1 (6)

ipi = i−1RTi

{i−1pi−1 + i−1ωi−1 × i−1pi

+ i−1ωi−1 × (i−1ωi−1 × i−1pi)}

+ 2(i−1RTi

i−1ωi−1) × (ez qi) + ez qi (7)isi = ipi + iωi × isi + iωi × (iωi × isi) (8)

Here, i−1Ri means orientation matrix, i−1pi representsposition vector from the origin of (i − 1)-th link to the oneof i-th, isi is defined as gravity center position of i-th linkand ezi

is unit vector that shows rotational axis of i-th link.However, velocity and acceleration of 4-th link transmit to8-th link and ones of 10-th link transmit to 11-th, 14-th and17-th link directly because of ramification mechanisms.

After the above forward kinematic calculation has beendone, contrarily inverse dynamical calculation procedures isthe next from top to base link. Newton equation and Eulerequation of i-th link are represented by Eqs. 9, 10 when iIi

is defined as inertia tensor of i-th link.

if i = iRi+1i+1f i+1 + mi

isi (9)ini = iRi+1

i+1f i+1 + iIiiωi + iωi × (iIi

iωi)

+ isi × (miisi) + ipi+1 × (iRi+1

i+1f i+1) (10)

On the other hand, since force and torque of 5-th and 8-th

links are exerted on 4-th link, effects onto 4-th link as:

4f4 = 4R55f5 + 4R8

8f8 + m44s4, (11)

4n4 = 4R55n5 + 4R8

8n8 + 4I44ω4 + 4ω4 × (4I4

4ω4)

+ 4s4 × (m44s4) + 4p5 × (4R5

5f5)

+ 4p8 × (4R88f8). (12)

Similarly, force and torque of 11-th, 14-th and 17-th linkstransmit to 10-th link directly. Then, rotational motion equa-tion of i-th link is obtained as Eq. 13 by making inner productof induced torque onto the i-th link’s unit vector ezi

aroundrotational axis:

τi = (ezi)T ini + diqi. (13)

However, when the supporting-foot (1-st link) is slipping(prismatic joint), the torque onto the 1-st link can be calcu-lated by following equation.

τ1 = (ez1)T 1f1 + kf1y1. (14)

Finally, we get motion equation with one leg standing as:

M(q)q + h(q, q) + g(q) + Dq = τ , (15)

Here, τ is input torque, M(q) is inertia matrix, h(q, q)and g(q) are vectors which indicate Coriolis force, centrifu-gal force and gravity. When the supporting-foot is slipping,the D = diag[kf1, d2, · · · , d17] is a matrix which meanscoefficients of joints and between foot and ground. Andq = [y1, q2, · · · , q17]T means the angle of joints and the rel-ative position between foot and ground.

Making floating-foot contact with ground, contacting-footappears with contacting-foot’s position zh or angle qe to theground being constrained. The constraints of foot’s positionand heel’s rotation can be defined as C1 and C2 respectively,these constraints can be written as follow, where r(q) meansthe contacting-foot’s heel or toe position in ΣW .

C(r(q)) =[

C1(r(q))C2(r(q))

]= 0 (16)

Then, robot’s equation of motion with external force fn,friction force ft and external torque τn corresponding to C1

and C2 can be derived as:

M(q)q + h(q, q) + g(q) + Dq

= τ + jTc fn − jT

t ft + jTr τn, (17)

where jc, jt and jr are defined as:

jTc =

„

∂C1

∂qT

«T„

1/

‚

‚

‚

‚

∂C1

∂rT

‚

‚

‚

‚

«

, jTt =

„

∂r

∂qT

«Tr

‖r‖ , (18)

jTr =

„

∂C2

∂qT

«T„

1/

‚

‚

‚

‚

∂C2

∂qT

‚

‚

‚

‚

«

. (19)

It is common sense that (i) fn and ft are orthogonal, and(ii) value of ft is decided by ft = Kfn (0 < K ≤ 1). Thedifferentiating Eq. (16) by time for two times, we can derivethe constraint condition of q.

(∂Ci

∂qT

)q+qT

{∂

∂q

(∂Ci

∂qT

)q

}= 0 (i = 1, 2) (20)

Making the q in Eqs. (17) and (20) be identical, we canobtain the equation of contacting motion as follow.

2

6

4

M(q) −(jTc − jT

t K) −jTr

∂C1/∂qT 0 0

∂C2/∂qT 0 0

3

7

5

2

6

4

q

fn

τn

3

7

5

=

2

6

6

6

6

4

fi − h(q, q) − g(q) − Dq

−qT

∂

∂q

„

∂C1

∂qT

«ff

q

−qT

∂

∂q

„

∂C2

∂qT

«ff

q

3

7

7

7

7

5

(21)

Here, since motion of the foot is constrained only verticaldirection, walking direction has a degree of motion. That is,contacting-foot may slip forward or backward depending onthe foot’s velocity in traveling direction.

3 VALIDATION OF MODEL3.1 Verification by Mechanical Energy

To verify this complex model, we used the mechanical en-ergy conservation law. Because to verify the conservation ofmechanical energy, the equation of motion must be correct.We make the model to do a free fall with the input torqueτ i = 0 and the viscous friction Di = 0. In this case, there isno friction. So, it will has no loss of energy during free fall.During the motion the mechanical energy will be saved at theinitial potential energy. To derive the mechanical energy, itis necessary to calculate all of the potential energy, rotationalenergy and translational energy.

3.1.1 Calculation of Mechanical EnergyIt is necessary to calculate the height of the center of grav-

ity of each link before the calculation of the potential energy.We use the homogeneous transformation matrix to calculateit as following equation.

W zGi =W zi +W zi+1 −W zi

2(22)

Here, W zGi is the height of the center of gravity of i-thlink which seen from the world coordinate. W zi is the heightof the joint which seen from the world coordinate. So, wecan calculate the potential energy as following equation.

Ep =17∑

i=1

mWi zig (23)

Here, Ep is the potential energy of the model. mi is themass of each link. g is the gravitational acceleration.

Then, we can calculate the rotational energy as followingequation.

Ek =17∑

i=1

12

W ωTi

W IiW ωi (24)

Here, Ek is the rotational energy of the model. Ii is themoment of inertia of each link.

Then, we can also calculate the translational energy as fol-lowing equation.

Ev =17∑

i=1

12mi

W rTgi

W rgi (25)

Here, Ev is the translational energy of the model, rgi isthe translational velocity of the center of gravity of i-th link.

Finally, the mechanical energy can be derived as followingequation.

EQ = Ep + Ek + Ev (26)

Stop

q1

Stop Slipy&&

q1

(a)Supporting-foot

surface contact (stop)

(c) Supporting-foot

surface contact (slip)

(b) Supporting-foot

line contact (stop)

(d) Supporting-foot

line contact (slip)

Slipy&&

Fig. 2. Ground state of the supporting-foot

3.1.2 Verification Simulation ExperimentFirst, the simulation which about the stopped state of

the supporting-foot with surface contact and line contact areshown in Fig.2(a), (b). As shown in Fig.3, the humanoidmodel is doing a free fall with the condition that withoutany viscous friction (Di=0) and input torque (τ input = 0).The Fig.4 shows the result of the mechanical energy whenthe supporting-foot is in the stopped and surface contactstate. And Fig.5 is about the mechanical energy during thesupporting-foot is in the stopped and line contact state. FromFig.4, 5, the total of mechanical energy is saved. So, it canbe seen that support-foot stopped model is correct.

Similarly, the simulation which about the slip state ofthe supporting-foot with surface contact and line contact areshown in Fig.2(c), (d). As well as Fig.3, the humanoid modelis also doing a free fall without any viscous friction and inputtorque. The Fig.6 shows the result of the mechanical energywhen the supporting-foot is in the slip and surface contactstate. And Fig.7 is about the mechanical energy during the

① ③ ④②

Fig. 3. Free-fall of humanoid model with the supporting-footin the stopped state

-600

-400

-200

0

200

400

600

800

1000

0 0.5 1 1.5 2 2.5

En

erg

y[J

]

Time[s]

Potential energy

Mechanical energy

Rotational energy

Translational energy

① ③ ④②

Fig. 4. Mechanical energy conservation during thesupporting-foot in the stopped and surface contact state

-600

-400

-200

0

200

400

600

800

0 0.5 1 1.5 2 2.5

En

erg

y[J

]

Time[s]

Potential energy

Mechanical energy

Rotational energy

Translational energy

① ③ ④②

Fig. 5. Mechanical energy conservation during thesupporting-foot in the stopped and line contact statesupporting-foot is in the slip and line contact state. FromFig.6, 7, the total of mechanical energy is saved. So, it canbe seen that support-foot slip model is correct.

3.2 Verification by The Position of The Overall Center

of GravityIn the simulation of free fall, if the supporting-foot can

slip on the ground (without any friction), the y-position ofthe overall center of gravity should not change. In order toconfirm that, the y-position of the overall center of gravitycan be calculated as following equation.

PGAy =m1PG1y + m2PG2y + · · · + m17PG17y

m1 + m2 + · · · + m17(27)

Here, PGAy is the y-position of the overall center of grav-ity, m1 ∼ m17 is the mass of each link, PG1y ∼ PG17y is they-position of center of gravity of each link.

-600

-400

-200

0

200

400

600

800

1000

0 0.5 1 1.5 2 2.5

En

erg

y[J

]

Time[s]

Potential energy

Mechanical energy

Rotational energy

Translational energy

① ③ ④②

Fig. 6. Mechanical energy conservation during thesupporting-foot in the stopped and surface contact state

-800

-600

-400

-200

0

200

400

600

800

1000

0 0.5 1 1.5 2 2.5

En

erg

y[J

]

Time[s]

Potential energy

Mechanical energy

Rotational energy

Translational energy

① ③ ④②

Fig. 7. Mechanical energy conservation during thesupporting-foot in the stopped and line contact state

The y-position of the overall center of gravity of free fallsimulation that supporting-foot in slip surface contact stateand slip line contact state are shown in Fig. 8, 9. FromFig. 8, 9, the y-position of the overall center of gravity isnot changed. So, it also can be seen that support-foot slipmodel is correct.

4 TRANSITION OF GAITSIn this research, humanoid walking with the transition of

gaits state as shown in Fig. 9. And the route of state transitionis determined by the walking motion of humanoid. In otherwords, it depends on the solution of the dynamics in Eq. (21).The dynamics and the state variable which depending on thegaits has been selected. And it will transit to the next statewhen the branching condition is satisfied.

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 0.5 1 1.5 2 2.5

En

erg

y[J

]

Time[s]

The position of the toe of the support leg

All position of the center of gravity

Fig. 8. The y-position of the overall center of gravity duringthe supporting-foot in the slip and surface contact state

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5

Y-a

xis[

m]

Time[s]

All position of the center of gravity

The position of the toe of the support leg

Fig. 9. The y-position of the overall center of gravity duringthe supporting-foot in the slip and line contact state5 BIPEDAL WALKING SIMULATION

By using the gait transition, the authors realized thebipedal walking of humanoid robot that just like Fig. 11.The humanoid was walking 1000 steps on the ground withfriction coefficient Kf = 0.7. The walking distance is424.13[m], the walking time is required to 710.35[s]. Bythe calculation, the average speed is 2.15[km/h], the lengthof stride is 0.42[m], and the period of walking is 1.42[s].

Fig. 11. The bipedal walking of humanoid robot

6 CONCLUSIONIn this paper, we showed a walking model of humanoid

including slipping, bumping, surface-contacting and point-contacting of foot, which dynamical equation is derived byNewton-Euler method with constraint condition, named asthe Extended Newton-Euler Method. For the future, we planto extend the calculation method to the case of more than twoconstraint conditions, and evaluate it by simulations.

fr <0

S

Slip

SStopj _y0j < èjftj>jfsj zh î 0

zh_y0

qeî0 S

_yh

j _yhj<"

Slip

S

Stop

S

*

fnS

jftj>jfsj

_y0

Slipj _y0j < è

S

Stop

fn

jftj>jfsj

Slip _y0

fr <0

S

zhî0

Sfr

S

fn<0

qeî0

Szh

j _y0j < è

Stop

fr <0

Slip

_yh

S

Stop

S

fr

fr fr fr

j _yhj<"

fr <0

qe

Slip

_yh

Sfr

qe

Stop

Sfr

j _yhj<" jfyj> jftj

qe

Slip

_yh

S

qe

Stop

S

j _yhj<" jfyj> jftj

fr <0 qeî0

Emerged gait Emerged gait Emerged gait Emerged gaitEmerged gait

Emerged gaitEmerged gaitEmerged gait

zh

zh î 0

Emerged gait

Emerged gait

zh

zhî0

qeî0

: Supporting-foot is switched from one foot to the other foot* S : Supporting-foot : Possible to fall down because of slipping

Fig. 10. States and gait transition including slip motion

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