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A biped with its wearable assist device: Numericalstudy of the efficiency for an optimal walking gait

Yannick Aoustin, Christine Chevallereau and Vigen Arakalian 1

France Nantes, IRCCyN, University of Nantes

June 30, 2014

1This work is supported by Région des Pays de la Loire, Project LMA andGérontopôle Autonomie Longévité des Pays de la Loire.

Yannick Aoustin, Christine Chevallereau and Vigen Arakalian A biped with its wearable assist device: Numerical study of the efficiency

Motivation

To design a wearable assist device for human.

To improve daily life for patients or elderlyTo avoid the musculoskeletal disorders for industrial workers

Desired performances:

To be able to compensate a part of the loads due the human’sweight.

To have an assist device with a energetic autonomy.

Good adaptation to the shape of human’s body.


Several realisations 1/2

prototype honda


Several realisations 2/2

ARGO Medical Technologies


Statement of the problem

Preliminary studies about a simple mechanical:

To consider a seven-link planar Biped with a wearable assist device.

To define a cyclic walking gait with for the biped without any assistdevice.

To compare the energy consumption of the biped with its wearableassist device for four cases.

The assist device is fully actuated.The hips of the assist device are actuated only.The knees of the assist device are actuated only.The ankles of the assist device are actuated only.

A each sampling time of the trajectory an optimal distribution of thetorques for the assisted biped are found.


The geometrical structure of the blped and its wearable

assist device.

(a) (b)

Γ1

Γ2

Γ3Γ4

Γ5

Γ6

Γ7

Γ8

Γ9

Γ10Γ11

Γ12


Generalized coordinates and torques

Both geometrical closed structures can be cut in any point.

The generalized coordinates is introduced, assuming a treegeometrical structure.

15 generalized coordinatesx = [q1, q2, q3, q4, q5, q6, q7, q8, q91, q92, q101, q102, q11, x , y ]

⊤.

q8, q91, q92, q101, q102, and q11 are devoted to the wearable assistdevice.

Torque vector, Γ number of components between six (no assistance)and 12 (fully assisted).


Physical parameters.

Mass (kg) Length (m) Inertia (kg .m2) center of mass (m)

Foot mf = 0.678 Lp = 0.207 I f = 0.012 spx = 0.0135and lp = 0.072 spy = 0.0321shoe Hp = 0.064Shin ms = 4.6 ls = 0.497 I s = 0.0521 ss = 0.324Thigh mt = 8.6 lt = 0.41 I t = 0.7414 st = 0.18Trunk mT = 16.5 lT = 0.625 IT = 11.3 sT = 0.386

Seat m3 = 2.0 l3 = 0.1 IT = 0.3 s3 = 0.05Upper m1 = 3.0 l1 = 0.392 I 1 = 0.04 s1 = 0.1127frameLower m2 = 2.0 l2 = 0.3645 I 2 = 0.02 s2 = 0.169link

Table : Physical parameters of the seven-link biped and of its walking assistdevice.


Matrix equations

Equations of the motion of the biped in single support

A(x)x + h(x, x) =[

D J⊤1 J⊤2

]

[

Γfc

]

+ J⊤

r1

[

r1

m1z

]

+ J⊤

r2

[

r2

m2z

]

, (1)

with the constraint equations,

Jrix + Jri

x = 0 for i = 1 to 2[

J1

J2

]

x +

[

J1

J2

]

x = 0.(2)

[

ri miz

]⊤, with i = 1 to 2, : resultant wrenches of the contact efforts

λi = fci=[fxi

, fyi,mzi

]⊤ : the wrench, for each loop closure.


The walking reference gait 1/3

An optimal cyclic walking gait

Each step: a single support phase and an impact.

The swing foot impacts the ground the other foot takes off theground

Biped’s joints are prescribed with third order polynomial functions

Definition of the coefficients of these polynomial functions takes intoaccount:

The cyclic propertiesThe impact equations

Parametric optimization algorithm is used to determine the neededcoefficients −→ optimization variables with constraints.

Non linear constraints (limits on ZMP, ground reaction, Torques).


Optimal distribution of the torques for the biped with the

wearable assist device 1/4

The biped equipped with its wearable assist device tracks the referencetrajectory defined previously.

Three cases

The biped is fully assisted There are 12 torques.

The biped is partially assisted: There are 6 torques + 2 provided bythe assit device in the hips.

The biped is partially assisted: There are 6 torques + 2 provided bythe assit device in the knees.

The biped is partially assisted: There are 6 torques + 2 provided bythe assit device in the ankles.




The optimization statement at each sampling period

A sampling period ∆ is defined such as: ∆ =T

N.

At each sampling period a new optimal process is stated to get anoptimal distribution of the torques.

Let matrix J⊥(12× 15) be such that [r1 m1z]⊤ belongs to the kernel

of the linear map represented with J⊥(12 × 15) J⊤r1

, (in singlesupport on leg 1: r2 = 0, m2z

= 0)

The dynamic equation of the biped and assist device is multipliedwith J⊥(12 × 15), we get

J⊥A(x)x + J

⊥h(x, x) = J

⊥[

D J⊤1 J⊤2

]

[

Γfc

]

. (3)




Definition of the optimization variables

The left handle side J⊥A(x)x + J⊥h(x, x) is well-known because itdepends on the walking reference gait and the physical parametersof the mechanical systems only.

The rank of this J⊥[

D J⊤1 J⊤2

]

is equal to 12.

Biped fully assisted in stance phase: There are 12 equations for 18unknown variables −→ 6 optimization variables.

Biped partially assisted: There are 12 equations for 14 −→ 2optimization variables.




The optimization statement

A sampling period ∆ is defined such as: ∆ =T

N.

At each sampling period a new optimal process is stated to get anoptimal distribution of the torques.

The criteria for k = 1, · · · ,N are:

C1k = m⊤

k mk .

with mk the vector torques of the bipedand

C2k = Γ⊤

k Γk .

with Γk the vector torques of the biped and its assist device.

Constraints: to limit the efforts of each loop closure for biped andits assist advice, vertical component of the ground reaction positiveand no rotation of the stance foot.


Reference walking gait

Numerical data

N = 50. Walking trajectory defined in 50 points.

Time of a step: 0.55 s.

Length of one step: 0.3575 m.

The velocity of the biped −→ : 0.65 m/s or 2.34 km/h


Three distributions for the impulsive torques 1/3

Fully

assisted

Hipsassisted

Kne

esassisted

Ank

lesassisted

200

400

600

Cost

Funct

ional

Part for Biped

Part for Assist device

Biped with Assist device

Biped without Assist device

Figure : The torques of the biped are minimized (Cost C1k): Histogram as afunction of different distributions for the torques.


Three distributions for the impulsive torques 1/3

Fully

assisted

Hipsassisted

Kne

esassisted

Ank

lesassisted

100

200

Cost

Funct

ional

Part for Biped

Part for Assist device

Biped with Assist device

Biped without Assist device

Figure : The torques of the biped and walking assist device are minimized (costC2k): Histogram as a function of different distributions for the torques.


Criteria as functions of the velocity 0.4 m/s and 0.9 m/s

(1.44 km/h and 3.24 km/h). (1/2)

(a)

0.4 0.5 0.6 0.7 0.8 0.90

100

200

300

400

500

600

700

800

900

(b)

0.4 0.5 0.6 0.7 0.8 0.90

100

200

300

400

500

600

700

(a) Fully assisted biped with C1k , and (b) Fully assisted biped with C2k .


Criteria as functions of the velocity 0.4 m/s and 0.9 m/s

(1.44 km/h and 3.24 km/h). (2/2)

(c)

0.4 0.5 0.6 0.7 0.8 0.90

50

100

150

200

250

300

350

400

(d)

0.4 0.5 0.6 0.7 0.8 0.90

50

100

150

200

250

300

350

400

450

500

(c) Biped with assistance in the hips with C1k and (d) Biped withassistance in the hips with C2k .


Conclusion and perspectives 1/2

Conclusion

Preliminary study with a simple bipedal robot and a simplegeometrical structure for the wearable assit device.

Full actuation of actuation at hip only are two interesting proposals.

actuation at the knee or ankle only are ineffective.

A study for several velocities confirm these properties.

In joints, which are not assisted, we can observe the torques of thebiped can increase little.


Conclusion and perspectives 2/2

Perspectives

Added research on the design of the assist device and the trajectoriesare necessary to get null torques with full actuation (paralyzedhuman subjects).

To use a walking gait with double support phases, single supportphases and impact as reference gait.

To insert flexible elements between human and its wearable assitdevice.

To deal with a 3D biped.

Climbing stairs


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