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Project and Implementation of a DynamicallyStable Gait in the Robot Nao

Universita degli Studi di Roma La SapienzaMaster in Mechanical Engineering

Rafael Rojas

Rafael Rojas Biped Locomotion in Nao Robot

1 Motivations.

2 Kajita simplification of gait dynamics.

3 Walking Pattern Generator.

4 The kinematic controller.

5 Implementation in Nao.


Motivations

Nao comes with build-in functions to make the robot walkgiven some parameters, such velocity or a position of arriving.

But we dont have access to the walking algorithm.

To modify the motion planner and make it robust, forexample, with respect to external disturbance, it is necessaryto have access to the walking pattern generator.

The development of a new walking pattern generator it isnecessary also for planning the motion of the robot throughobstacles.


The Kajita simplified dynamics

zcgx + x = ZMPx

zcgy + y = ZMPy (1)

This approach have a linear relationship between the state of thesystem and the Zero Moment Point (ZMP) position in thelocomotion plane.


WPG

The Walking Pattern Generator (WPG) is a block, where afootstep sequence with a ZMP trajectory are transformed into atrajectories of the feed and the center of Mass (CoM) that allowsan humanoid robot to walk with dynamic stability.


WPG: Preview Controller

Our WPG is based on a feedback controlled system with a controllaw noted as preview controller.


WPG: Preview Controller

The control law is:

u(k) = GIk

i=0

e(i) Gxx(k)NLl=1

Gd(l)ZMPd(k + l)

and the values of GI , Gx and Gd(l) are calculated to minimize theperformance index:

J =i=k

[eT (i)Qee(i) + x

T (i)Qxx(i) + uT (i)Ru(i)

]this is done through the resolution of the discrete algebraic riccatiequation (DARE).


WPG: our variables

Our WPG have the follow variables that we can modify:

time of double support Tds

time of single support Tss

discretization time T

height of the car of the kajita model zc

the trajectory of the swinging foot

the trajectory of the ZMP


DARE risolution

with a discretization time of T = 0.01 s, Nl = 150 future samplesof ZMPd and zc = 0.31

0

500

1000

1500

2000

0 0.2 0.4 0.6 0.8 1 1.2

GI = 1149.635782.

Gx = 42986.459193 8251.545731 114.108039.

Gd is in the figure.


WPG: ZMP trajectory


WPG: CoM trajectory

-0.06

-0.04

-0.02

0

0.02

0 2 4 6 8 10 12 14

posizione del CoM lungo x0

0.2

0.4

0.6

0.8

1posizione del CoM lungo x


WPG: left foot trajectory

0 0.050.10.15

0.20.250.30.35

-0.1-0.05

00.05

0.10

0.01

0.02

0.03

0.04

0.05

0

0.002

0.004

0.006

0.008

0.01

0.012

0 2 4 6 8 10 12 14

z0.042

0.043

0.044

0.045

0.046

0.047

0.048

0.049

0 2 4 6 8 10 12 14

y

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4 x


WPG: rigth foot trajectory

0 0.050.10.15

0.20.250.30.35

-0.1-0.05

00.05

0.10

0.01

0.02

0.03

0.04

0.05

0

0.002

0.004

0.006

0.008

0.01

0.012

0 2 4 6 8 10 12 14

z

-0.4

-0.2

0

0.2

0 2 4 6 8 10 12 14

y

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4 x


WPG: trajectory in the space

00.1

0.20.3

0.40.5

-0.2-0.15

-0.1-0.05

00.05

0.10.15

0.20

0.1

0.2

0.3

0.4

0.5


Kinematic Controller

the WPG implicitly splits the problem of the dynamically stablegait in two parts:

a dynamic part solved by the WPG, that generates stabletrajectories

a kinematic part of solving the inverse kinematic problem oftracking the trajectories generated by the WPG.


Kinematic Control: resolution of kinematic redundancy

the general solution of the inverse kinematicsfor a task ti is:

q = J#ti ti+Piuj where Pi = IJ#ti Jti and uj R

n

the vector uj Rn can be used to trackanother task tj :

uj = (JtjPi )#[tj JtjJ#ti ti

]+ Pjuk

with Pj = I (JtjPi )# (JtjPi )


this reasoning can be usedrecursively to solve and array of Ntask t1, t2, , tN sorted by priority,defining:

i = i1Pi with 0 = I

Jti = Jti i1 with Jt1 = Jt1

Pi = I J#ti Jti ,vi =i

k=1 k1J#tk

(tk Jkvk1

)with v0 = 0.

q =Ni=1

i1J#ti

(ti Jivi1

)after, we need to integrate numerically: qk+1 = qk + q(k)T


General View of the Project


Simulation in Webots

Webots is an environment of develop, design and dynamicsimulation of mobile robots.


Simulation: Indici di prestazioni

Now we can show an example of the performance of our WPG andkinematic controller on the physic model of Nao that we can foundin webots.The performance index that we have used is:

Total Square Torque: TST = QTQ,

where Q is the vector of actual torque in the motors.


Example: one trajectory of the ZMP


Example

(Loading Circle-m-increase3.mp4)


sim2video.aviMedia File (video/avi)

Example: performance index

0

100

200

300

400

500

0 50 100 150 200 250 300 350 400 450


Integration in Nao

We start the implementation of our software in NaoQi from onemodule example named fastgetsetexample that uses themodule DCM. We have modified the lines from 347 to 385 andthere we have put the next pseudocode :

1 get DCM time and set in the variable command

2 get the feedback of the motors of nao withfMemoryFastAccess->GetValues(sensorValues);

3 compute the WPG actual desired position and velocities

4 resolve the inverse kinematic problem as we shown below

5 send the desired position to Nao withdcmProxy->setAlias(commands);


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