rotation αααα

αααα motor 1

Epycloidal gear

-0.2 -0.1 0 0.1 0.2 0.3-0.1

0

0.1

0.2

0.3

0.4

0.5

X [m]

Y

[m]

+0.5e0

+0.6e0

+0.7e0

+0.8e0

-0.2 -0.1 0 0.1 0.2 0.3-0.1

0

0.1

0.2

0.3

0.4

0.5

X [m]

Y

[m]

+0.8e0

+1.0e0

+1.2e0

+1.4e0

λ

Drill tip movement λλλλ

Linear track

slideBall skrew

motor 3

Kinematics, dynamics and control of a hybrid

parallel-serial redundant manipulator

Diego Tosi, Giovanni Legnani, Pier Luigi Magnani

Dip. Ing. Meccanica, Università di Brescia

Via Baranze 38, 25123 Brescia, Italy

diego.tosi@ing.unibs.it giovanni.legnani@ing.unibs.it

Applications

SURGICAL ROBOT

• high precision and dexterity

• mini-invasive surgery

• easier surgical procedure planning

• intra-surgical co-ordination with other

medical equipment

Pedicle screws

Metal plate

=

gz

gy

gx

zzx

zyx

zyx

V

V

V

aaa

aaa

aaa

q

q

q

323

222

111

3

2

1

&

&

&

iii

iu

a λλ ⋅

=1

gJVQ =&

Slide:

Pi position

ui unit vector

linear velocity

Vi vector velocity

linear acceleration

Ai vector acceleration

platform:

Vgi vector velocity

Agi vector accelerationiq&&

iq&

leg:

Li leg vector

li length

λλλλi unit vector

wi unit vector of velocity

angular velocity (module)

angular velocity (vector)

angular acceleration (module)

angular acceleration (vector)

iΩiω

iΩ&iω&

=

z

y

xT

f

f

f

J

f

f

f

3

2

1

gripper forcesactuator forcesgripper velocityactuator velocity

iiii quPP += *

lL ii λ=

iii quV &=

iii quA &&=

gigigi AVP ,,

iii wω=Ω iiii

igi

quPP

lPP

+=

=−*

( ) lwquLquV iiiiiiiiigi ωλ×+=×Ω+= &&

0=⋅ iiw λii

giii

u

Vq

⋅

⋅=λ

λ&

( ) iiiiiiiiiiiiiiigi llwquLLquA λωλω 2−×+=×Ω×Ω+×Ω+= &&&&&&

( )ii

iigiii

u

lAq

⋅

+⋅=

λ

λωλ 2

&&

l

VV igii

−=ω

slide

leg

moving platformposition analysis

velocity analysis

acceleration analysis

• 3 d.o.f.

• 3 motors

• isostatics

• pure translation

• 3 d.o.f.

• 4 motors• over-constrained

• structure preloaded toreduce backlash andsingularity

• improved workspace

• 4 d.o.f.

• 4 motors• isostatics

• 4 dof “complex”rototranslation

• improve rotation ofserial wrist

• 4 d.o.f. (type B)

• 4 motors

• isostatics

•translation plusrotation aroundaxis u

u

(in cooperation with Politecnico di Milano)

36 parameters

24 significant combinations

of parameters

13° 25°

ideal geometry

with errors

13°

25°

σσσσi

104

102

100

10-2

10-4calibration simulations predict the following accuracy

5mm / 50 mrad before calibration

0.5 mm / 3 mrad after calibration

effect of significant structural errors (left) and non-significant errors (right)

Parallel structure

3 pairs of parallel rods

• spherical joints at the ends

• slides on guides inclined of 30°

• 3+1 axes

The platform

translates

Serial wrist

• α rotation

• β lifting

• λ drill tip movement

(redundant)

roof

Mobile

platform

slides

Spherical

joints

α

β

λ

INTERVERTEBRAL PEDICLE

SCREW INSERTION

working space: 400x400x300 mm

rotations: +\- 40°

velocity: 1.5 m/s

acceleration: 20 m/s2

Kinetostatic analysis

Hybrid kinematic structure

rods

Parallel part re-configurability

Geometry optimisation

Parallel structure

supporting frame

high mobility

spherical joints

support for the

serial wrist

precision plate for

the laser calibration

of the slides

Brushless

linear motor

Serial wrist

flexible cable support

linear tracks

permanentmagnets

optical encoder

support for

spherical joints

Calibration is performed moving the

axis and measuring the x-y position of

the incoming beam

precision laser support (8 positions)

encoder

lift ββββ

ββββ

motor 2

Calibration

-0.2 -0.1 0 0.1 0.2 0.3-0.1

0

0.1

0.2

0.3

0.4

0.5

X [m]

Y

[m]

+1.6e0

+1.7e0

+1.8e0+1.9e0

-0.2 -0.1 0 0.1 0.2 0.3-0.1

0

0.1

0.2

0.3

0.4

0.5

X [m]

Y

[m]

+3.0e0

+2.5e0

+2.0e0

Det(J) σmin(J) σmax(J) cond(J)

laser source

laser beam

x-y sensor

D. Tosi, G. Legnani, Calibration of a Parallel-Serial Hybrid Redundant Manipulator, Proc.

of ISR 2003, Chicago, 2-5 June 2003

-H. Giberti, P. Righettini, D. Tosi, G. Legnani, R. Adamini, Progetto di “CHEOPE”: un

manipolatore ridondante ibrido palallelo-seriale, Proc. of AIMETA 2003, Ferrara 9-12 Sept.

2003

-D.Tosi, G. Legnani, P. L. Magnani, Calibration of “CHEOPE” a parallel-serial redundant

manipulator, Proc. of AIMETA 2003, Ferrara 9-12 Sept. 2003

References