screw theory for ecological biomechanics

1. Screw

2. The principle of duality
• This study in projective geometry reveals that the
principle of duality applies to the screw.
• Here, the screw is demonstrated to be an element of
a projective three-dimensional space (P3), right
alongside the line.
• Dual elements for the screw and line are also
revealed (the nut and spline).
• Reciprocity is demonstrated for a pair of screws, and
incidence is demonstrated for screw and its dual
element. Reciprocity and incidence are invariant for
projective transformations of P3, but only incidence
is invariant for the more general linear
transformations of screws.

3. ‘‘A screw is a screw is a screw.’’
• The screw is an ingenious and useful element of three-
dimensional space. You can use it to model the general
infinitesimal displacement of a rigid body, or you can use it
to model the general loading exerted on a body.
• One body twists relative to another body on a screw. A
body applies a wrench to another body along a screw.
• One of the most appealing things here is that the screw
handles all cases of instantaneous displacement and all
cases of loading, all under one umbrella.
• Screws of zero pitch are lines, which are used to model
pure rotations and pure forces. Screws of infinite pitch (or
of zero pitch on lines at infinity) are used to model pure
translations and pure couples.
• The screw therefore has been poised for more than 100
years to unify the sciences of kinematics and statics.

4. Featherstone’s Vampire
The lack of duality The concept of “aggregate”
Griffis M, Rico JM. 2003. The nut in screw theory. Journal of Robotic Systems 20:437-47
6.
Plücker J, Klein F. 1868. Neue geometrie des raumes gegründet auf die betrachtung der
geraden linie als raumelement Erste-[zweite] Abtheilung. In. Leipzig: B. G. Teubner.

5. Helicoidal velocity field

6. Velocity Twist
describes dynamics with physically meaningful quantities

7. The velocity twist of a rigid body expressed at
a point P
P
P
v
v

 
  
 
Where v
P
is the linear velocity vector of the
body expressed at P and  the angular velocity
of the rigid body.

8. The information in the array
• This vector combination conveys enough
information to describe the motion as a
simultaneous rotation and translation ab
out an axis in space.
• The instantaneous path followed by any
point on the rigid body through time is
a helical (or helicoidal) curve.

9. The screw pitch
 By definition, the linear vector Q
v of any
point Q lying on the twist axis is parallel to
the angular velocity .
 The screw pitch  is defined by
Q
v 
 The pitch is the scalar factor between the
linear velocity Q
v and the angular velocity
vector .

10. The linear velocity vector at
reference point P is
P
rv    
Where ris the position vector from point P to
point Q.

11. The Spatial Velocity is split into
two parts
0P
r
v
 

   
    
   
which represents a rotation of  about an axis
located by r and a translation  parallel to
the axis of 

12. The geometrical information
 The geometrical information in P
v is a screw
axis located by rin the direction of and
the pitch, , which has units of length
representing the translation along the axis
per revolution.
 A pitch of zero indicates a purely rotational
motion and an infinite pitch indicates a
purely translational motion.

13. A line vector and a free vector
0P
r
v
 

   
    
   
 The first part is associated with a line vector
since it contains the orientation and
location of the screw axis.
 The second part is associated with a free
vector because its components are
independent to the reference point.

14. Screw Two

15. Acceleration Twists
• Rigid body accelerations do not form helical ve
ctor fields. The reason is that the linear acceler
ation of any point on a rigid body contains vel
ocity related terms.
• This is called the material acceleration and it is
the total derivative of the velocity vector such t
hat
• m d
v
dt
a 

16. The local acceleration
• Assuming the velocity is a function of po
sition and time, then the mate
rial acceleration is split into two parts
• The local acceleration is defined by
( , )v v p t
m v v dp
t p dt
v
a v
p
a  
 
 

 

v
t
a 



17. A small change in the position
vector changes the velocity vector
• The change of the velocity vector while holding everyt
hing constant and changing the position of the referen
ce point is calculated from the velocity vector expansio
n
• Using the position vector of the screw axis and the po
sition of the reference point, then
v r   
r q p 

18. ( , ) ( , )
[ ( ) ] [ ( ) ]
v v p p t v p t
q p p q p
p
p
   


   
        
 
 
Therefore
v
p

 

A small change in the position vector ch
anges the velocity vector as

19. The material acceleration
• To relate the material acceleration with t
he local acceleration for a rigid body rot
ating simplifies to
• For particles and rigid bodies, forces thro
ugh the center of mass and material acc
elerations are related by a scalar mass su
ch that
m a va   
mF ma

20. The linear equation of motion
• The linear equation of motion becomes
• The local vector acceleration does form a
helical vector field and can be represente
d by a screw quantity. Together with the
angular acceleration vector they form th
e spatial acceleration.
F ma mv  

21. Spatial acceleration
• The acceleration twist of a rigid body ex
pressed at a point P is
• The acceleration is also a screw quantity
and can be split into two parts
P
P
a
a

 
 
 
0
r ha  

  
  
  
    
 

22. The spatial acceleration vs.
the material acceleration
• By definition, the spatial acceleration me
asures a different quantity from the mate
rial acceleration.
• The material acceleration measures the a
cceleration of a particular point on the ri
gid body, while spatial acceleration meas
ures the acceleration state of the entire ri
gid body at a fixed location in space.

23. Material acceleration of a constant
spinning disk

24. Spatial acceleration of a constant
spinning disk is zero

25. The distinction
• The distinction between measuring a par
ticular point on a body, and measuring t
he entire body at a particular location is
crucial to the understanding of screw the
ory.
• The velocity twist is a function of positio
n, orientation, and time. The change with
time is the spatial acceleration
v
t
a 



26. An infinitesimal screw motion
• The change of position and orientation is
expressed through an infinitesimal screw
motion
p


 
   


27. If the reference frame is moving
• If the reference frame is moving along wi
th the body then
• The total derivative of the velocity twist i
s then split in two parts
v t  
dv v v d
tdt dt
va v



  
 
 


28. 6×6 matrix
• Also, since the angular velocity vector do
es not change with position
v v
pv
p

 

  
    
   
   
v
p

 

0
p




29. Change of any arbitrary vector A
with an infinitesimal rotation
• If the vector is fixed and the reference fr
ame changes then the apparent change i
n is
A A  
A A
A


  
 
A

30. • Finally, assembling all the partials togeth
er in spatial form
Applying substituting vandfor Ayields
v
v

 


 


 

0
( )T
v
v
v


 
 
   
 

31. The spatial cross product operator
v×
• Simplifies
0
v
v


  
   
dv v v d
tdt dt
va v



  
 
 

( )dv Ta v v
dt
  

32. in component form
v a vd
dt

 
    
   
   
m a va   