A special kind of mechanical event involving two detached objects and two successive displacements is an articulation, in which the first displacement causes the second. It is a superordinate event with two subordinate events.

1. Screw Kinematics

2. THE CAUSATION OF EVENTS
• A special kind of mechanical event involving
two detached objects and two successive
displacements is an articulation, in which the
first displacement causes the second. It is a
superordinate event with two subordinate
events.
• Joints are modeled using unit twists.
• The question of whether or not such
causation could be perceived.

3. A linear chain of rigid bodies

4. Velocity Kinematics
• Twists are used to describe the motion of rigid bodies.
The relative motion twist between two joined bodies
is split into two parts.
• The unit twist conveys the geometrical information;
whereas, the constitutive portion is the magnitude. If
the joint has a single degree of freedom then it can be
modeled by a screw. For example, a revolute joint is
modeled by a zero pitch screw and a prismatic joint by
an infinite pitch screw. The joint axis thus represented
by a unit twist s. The twist magnitude represents the
joint speed.
rv sq &
where s is a unit twist and 𝑞 is the twist magnitude

5. • This notation is extended to multiple degrees of
freedom with multiple sequential screws, such that
1 1 2 2
...r
k k
s q s q s qv    & & &
Where k is the number of degrees of freedom, 1
s
through k
s the unit twists representing each joint,
and 1
q&through k
q&each joint speed.
Equation 1 1 2 2
...r
k k
s q s q s qv    & & &
is compacted into
r sqv  &
Where s is the 6 k joint matrix
1 2 ... ks s ss   
And q&the 1k joint speed matrix

6. A slider pin joint model

7. And q&the 1k joint speed matrix
1
2
k
q
q
q
q
 
 
 
 
 
 

&
&
M
&
&
Each column of s represents a base screw for the subspace of relative
motions between joined bodies.
A slider pin joint is modeled with the slider in the x the direction and
the pin about zaxis. The individual joint axes are
1
0
0
0
0
0
x
s
 
 
 
 
  
 
 
 
  
2
0
0
0
0
0
z
s
 
 
 
 
  
 
 
 
  
The 6x2 matrix joint space is
1 2
0
0
s s
x
z
s   
 
  
 


8. The Basis Screw
• Joints are modeled as the basis screw of the
relative motion between bodies.
• The velocity difference between two
successive bodies is proportional to the screw
axis of the joint between them.
• The relative acceleration between bodies
contains a part along the joint axes as well as
bias acceleration in other direction.

9. The knee joint connects femur to tibia
1i i i i
s qv v
  &

10. Acceleration Kinematics
The acceleration kinematics are defined by
substituting 1i i i i
s qv v
  &into v
t
a 

 for each body i
The change of the velocity twist with time is the
spatial acceleration.
1
1
1
( )
( )
i i
i i i
i i i
i
i i i i
v
t
v s q
t
a s q
t
s
a s q q
t
a







 


 


  

&
&
&& &

11. where i
a and 1i
a
are the spatial accelerations of
bodies i and i-1. i
s is the joint axis, iq&the joint
speed, and iq&&the joint acceleration. In order to
calculate the rate of change of i
s the joint axis
is expanded from
0P
r
v
 

   
    
   
into
i
e r e
s
e
   
  
 
Where is the joint axis pitch, e is its direction, and
r its location. It is known from
[1]
1. Featherstone, R. Robot dynamics algorithms.
Kluwer: 1987.
The rate of change of any vector e fixed to a
moving rigid body is
e
e
t


 


12. The rate of change of the joint axis
( ) ( )
( ) ( )
0
i
e r es
et t
e r e
t t
e
t
r
e r e e
t
e
v e r e
e


  

 

   
     
  
   
  
 
  
 
       
 
 
      
       

13. Which is generalized as
i
i
s
v s
t

 

Where v is the velocity twist
And v is the spatial cross product operator.
0
v
v


  
    
Let joint i connect body i to body i-1, then the
accelerations are related by
1i i i i i i ia a s q v s q   && &
The bias acceleration term i i iv s q &is the
convective acceleration of the joint is .