Notes_04_03
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Two-Dimensional Kinematics
Position
x i 
ri   
q i      y i 
  i   
 i
ri P  ri   A i s i ' P
x i  P x i  C i
     
y i 
y i   Si
 S i  x i  ' P
 
C i  y i 
s i P  A i s i ' P
s i ' P  A i T s i P
C i
A i   
 S i
 S i 
C i 
Velocity
r 
q i    i 
 i 
ri P  ri   A i s i ' P
A    CS
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Bi   

 Si
 C i
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 C i  
  i Bi    i R A i 
 S i 
 C i 
 R A i   A i R 
 S i 
ri P  ri    i Bi s i ' P  ri    i R A i s i ' P
Acceleration
r 
qi    i 
 i 
ri P  ri   A i s i ' P
0  1

1 0 
R   
R 2  
1
0
0
 1
Notes_04_03
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A    R A 
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A    R A    R  A 
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A    R A    R   R A 
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A    B    A 
2
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ri P  ri   i Bi s i ' P  i 2 A i s i ' P
Jerk
r
qi    i 
 i 
riP  ri  A i s i ' P
A    B    A 
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A    B    B   2  A    A 
2
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A    B    R A  2  A    A 
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A    B    R  R A   2  A     R A 
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A      B   3  A 
3
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Snap

q   r
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 i 
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r  r A s '
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A      B   3  A 
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A     3  B      B  3
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A i   3 ii A i   3 i i A i 
Notes_04_03
3 of 9
A     3  B      RA  3 A   3  A   3  A 
A     3  B      R RA   3 A   3  A   3   RA 
A     6   B   4    3   A 
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Partial derivatives
ri 
P
 ri   A i s i '
 x i P   x i   C i
 P    
 y i   y i   S i
P
r  
r  
1
 
0
r  
r  
 S i

 C i
 C i  x i ' P 
P
 P   Bi s i '

 S i  y i ' 
P
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xi
P
i
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P
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yi
0
 
1
P
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ri
 S i   x i ' P 


C i   y i ' P 
1 0

  I 2 
0 1 
r 
qi    i 
 i 
r    r   r   
P
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P
qi
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r     I 
P
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qi
P
ri
2
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i
Bi s i ' P

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2
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2
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r  
P
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qj
 0 2 x 3 
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Notes_04_03
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Although there are no formal definitions for terminology, higher derivatives of position
are often denoted as snap (fourth derivative), crackle (fifth derivative) and pop (sixth derivative).
While snap has been considered in the fields of unmanned air vehicle (UAV) trajectory planning
[18], biomechanics [19, 20], hand writing analysis [21] and astrophysics [22], effectively no snap
kinematics have been presented for multibody chains.
[18] Mellinger, D., and Kumar, V., 2011, “Minimum Snap Trajectory Generation and Control
for Quadrotors”, IEEE Int. Conf. on Robotics and Automation, p. 2520-2525, Shanghai,
China.
[19] Wiegner, A.W., and Wierzbicka, M.M., 1992, "Kinematic Models and Human Elbow
Flexion Movements: Quantitative Analysis”, Exp. Brain. Res., 88:665-673.
[20] Novak, K.E., Miller, L.E. and Houk, J.C., 2000, “Kinematic Properties of Rapid Hand
Movements in a Knob Turning Task”, Exp. Brain Res., 132:419–433.
[21] Edelman, S., and Flash, T., 1987, “A Model of Handwriting”, Biol. Cybern. 57: 25-36.
[22] Russo, J.G., and Townsend, P.K., 2009, “Relativistic Kinematics and Stationary Motions”,
J. Physics A: Math. Theory. 42-445402.
Angular jerk of the output link for the four bar is shown in Fig. 2. Error between the
explicit geometric jerk solution and the numerical jerk simulation is provided in Fig. 3 for
assembly tolerance used to terminate the iterative Newton-Raphson numerical position solution
of 1.0e-12. Root-mean-square error (RMSE) for this jerk simulation was 1.7e-13. Angular snap
of the output link for the four bar is shown in Fig. 4. Error between the explicit geometric snap
solution and the numerical snap simulation for the same assembly tolerance is provided in Fig. 5
with RMSE of 1.1e-12. Similar results were observed for the inverted slider crank with both
drivers.
When RMSE for snap is normalized by the maximum absolute value for snap and
compared to similar normalized errors for velocity, acceleration and jerk, all are approximately
the same as shown in Fig. 6. Normalized RMSE between explicit geometric solutions and planar
numerical simulations was strongly related to assembly tolerance for values larger than 1.0e-11
also shown in Fig. 6.
List of Figures
Figure 1 – Four bar and inverted slider crank planar mechanisms
Figure 2 – Angular jerk for the output link of the four bar mechanism
Figure 3 – Jerk error between explicit geometric solution and numerical simulation for the output
link of the four bar mechanism
Figure 4 – Angular snap for the output link of the four bar mechanism
Figure 5 – Snap error between explicit geometric solution and numerical simulation for the
output link of the four bar mechanism
Figure 6 – Normalized RMSE for planar four bar numerical simulation
Notes_04_03
Figure 1 – Four bar and inverted slider crank planar mechanisms
Figure 2 – Angular jerk for the output link of the four bar mechanism
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Notes_04_03
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Figure 3 – Jerk error between explicit geometric solution and numerical simulation for the output
link of the four bar mechanism
Notes_04_03
Figure 4 – Angular snap for the output link of the four bar mechanism
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Notes_04_03
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Figure 5 – Snap error between explicit geometric solution and numerical simulation for the
output link of the four bar mechanism
Notes_04_03
Figure 6 – Normalized RMSE for planar four bar numerical simulation
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