Computational Motor Control: Kinematics & Dynamics (JAIST summer course) This is lecture 1 note for JAIST summer school on computational motor control (Hirokazu Tanaka & Hiroyuki Kambara). Lecture video: https://www.youtube.com/watch?v=8nk4DlpAaS8

1. Computational Motor
Control Summer School
01: Kinematics, Dynamics,
and Coordinate
transformation.
Hirokazu Tanaka
School of Information Science
Japan Institute of Science and Technology

2. Kinematics, dynamics and coordinate transformations.
In this lecture, you will learn…
• Kinematics
• Redundancy (ill-posed) problem
• Dynamics
• Equations of motion: Euler-Lagrange & Newton-Euler
methods
• Neurophysiology: Gain fields in parietal cortex
• Coordinate transformation in the brain
• Human psychophysics experiments

3. Kinematics: two coordinate systems for the body.
Atkeson (1989) Ann Rev Neurosci
(x, y): Cartesian coordinates
“Extrinsic coordinates”
(θ1, θ2): Joint angle coordinates
“Intrinsic coordinates”

4. 1.1. Forward kinematics from joint angles to Cartesian positions.
 
 
1 1 2 1 2
1 1 2 1 2
cos cos
sin sin
x l l
y l l
  
  


 

 
Atkeson (1989) Ann Rev Neurosci
Forward kinematics
= Computation of extrinsic coordinates
from intrinsic coordinates

5. 1.1. Inverse kinematics from Cartesian positions to joint angles.
2 2 2 2
1 2
1
1 2
2 2
2
1 1 2
arccos
2
sin
arcta arctann
cos
y
l
x l l
l
l l
l
y
x




  









Atkeson (1989) Ann Rev Neurosci
Inverse kinematics
= Computation of intrinsic coordinates
from extrinsic coordinates

6. Ill-posedness: many-to-one mappings from joints to positions.

7. 1.2. Dynamics = Equations of motion in joint angle and space coordinates.
 
   
   
2 2 2
1 1 2 1 1 2 2 2 1 2 1 2 2 1
2
2 2 2 2 1 2 2 2 2 1 2 1 2 2 2
2 2
2 2 2 2 2 1 2 2 1 2 2 2 2
2
2 1 2 1 2
2 cos
cos 2 sin
cos
sin
I I m r m r m l m l r
I m r m l r m l r
I m r m l r I m r
m l r
  
     
   
 
     
    
    

2 1
1 2 20 20 21 21 1 10 10 10 102 2
2 1
2
2 2 21 20 21 212
2
Z
Z
I I
m X A X A m X A X A
r r
I
m X A X A
r


 
        
 
 
    
 
Euler-Lagrange method: Dynamics based on joint angles
Newton-Euler method: Dynamics based on spatial vectors
Tanaka & Sejnowski (2013) J Neurophysiol

8. Dynamics based on Joint angles: Euler-Lagrange method.
   0
, ,
ft
xS x x L x dt 
     
   
0
0 0
0
0
, ,
, , ,
f
f f
f
f
t t
t
t t
t
t
L x dt L x dt
L
S x x S x x x x S x x
x
L
dt
x x
L d L L
dt
x dt
x x x
x
x
x
x x
x
  
 
 
 


 
  
    
    
        
   
 

 


0
d L L
dt x x
  
  
   0
0
ft t t
L L
x
x
x
x 
 
 
 
 
Principle of least action
Action:  ,L x xLagrangian:
Euler-Lagrange equation:

9. Dynamics based on Joint angles: Euler-Lagrange method.
 0 1,2
ii
d L L
i
dt  
  
   
  
     
2
2 2
2 2
1 1 1
1 1
,
2 2
i
i i i i i i j
i i j
L m X Y I  
  
 
       
 
  
1 1 1
1 1 1
cos
sin
X r
Y r





 
 
2 1 1 2 1 2
2 1 1 2 1 2
cos cos
sin sin
X l r
Y l r
  
  
   

  
       
      
 
2 2
1 1 1 1 1
2 2
1 1 2 1 2 1 1 2 1 2
2
2
1
2
21 21
1
, cos sin
2
1
cos cos sin sin
2
1 1
2 2
i i
d d
L m r r
dt dt
d d
m l r l r
dt dt
I I
 
     
  
 
 
                  
    
       
     
  

10. Dynamics based on Cartesian vectors: Newton-Euler method.
 
1 ,0
e
1 , 1 ,0 , 1 1
( 1, , )
i i i i
i i i i i i i i i i i i i i
F F m A
i n
m X A X F    

   
  

        I I
 
2 2 20
2 2 21 20 2 2 2 2 2
F m A
m X A   
 

     I I
 
1 1 1 10
e
1 2 1 10 10 10 2 1 1 1 1 1
F F m A
m X A X F    
  

        I I
Otten (2003) Phil Trans R Soc Lond B

11. Dynamics based on Cartesian vectors: Newton-Euler method.
10 10 10 10 10 10
1 1 10 10 1 12 2 2
1 1 1
21 21 21 21 21 21
2 20 20 2 22 2 2
2 2 2
X A X V X V
m X A
r r r
X A X V X V
m X A
r r r

     
       
   
     
       
   
I I
I I
21 21 21 21 21 21
2 2 21 20 2 22 2 2
2 2 2
X A X V X V
m X A
r r r

     
       
   
I I
Tanaka & Sejnowski (2013) J Neurophysiol
Equations of motion in general 3D movements:

12. Dynamics based on Cartesian vectors: Newton-Euler method.
10 10 21 21
1 1 10 10 1 2 20 20 22 2
1 2 Z
X A X A
m X A I m X A I
r r

  
      
 
21 21
2 2 21 20 2 2
2 Z
X A
m X A I
r

 
   
 
Tanaka & Sejnowski (2013) J Neurophysiol
Equations of motion in 2D planar movements:

13. Coordinate transformation problem in the brain.
Kalaska & Crammond (1992) Science

14. Neural pathway for visually guided reaching movements.
Haggard (2008) Nature Rev Neurosci

15. Equilibrium-point control: the brain may not solve the dynamics.
Feldman (2009) Encyclopedia of Neuroscience
initial
equilibrium
new
equilibrium
final
equilibrium

16. Joint stiffness indicates that the brain “solves” the arm dynamics.
Gomi & Kawato (1996) Science

17. Neurophysiology from posterior parietal cortex (area 7a).
Andersen et al. (1985) Science

18. Neurophysiology from posterior parietal cortex (area 7a).
Andersen et al. (1985) Science
Retinal position
Spikecounts
Position (head-centered, h)
Gazedirection(e)
   (response) g e f r 
e: gaze direction r: retinal position

19. Gain fields as an intermediate step for coordinate transformation?
total activity
background activity (due to eye position)
visually evoked activity Zipser & Andersen (1988) Nature

20. Multilayer neural network model of coordinate transformation.
Zipser & Andersen (1988) Nature

21. Hidden layer units exhibit gain field modulation.
Zipser & Andersen (1988) Nature
Experiment
Model

22. How gain fields work for coordinate transformation.
Kakei et al. (1999) Science; Kakei et al. (2003) Neurosci Res

23. Parietal and motor cortices exhibit distinct coordinate systems.
Haggard (2008) Nature Rev Neurosci
PRR: eye-centered reference frame (+
gain fields)
Buneo et al. (2002) Nature
PMd: relative position representation
Pesaran et al. (2006) Neuron
Area 5d: hand-centered reference
frame
Bremner & Andersen (2012) Neuron

24. Human psychophysics for coordinate transformation: pointing.
Soechting & Flanders (1989) J Neurophysiol
Intrinsic coordinates
η: yaw of upper arm
θ: elevation of upper arm
α: yaw of forearm
β: elevation of forearm
Extrinsic coordinates
(x, y, z): three-dim position
𝑅2
= 𝑥2
+ 𝑦2
+ 𝑧2
tan χ =
𝑥
𝑦
tan 𝜓 =
𝑧
𝑥2 + 𝑦2

25. Extrinsic-intrinsic transformation is linearly approximated.
Soechting & Flanders (1989) J Neurophysiol
Reaching toward remembered target in dark (inaccurate)
Reaching toward visible target in dark (accurate)

26. Human psychophysics: Motion planning in visual space.
Flanagan & Rao (1995) J Neurophysiol

27. Summary
• Movements (kinematics) and equations of motion
(dynamics) are described either in external coordinates
(i.e., external position) or in internal coordinates (i.e.,
joint coordinates).
• Mechanisms of coordinate transformation have been
examined in human psychophysics.
• Movement planning appears to be processed in visual
coordinates.
• Gain fields are the neural mechanisms for coordinate
transformations.

28. References
• Atkeson, C. G. (1989). Learning arm kinematics and dynamics. Annual Review of Neuroscience, 12(1), 157-183.
• Hollerbach, J. M., & Flash, T. (1982). Dynamic interactions between limb segments during planar arm movement. Biological
cybernetics, 44(1), 67-77.
• Morasso, P., Casadio, M., Mohan, V., Rea, F., & Zenzeri, J. (2015). Revisiting the body-schema concept in the context of
whole-body postural-focal dynamics. Frontiers in Human Neuroscience, 9.
• Tanaka, H., & Sejnowski, T. J. (2013). Computing reaching dynamics in motor cortex with Cartesian spatial coordinates.
Journal of neurophysiology, 109(4), 1182-1201.
• Andersen, R. A., Essick, G. K., & Siegel, R. M. (1985). Encoding of spatial location by posterior parietal neurons. Science,
230(4724), 456-458.
• Zipser, D., & Andersen, R. A. (1988). A back-propagation programmed network that simulates response properties of a
subset of posterior parietal neurons. Nature, 331(6158), 679-684.
• Chang, S. W., Papadimitriou, C., & Snyder, L. H. (2009). Using a compound gain field to compute a reach plan. Neuron,
64(5), 744-755.
• Soechting, J. F., & Flanders, M. (1989). Sensorimotor representations for pointing to targets in three-dimensional space.
Journal of Neurophysiology, 62(2), 582-594.
• Flanagan, J. R., & Rao, A. K. (1995). Trajectory adaptation to a nonlinear visuomotor transformation: evidence of motion
planning in visually perceived space. Journal of Neurophysiology, 74(5), 2174-2178.
• Andersen, R. A., Snyder, L. H., Bradley, D. C., & Xing, J. (1997). Multimodal representation of space in the posterior parietal
cortex and its use in planning movements. Annual Review of Neuroscience, 20(1), 303-330.
• Batista, A. P., Buneo, C. A., Snyder, L. H., & Andersen, R. A. (1999). Reach plans in eye-centered coordinates. Science,
285(5425), 257-260.
• Graziano, M. S., Yap, G. S., & Gross, C. G. (1994). Coding of visual space by premotor neurons. SCIENCE-NEW YORK THEN
WASHINGTON-, 1054-1054.
• Graziano, M. S., & Gross, C. G. (1998). Spatial maps for the control of movement. Current Opinion in Neurobiology, 8(2),
195-201.
• Buneo, C. A., Jarvis, M. R., Batista, A. P., & Andersen, R. A. (2002). Direct visuomotor transformations for reaching. Nature,
416(6881), 632-636.

29. Exercise
1. Derive the EOMs of two-link arm model using Joint
angle representation (i.e., Euler-Lagrange method).
2. Confirm that the EOMs derived in the Newton-Euler
method equals to the EOMs derived in the Euler-
Lagrange method.