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Computational Motor Control Summer School
01: Kinematics, Dynamics, and Coordinate transformation.
Hirokazu Tanaka
School of Information Science
Japan Institute of Science and Technology
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
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”
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
1.1. Inverse kinematics from Cartesian positions to joint angles.
2 2 2 2
1 21
1 2
2 22
1 1 2
arccos2
sinarcta 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
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 11 2 20 20 21 21 1 10 10 10 102 2
2 1
22 2 21 20 21 212
2
Z
Z
I Im X A X A m X A X A
r r
Im 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
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
Ldt
x x
L d L Ldt
x dt
x x x
x
x
x
x xx
0d L L
dt x x
0
0
ft t t
L L
xx
xx
Principle of least action
Action: ,L x xLagrangian:
Euler-Lagrange equation:
Dynamics based on Joint angles: Euler-Lagrange method.
0 1,2ii
d L Li
dt
2
2 22 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
22
1
2
21 21
1, cos sin
2
1cos cos sin sin
2
1 1
2 2
i i
d dL m r r
dt dt
d dm l r l r
dt dt
I I
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 Ai 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
Dynamics based on Cartesian vectors: Newton-Euler method.
10 10 10 10 10 101 1 10 10 1 12 2 2
1 1 1
21 21 21 21 21 212 20 20 2 22 2 2
2 2 2
X A X V X Vm X A
r r r
X A X V X Vm X A
r r r
I I
I I
21 21 21 21 21 212 2 21 20 2 22 2 2
2 2 2
X A X V X Vm X A
r r r
I I
Tanaka & Sejnowski (2013) J Neurophysiol
Equations of motion in general 3D movements:
Dynamics based on Cartesian vectors: Newton-Euler method.
10 10 21 211 1 10 10 1 2 20 20 22 2
1 2 Z
X A X Am X A I m X A I
r r
21 212 2 21 20 2 2
2 Z
X Am X A I
r
Tanaka & Sejnowski (2013) J Neurophysiol
Equations of motion in 2D planar movements:
Equilibrium-point control: the brain may not solve the dynamics.
Feldman (2009) Encyclopedia of Neuroscience
initial equilibrium
new equilibrium
final equilibrium
Neurophysiology from posterior parietal cortex (area 7a).
Andersen et al. (1985) Science
Retinal position
Spik
e co
un
ts
Position (head-centered, h)G
aze
dir
ecti
on
(e)
(response) g e f r
e: gaze direction r: retinal position
Gain fields as an intermediate step for coordinate transformation?
total activity
background activity (due to eye position)
visually evoked activity Zipser & Andersen (1988) Nature
How gain fields work for coordinate transformation.
Kakei et al. (1999) Science; Kakei et al. (2003) Neurosci Res
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 representationPesaran et al. (2006) Neuron
Area 5d: hand-centered reference frameBremner & Andersen (2012) Neuron
Human psychophysics for coordinate transformation: pointing.
Soechting & Flanders (1989) J Neurophysiol
Intrinsic coordinates
η: yaw of upper arm θ: elevation of upper armα: yaw of forearmβ: elevation of forearm
Extrinsic coordinates
(x, y, z): three-dim position
𝑅2 = 𝑥2 + 𝑦2 + 𝑧2
tan χ =𝑥
𝑦
tan𝜓 =𝑧
𝑥2 + 𝑦2
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)
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.
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.