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

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

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:

Coordinate transformation problem in the brain.

Kalaska & Crammond (1992) Science

Neural pathway for visually guided reaching movements.

Haggard (2008) Nature Rev Neurosci

Equilibrium-point control: the brain may not solve the dynamics.

Feldman (2009) Encyclopedia of Neuroscience

initial equilibrium

new equilibrium

final equilibrium

Joint stiffness indicates that the brain “solves” the arm dynamics.

Gomi & Kawato (1996) Science

Neurophysiology from posterior parietal cortex (area 7a).

Andersen et al. (1985) Science

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

Multilayer neural network model of coordinate transformation.

Zipser & Andersen (1988) Nature

Hidden layer units exhibit gain field modulation.

Zipser & Andersen (1988) Nature

Experiment

Model

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)

Human psychophysics: Motion planning in visual space.

Flanagan & Rao (1995) J Neurophysiol

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.

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.