“One should beware of m
a
them
aticians and all who m
a
ke
em
pty prophecies. T
he danger already exists that the
m
a
them
aticians have m
a
de a covenant with the devil to darken
the spirit and confine m
a
n in the bonds of Hell”
St Augustine, Bishop of Hippo, circa 400 A.D
“To m
ove things is all that Mankind can do…For such
the
sole exe
cu
t
a
n
t is m
u
scle
, whether
in
whisper
i
ng
a sy
llab
le o
r
in felling a f
orest”
Charles Sher
ington, 1924
O
p
t
i
m
i
za
t
i
o
n
Pr
i
n
ci
p
l
es
i
n
M
o
t
o
r Co
n
t
ro
l
1
Optimization Principles in Motor Control
Prof. Dava J. Newm
an
Joseph H. Saleh
Depa
rt
m
e
nt
of
Aer
o
nau
t
i
c
s
a
n
d
Ast
r
o
n
aut
i
c
s
Ma
s
s
a
ch
us
e
t
ts
I
n
s
titu
t
e of
T
e
ch
no
log
y
O
p
t
i
m
i
za
t
i
o
n
Pr
i
n
ci
p
l
es
i
n
M
o
t
o
r Co
n
t
ro
l
2
Outline
?
R
eview
of M
u
scle Cont
racti
o
n
–
F
rom AP generation
to con
t
raction of fib
e
rs
–
M
uscle proprio
ceptors (spindles
and Golgi tendo
ns)
–
A
fferent
and eff
e
rent axons
?
T
h
e
M
u
scle Si
m
u
lin
k m
o
d
e
l
?
R
eflecti
o
ns on Mode
ls
?
O
p
tim
ization
Pri
n
ci
p
l
es in
Motor
Con
t
rol
–
U
nderstanding the
fundamental question in
biom
echanics
–
A
re all motor b
e
havior op
timal
in some sense?
–
K
inem
atic
vers
u
s
d
y
n
a
m
i
c obj
ect
ive fun
c
tions
?
P
r
e
-
P
ro
gr
am
med
M
u
scle Resp
on
se Du
ring
D
o
wn
w
a
r
d Jum
p
s
O
p
t
i
m
i
za
t
i
o
n
Pr
i
n
ci
p
l
es
i
n
M
o
t
o
r Co
n
t
ro
l
3
Muscles: Effectors of the Motor System
?
The m
ajor out
p
ut
of the ela
b
orate i
n
form
ation proces
si
ng t
h
at ta
ke
s
place
in
o
u
r
brai
n
i
s
t
h
e g
e
n
e
rati
o
n
of
a con
t
ractile force i
n
our sk
eletal m
u
scles.
?
M
uscl
e
fasci
c
u
l
us
–
M
us
cle fib
e
r
?M
y
o
f
i
b
r
i
l
–
S
arcom
e
re
?
E
ach
m
u
scle f
i
b
e
r
is i
n
n
e
rv
at
ed
b
y
on
ly
on
e m
o
to
r neuro
n
, althou
gh
each
m
o
to
r
n
e
ur
on
i
n
n
e
rv
ates a
num
b
e
r
of
m
u
scl
e
f
i
b
e
r
s
?
T
he m
o
t
o
r
ne
u
r
o
n
an
d al
l
t
h
e
fi
ber
s
i
t
i
nne
r
v
at
es i
s
cal
l
e
d a
m
o
t
o
r
u
n
i
t
(t
he
sm
al
l
e
st
f
u
nct
i
ona
l
uni
t
c
ont
r
o
l
l
e
d
by
t
h
e m
o
t
o
r
sy
st
em
)
O
p
t
i
m
i
za
t
i
o
n
Pr
i
n
ci
p
l
es
i
n
M
o
t
o
r Co
n
t
ro
l
4
Muscles: Effectors of the Motor System
?
T
he num
ber of m
uscle fibers innervated by one m
otor
neuron is called the innervation ratio. The innervation ratio can vary between 10 and 2000
?
A
low innervation ratio indicates a g
r
eater cap
acity for
f
i
nely g
r
ad
ing the m
u
scle to
tal f
o
rc
e
O
p
t
i
m
i
za
t
i
o
n
Pr
i
n
ci
p
l
es
i
n
M
o
t
o
r Co
n
t
ro
l
5
Muscles: Effectors of the Motor System
A sim
plified sequence from
AP gene
ration to m
uscular contraction
?
M
otor neuron f
i
r
e
s an
action po
tential
?
I
t propag
a
tes
do
wn the m
o
tor
ax
on unti
l
i
t
r
each
e
s
the n
e
uro-m
u
s
c
ular
junc
tion
?
I
t tr
iggers an
AP in
the
m
u
scle fi
ber
?
T
his AP is prop
agated
rapid
l
y
ov
er th
e surface of
the f
i
ber
and
con
ducted
into
the
my
ofi
b
ri
l by
mea
n
of
t
h
e T
-
t
ubule
sy
st
e
m
?
T
his in
turn
rel
e
ases Ca
++
from the Sarcop
lasmic Reticulum (SR)
-the SR serv
es as a
store of C
a
++
?
T
his in
turn
trigg
e
rs the
c
y
c
lic
m
o
tion of
M
y
osin
h
eads,
att
aching
a
nd det
aching
on
the Ac
tin
f
ilam
e
nts, thus
form
ing cross-bridges an
d gener
a
ting
the
pulling
force
?
C
a
++
are pumped back
to
the SR
O
p
t
i
m
i
za
t
i
o
n
Pr
i
n
ci
p
l
es
i
n
M
o
t
o
r Co
n
t
ro
l
6
Muscles: Effectors of the Motor System
?
T
he
fo
rce
o
f
c
o
nt
ract
i
o
n
de
pe
nd
s
on
t
h
e l
e
n
g
t
h
o
f
t
h
e m
u
scl
e
(l
e
n
gt
h-
t
e
n
s
i
o
n
relati
on
sh
i
p
)
?
T
he
fo
rce
o
f
c
o
nt
ract
i
o
n
al
s
o
depe
n
d
s
o
n
t
h
e
rel
a
t
i
v
e
rat
e
s
o
f
m
ovem
e
nt
o
f
t
h
e
Actin and
My
osin filam
e
n
t
s
(t
en
si
on
-v
el
o
c
it
y relati
on
sh
i
p
,
Hill’s curv
e)
?
M
ot
or
u
n
i
t
s a
r
e rec
r
ui
t
e
d i
n
a
fi
xe
d
or
de
r
f
r
o
m
t
h
e weake
s
t
t
o
t
h
e
st
ro
n
g
est
(Hen
n
e
m
a
n
size pri
n
ci
p
le): Th
e weak
est i
n
pu
ts recru
it
th
e slo
w
un
its wh
ich
gene
rate
t
h
e s
m
allest f
o
rc
e a
n
d
a
r
e m
o
st
re
s
i
stan
t t
o
fati
g
u
e
. The
f
a
st
fat
i
gue
-
resi
st
a
n
t
a
r
e re
cr
ui
t
e
d next
, f
o
l
l
o
we
d by
t
h
e f
a
st
fat
i
ga
bl
e un
i
t
s
w
h
i
c
h ge
ner
a
t
e
the st
rongest
force.
O
p
t
i
m
i
za
t
i
o
n
Pr
i
n
ci
p
l
es
i
n
M
o
t
o
r Co
n
t
ro
l
7
Muscles: Effectors of the Motor System
?
M
uscl
e
P
r
op
ri
ocep
t
o
rs
(s
pi
nd
l
e
s a
n
d
G
o
l
g
i
t
e
nd
o
n
s
)
There
are
differ
e
nt t
y
p
e
s of r
ecep
tors which r
e
spond to
light
, soun
d, odor
, he
at
,
touch,
pain
,
etc
.
The r
ecep
tors
which
lead
to
cons
cious
s
e
ns
at
ions
are
cal
led
exteroc
e
p
t
ors
,
those which
are n
ot responsible fo
r conscious sens
ation
are called-
prim
ar
y in m
o
to
r funct
i
ons-
a
re
c
a
lled
p
r
op
rio
c
e
p
tors
–
S
pindle org
a
ns
Those a
r
e s
t
ret
c
h
r
ecepto
rs s
cat
te
r
e
d
deep within
al
l
m
u
scles
.
The
y
a
r
e usu
a
ll
y
at
tach
ed
in parallel with a
m
u
scle f
i
ber
,
and ther
efore
experience the sa
m
e
r
e
lative length change.
Spindles give
inform
ation about its
length and r
a
te of ch
ange of
its
length
–
G
olgi tendon
They
are found
ver
y
close to
th
e junction b
e
tween tendon
and
muscle f
i
bers.
They
are p
l
aced
in series with
the muscle fib
e
rs an
d respond to
th
e
tendon str
e
tch
which a
ccom
p
an
ies a
m
u
scle
tens
ion.
Thus th
e
y
ar
e
for
ce t
r
an
s
d
ucers
for
the
musc
l
e
.
O
p
t
i
m
i
za
t
i
o
n
Pr
i
n
ci
p
l
es
i
n
M
o
t
o
r Co
n
t
ro
l
8
Muscles: Effectors of the Motor System
?
The nerve
axons whic
h run out of
t
h
e spi
n
al
cord
a
r
e
call
ed
efferent
,
the
ones
that
c
arry inform
ation t
o
t
h
e c
o
rd a
r
e
afferent
?
G
r
o
u
p
I
a
f
fe
re
nt
fi
be
rs ha
ve
l
a
r
g
e di
am
et
ers t
h
e
r
e
f
o
r
e rel
a
t
i
vel
y
hi
g
h
co
ndu
cti
o
n v
e
l
o
cities. Th
ey
bri
n
g inform
ation
fro
m
th
e sp
i
n
d
le
(Ia)
a
n
d
th
e
g
o
l
g
i
(Ib
) to
th
e
co
r
d
?
T
he e
f
fere
nt
whic
h i
nne
rvate
the m
a
in m
u
scl
e
m
a
ss are
the
α
, and
those t
h
at
se
rve the
intra
f
usa
l
fi
be
rs
wit
h
i
n
t
h
e s
p
i
n
dles
are
called
γ
?
T
he s
t
re
tch
re
fl
ex, c
o
-acti
v
ati
o
n
o
f
α
-m
n a
n
d
γ
-m
n
O
p
t
i
m
i
za
t
i
o
n
Pr
i
n
ci
p
l
es
i
n
M
o
t
o
r Co
n
t
ro
l
9
Muscles: Effectors of the Motor System
St
ret
c
h reflex st
i
ffness
?
U
ntil
recent
l
y,
it was
s
u
ppose
d t
h
at
the
tendon orga
n se
rve
d
as
a
sensor
wh
ich
t
u
rn
ed
off m
u
scle acti
vity (i
nh
ibited
α
-m
n) whe
n
m
u
scle fo
rce
r
o
se
b
e
yond
saf
e
lev
e
l
s
?
A
ff
er
en
t
acti
v
it
y fr
o
m
bo
t
h
sp
i
n
dles and
Go
l
gi tendo
ns
b
a
lance in su
ch
a way th
at
n
e
it
h
e
r m
u
scle
force nor m
u
scle
l
e
n
g
th sh
ou
l
d
be
consi
d
ered as c
o
ntrolle
d
quant
ity, rathe
r
the
i
r rat
i
o (t
he
sti
ffnes
s
or
chan
ge
i
n
f
o
rc
e pe
r c
h
a
nge
i
n
l
e
n
g
t
h
) a
ppea
r
s t
o
be
fi
xed
b
y
t
h
e
s
t
r
et
c
h
re
fle
x
O
p
t
i
m
i
za
t
i
o
n
Pr
i
n
ci
p
l
es
i
n
M
o
t
o
r Co
n
t
ro
l
1
0
Muscles: Effectors of the Motor System
?
Th
e s
e
n
s
orimot
or co
rtex
is at th
e top
of
the
chain of command in
the sensorimotor
area
of th
e
cereb
ral
cortex
.
There
is
a
s
p
ec
ial
ized
area
in
the
cer
eb
ral
cortex
devot
e
d
to movement of
the
limbs (1691,
the
case of
a knight with
a fractu
red skull and
paraly
sis of
the left sid
e
of
the bo
d
y
)
–
T
he fraction of the cer
e
b
r
al co
r
t
ex co
ntr
o
lling each par
t
of the
body
is
by
n
o
m
e
ans
pr
opor
tional to
the size of that par
t
–
I
f the cerebral
cortex is
r
e
m
oved, the
ani
m
al continues to display all
the
lo
co
m
o
tion
r
e
flexes, but cannot lear
n new skills
?
Basal ganglia
ar
e a
s
e
t
of s
p
ec
ial
ized
nerve
ce
lls
i
n
the
brain
s
tem
.
?
Cereb
ellu
m
is a major focus
of incoming sensor
y
information
.
Th
e information
reaching
th
e c
e
r
e
bellum
h
a
s
to
d
o
with
length
,
fo
rce,
velo
cit
y
of
m
u
s
c
les
and
position of
join
ts.
O
p
t
i
m
i
za
t
i
o
n
Pr
i
n
ci
p
l
es
i
n
M
o
t
o
r Co
n
t
ro
l
1
1
On Models and Ot
her
Demons
What do you think of the following quotes? ?
“
I
f
a
ki
nem
a
t
i
c
o
b
j
ect
i
v
e
f
unc
t
i
o
n ca
n
be
f
o
u
n
d
t
h
at
l
e
a
d
s
t
o
o
p
t
i
m
al
tra
j
ect
orie
s t
h
a
t
accurate
ly re
produce t
h
e
pa
tterns
of
obse
rved be
ha
vi
or, i
t
im
p
lies t
h
at t
h
e brai
n
i
g
n
o
res no
n-k
i
n
e
m
a
tic fact
ors in selecting
an
d
re
pr
o
duc
i
n
g t
h
at
be
ha
vi
o
r”
?
“If a
dynam
i
c
objecti
v
e
funct
i
on ca
n
be
found t
h
at lea
d
s
t
o
optim
al tra
j
ect
ories
th
at accurately
rep
r
od
u
c
e t
h
e
p
a
ttern
s
of
o
b
serv
ed b
e
h
a
v
ior, it
im
p
lies t
h
at
th
e
br
ai
n
co
ns
i
d
e
r
s
dy
nam
i
c
fact
o
r
s
i
n
sel
e
c
t
i
n
g
and
rep
r
od
uci
n
g t
h
at
be
hav
i
o
r
”
O
p
t
i
m
i
za
t
i
o
n
Pr
i
n
ci
p
l
es
i
n
M
o
t
o
r Co
n
t
ro
l
1
2
Optimization Principles in Motor Control
Fundam
e
ntal question in biom
echanics
?
T
he
hum
an l
i
m
b
s a
r
e
i
n
v
o
l
v
ed
i
n
a
pr
o
d
i
g
i
o
u
s
va
ri
e
t
y
o
f
t
a
s
k
s.
M
ovem
e
nt
s
tend
t
o
b
e
gr
acefu
l an
d usuall
y inv
o
l
v
e
m
a
ny lim
b
seg
m
ents
?
D
ifferent tasks typ
ically
requ
i
r
e
–
d
ifferent
sequen
c
ing of
m
u
scle
a
c
tiva
tion
and
lim
b m
o
tion
–
different inform
ation
from sensors
?
H
ow a
r
e
t
h
e
s
e
m
ovem
e
nt
s o
r
gan
i
zed
?
F
u
nd
am
ent
a
l
q
u
est
i
on
i
n
bi
om
ech
ani
c
s
:
Which m
u
scles
are
used and
i
n
what
patte
rn?
[Bernste
in,
The Co-
O
rdination and R
e
gulation of Mo
vement
.
Per
ga
m
on
Press,
1967]
O
p
t
i
m
i
za
t
i
o
n
Pr
i
n
ci
p
l
es
i
n
M
o
t
o
r Co
n
t
ro
l
1
3
Optimization Principles in Motor Control
?
O
n
e
wid
e
ly
used
m
a
th
em
atical too
l
is
op
tim
ization
t
h
eory
Objecti
v
e:
to discove
r
princ
i
ples t
h
at
gui
d
e
goal
-direc
ted mot
o
r be
ha
vi
or
?
F
ou
r co
m
p
o
n
e
n
ts to an op
tim
ization
prob
lem
:
1.
An objective fun
c
tion
that quan
t
ifies what is
to be regard
ed as
optimum (also called
performance fun
c
tion
or cost fun
c
tion)
2.
A d
y
n
a
m
i
c s
y
s
t
e
m
that
is to
be
c
ontrolled
3.
A set of
contro
ls that ar
e
available for modulation
4.
An algorithm capable of find
ing
an an
aly
t
ical or
numerical solution (tools of
variational calcu
l
us)
?
G
ive
n
a m
o
del
of m
u
sculo-s
k
e
l
etal
dynam
i
cs, optim
ization t
h
eory
re
-m
aps
Ber
n
stei
n
’
s
pr
ob
lem
of
ch
oo
si
n
g am
o
ng
an
i
n
fin
i
ty of
po
ssi
b
le
p
a
ttern
s of
m
u
scl
e
act
i
v
at
i
o
n
i
n
t
o
an
eq
ui
val
e
nt
p
r
o
b
l
e
m
o
f
c
h
oo
si
n
g
a
m
ong a
n
i
n
fi
ni
t
y
o
f
pe
rf
o
r
m
a
nce c
r
iteria
O
p
t
i
m
i
za
t
i
o
n
Pr
i
n
ci
p
l
es
i
n
M
o
t
o
r Co
n
t
ro
l
1
4
Optimization Principles in Motor Control
?
O
pt
i
m
i
zat
i
o
n
-
b
a
sed
m
odel
s
ha
ve
been
d
e
vel
o
ped
t
o
a
d
d
r
e
ss
t
h
e “e
xces
s
deg
r
ee
s of
free
dom
”
p
r
ob
l
e
m
?
R
ecall Bernstei
n
que
sti
o
n: How does
the m
o
t
o
r syst
em
select the
b
e
h
a
v
i
or it
uses
fro
m
th
e
infi
n
ite
nu
m
b
er
of
p
o
ssi
b
i
l
ities open
t
o
i
t
?
–
I
n m
a
them
ati
cal
parlanc
e
,
this
is
an i
ll-posed prob
lem
in
the
sense
that
m
a
n
y
solutions ar
e pos
sible
–
F
or example, most limb segments
are moved b
y
a
larger
number of
muscles
than
appear
to
b
e
ne
ces
s
a
r
y
–
T
o reach
a
cup o
f coffee, the han
d may
move alo
ng an
infinity
of
paths
?
R
ephrasi
n
g t
h
e
cent
r
al
ques
tion: How does
the m
o
tor sys
t
em chooses
val
u
es
f
o
r t
h
e l
a
r
g
e
num
be
r
of
pa
ram
e
t
e
rs
t
h
a
t
can
be
co
nt
r
o
l
l
e
d
i
n
o
r
de
r
t
o
pe
rf
o
rm
a g
o
al
-o
ri
e
n
t
e
d m
ovem
e
nt
?
O
p
t
i
m
i
za
t
i
o
n
Pr
i
n
ci
p
l
es
i
n
M
o
t
o
r Co
n
t
ro
l
1
5
Optimization Principles in Motor Control
?
N
eed
t
o
m
a
k
e
ex
plicit and
q
u
an
titat
i
v
e
h
y
p
o
th
eses ab
ou
t
the go
al
of
m
o
tor acti
o
ns
?
A
re all m
o
tor beh
a
vior
n
ecessarily
op
tim
al in
so
m
e
sense
?
Mayb
e!
?
O
ne a
p
peal
i
n
g
po
ssi
bi
l
i
t
y
i
s
t
h
at
t
h
e
ne
r
v
ou
s
sy
st
em
has e
v
o
l
ve
d
t
o
select “
s
olut
ions”
tha
t
a
r
e i
ndeed “
o
ptim
al”: the
hypot
h
ese
s
is t
h
at
in
pe
rf
o
r
m
i
ng a
m
o
t
o
r
t
a
s
k
,
t
h
e
C
N
S p
r
od
uce
s
co
o
r
di
nat
e
d ac
t
i
o
ns
t
h
at
m
i
ni
m
i
ze som
e
m
easure
o
f
pe
r
f
o
r
m
a
nce
(e
ff
or
t
,
sm
oot
hne
s
s
, et
c
.)
O
p
t
i
m
i
za
t
i
o
n
Pr
i
n
ci
p
l
es
i
n
M
o
t
o
r Co
n
t
ro
l
1
6
Optimization Principles in Motor Control
?
K
i
n
em
at
i
c
s ve
r
s
u
s
dy
nam
i
cs
o
b
j
ect
i
v
e f
unc
t
i
on
s
?
–
K
inem
atics r
e
fer
s
to th
e
tim
e cou
r
se of an
obj
ect
(
position,
velo
cit
y
,
acce
lera
tion
,
e
tc
.
)
–
D
y
n
amics r
e
fers
to v
a
riab
les such as forces and
torques
?
E
ven
si
n
g
l
e
de
gr
ee
of
f
r
ee
d
o
m
can be
pe
r
f
o
r
m
e
d i
n
a
va
ri
e
t
y
o
f
way
s
:
–
P
ath is
constrain
t
–
S
peed along th
e
path
can v
a
r
y
(
t
r
a
jector
y
)
?
T
wo di
f
f
ere
n
t
t
y
pes o
f
ob
je
ct
i
v
e f
unct
i
o
n
s
ha
ve been
p
r
o
p
o
s
ed, t
h
ey
re
flect the
tw
o m
a
jo
r
c
o
m
p
eting
t
h
eo
rie
s
of
m
o
tor co
nt
ro
l:
Kinematic objective function
D
y
namic objective function
O
p
t
i
m
i
za
t
i
o
n
Pr
i
n
ci
p
l
es
i
n
M
o
t
o
r Co
n
t
ro
l
1
7
Optimization Principles in Motor Control
Kinem
a
tic objective function, single-joint m
ovem
e
nts
?
T
hey
ar
e c
h
ara
c
t
e
ri
ze
d
by
s
i
n
g
l
e
-pea
ke
d,
bel
l
-s
ha
ped
s
p
ee
d
p
r
o
f
i
l
es.
It
was
po
st
ul
at
e
d
(H
o
g
an
,
19
8
4
)
t
h
at
v
o
l
u
nt
a
r
y
m
ovem
e
nt
s a
r
e
m
a
de t
o
be a
s
sm
oo
t
h
a
s
p
o
ssi
bl
e
?
A
qu
an
titati
v
e
m
easu
r
e of
sm
o
o
thn
e
ss
i
s
n
e
ed
ed, o
n
e
su
ch
m
easu
r
e
i
s
the squa
re
d
m
a
gnitude of
t
h
e j
e
rk (ra
te of
c
h
a
nge
of
accele
r
a
tion or
th
i
r
d tim
e d
e
ri
v
a
ti
v
e
of
p
o
siti
o
n
) O
p
t
i
m
i
za
t
i
o
n
Pr
i
n
ci
p
l
es
i
n
M
o
t
o
r Co
n
t
ro
l
1
8
Optimization Principles in Motor Control
2
t
1
?
d
3
θ
?
J
=
t
∫
0
? ??
dt
3
? ??
dt
θ
()
is
t
h
e jo
i
n
t
angle.
Using
v
a
riat
ion
a
l
calcu
l
u
s, th
e un
iqu
e
tim
e
h
i
story
t
of
joint
positi
ons
that
m
i
nim
i
zes t
h
is
pe
rform
ance m
easure m
ay be
de
ri
ve
d a
n
al
y
t
i
cal
l
y
θ
()
t
=
c
0
+
c
1
t
+
c
2
t
2
+
c
3
t
3
+
c
4
t
4
+
c
5
t
5
c
i
are uns
p
eci
fi
ed
c
o
e
f
ficients whose val
u
es
a
r
e dete
rm
ined by
t
h
e
co
nd
it
ion
s
at t
h
e
b
e
ginn
i
n
g an
d end
of
m
o
vem
e
n
ts (bo
undar
y
co
nd
it
ion
s
)
O
p
t
i
m
i
za
t
i
o
n
Pr
i
n
ci
p
l
es
i
n
M
o
t
o
r Co
n
t
ro
l
1
9
Optimization Principles in Motor Control
?
W
he
n
t
h
e
m
o
v
e
m
e
nt
i
s
as
s
u
m
e
d t
o
be
gi
n
at
r
e
st
i
n
on
e
po
si
t
i
o
n
a
n
d
en
d at
rest
i
n
a
not
he
r,
t
h
e
“m
i
n
i
m
um
jer
k
”
o
r
“m
axi
m
u
m
sm
oot
h
n
e
ss” m
ovem
e
nt
t
u
rn
s ou
t
t
o
ha
ve
t
h
e sm
oot
h, un
i
-m
o
d
a
l
, bel
l
-s
ha
ped
ve
l
o
ci
t
y
pr
o
fi
l
e
t
y
pi
ca
l
m
o
st expe
rim
e
ntal
obse
rvati
o
ns
?
T
he m
a
xi
m
u
m
sm
oot
h
n
e
ss
hy
po
t
h
e
s
i
s
i
s
rea
d
i
l
y
gene
ral
i
ze
d t
o
m
u
l
t
i
-
j
oi
nt
m
o
tio
ns.
O
p
t
i
m
i
za
t
i
o
n
Pr
i
n
ci
p
l
es
i
n
M
o
t
o
r Co
n
t
ro
l
2
0
Optimization Principles in Motor Control
Kinem
a
tic
o
bjective function, m
ulti-joint m
ovements
?
T
he ob
j
ect
i
v
e f
unc
t
i
o
n
ca
n be
wri
t
t
en
as f
o
l
l
o
w
s
i
n
t
h
e
C
a
rt
e
s
i
a
n
coo
r
di
nat
e
f
r
a
m
e of t
h
e
ha
nd
:
J
=
∫
t
1
? ?
?? ?
d
3
x
? ?
2
+
?? ?
d
3
y
? ?
2
? ?
×
dt
t
0
? ?
?
dt
3
??
?
dt
3
??
??
?
A
ss
um
i
ng t
h
e
m
ovem
e
nt
st
a
r
t
an
d
e
n
d
at
zer
o vel
o
ci
t
y
f
r
o
m
(x
0
, y
0
) to
(x
f
, y
f
) at
tim
e t
f
(
τ
= t/t
f
)
x
(
τ
)
=
x
0
+
(
x
0
?
x
f
)
(
15
τ
4
?
6
τ
5
?
10
τ
3
)
y
(
τ
)
=
y
0
+
(
y
0
?
y
f
)
(
15
τ
4
?
6
τ
5
?
10
τ
3
)
O
p
t
i
m
i
za
t
i
o
n
Pr
i
n
ci
p
l
es
i
n
M
o
t
o
r Co
n
t
ro
l
2
1
Optimization Principles in Motor Control
?
T
he m
a
xim
u
m
s
m
oothness theory yields in the m
ulti-joint
m
ove
m
e
nt several explicit predictions:
1.
Tra
j
ect
ories
of
the l
i
m
b
s a
r
e s
t
rai
g
ht li
ne
pat
h
s
2.
The
t
a
n
g
ent
i
al
vel
o
ci
t
y
al
on
g t
h
at
pat
h
i
s
sm
oo
t
h
an
d u
n
i
-
m
oda
l
3.
The s
h
a
p
e
of t
h
e lim
b t
r
a
j
ec
t
o
ries a
r
e
inva
ri
ant
unde
r
tra
n
s
l
ation,
rotati
o
n
, an
d am
p
litu
d
e
scal
ing
?
T
hese predictions are in agreem
ent with experim
e
ntal
observations
O
p
t
i
m
i
za
t
i
o
n
Pr
i
n
ci
p
l
es
i
n
M
o
t
o
r Co
n
t
ro
l
2
2
Optimization Principles in Motor Control
Lim
itations of the ki
nem
a
tic objecti
v
e functions
?
A
trou
bling
asp
ect
of t
h
is t
h
eo
ry
is
th
at
it
imp
lies t
h
at at
high
er lev
e
ls i
n
the m
o
t
o
r syste
m
, the brai
n
does not
ta
ke i
n
t
o
account a
n
y dynam
ic
consi
d
erati
o
ns
suc
h
a
s e
n
ergy
re
qui
re
d, the
l
o
ads
on t
h
e
lim
b segm
ent
s
or
th
e
fo
rce an
d fatigu
e
lim
itati
on
s
of t
h
e
n
e
urom
u
s
cu
lar system
?
I
n oth
e
r
word
s, it im
p
lies t
h
at
th
e
brai
n
d
e
term
in
es th
e “op
ti
m
al”
tra
j
ect
ory i
nde
pende
n
tly
of t
h
e physical
syst
em
that will
ge
ne
rate t
h
e
m
ovem
e
nt
, i
.e.
, t
h
e l
i
m
b
!
“It seems ver
y
strange
that the op
timal
trajector
y
of our movement is
determined
perfectly
indep
e
ndent of th
e d
y
n
a
mic
quantities such
as arm
length
,
pay
l
oad
,
motor command, torqu
e
or
exter
n
al for
ce, etc.”
Y.
Uno
a
nd M
.
Kawato,
1989
O
p
t
i
m
i
za
t
i
o
n
Pr
i
n
ci
p
l
es
i
n
M
o
t
o
r Co
n
t
ro
l
2
3
Optimization Principles in Motor Control
Lim
itations of the
k
inem
atic
objective functions (suite)
?
T
he t
r
a
j
ect
o
r
i
e
s de
ri
ve
d
f
o
r
t
h
e m
i
ni
m
u
m
jer
k
m
odel
a
r
e i
n
va
ri
an
t
wi
t
h
re
spect
t
o
t
h
e
re
gi
on
o
f
t
h
e
wo
r
k
-s
pac
e
an
d i
n
depe
n
d
e
nt
o
f e
x
t
e
rna
l
fo
rce
s
?
T
he m
i
ni
m
u
m
je
r
k
m
odel
de
t
e
rm
i
n
es t
r
a
j
ect
or
i
e
s
i
r
re
spec
t
i
ve
o
f
gra
v
i
t
y
?
T
o
ci
rcu
m
v
e
n
t
th
is prob
lem
with
i
n
t
h
e
fram
ework of
op
tim
ization
t
h
eo
ry,
a
seco
n
d
t
y
pe
o
f
ob
j
ect
i
v
e fu
nct
i
o
n
s
wa
s f
o
rm
ul
at
e
d
base
d o
n
dy
nam
i
c
va
ri
abl
e
s
(joint t
o
rques
,
m
u
scle forces
,
etc.)
O
p
t
i
m
i
za
t
i
o
n
Pr
i
n
ci
p
l
es
i
n
M
o
t
o
r Co
n
t
ro
l
2
4
Optimization Principles in Motor Control
Dyna
m
i
c objective function
?
M
odels using a dynam
i
c objective function in m
ovem
e
nts
assum
e tha
t
the CNS solves th
e th
re
e
f
o
llowing
com
putational problem
s at different levels:
1.
Det
e
rm
i
n
at
i
o
n of
a desi
re
t
r
a
j
ect
o
r
y
2.
Tran
s
f
o
r
m
a
t
i
o
n
of
v
i
s
u
al
c
o
o
r
d
i
na
t
e
s
o
f
t
h
e
desi
re
d t
r
a
j
ec
t
o
ry
t
o
bo
dy
c
o
o
r
di
nat
e
3.
Gene
rat
i
on
o
f
m
o
t
o
r
com
m
a
nds
(
f
o
r
ces a
n
d t
o
r
q
ues
)
t
o
rea
l
i
ze t
h
e
desi
re
d t
r
a
j
ec
t
o
ry
O
p
t
i
m
i
za
t
i
o
n
Pr
i
n
ci
p
l
es
i
n
M
o
t
o
r Co
n
t
ro
l
2
5
Optimization Principles in Motor Control
Dyna
m
i
c objective function, m
ulti-joint m
ovem
e
nts
?
O
ne dy
nam
i
c
o
b
j
ect
i
v
e f
unc
t
i
on
p
r
op
o
s
ed
i
s
t
h
e f
o
l
l
owi
n
g
:
t
f
n
2
J
=
∫
∑
? ?
dz
i
??
dt
t
0
i
?
dt
?
?z
i
is t
h
e m
o
t
o
r
comm
and
fe
d t
o
t
h
e i
-
t
h
ac
tua
t
or
(m
uscle
)
out of
n ac
tuat
ors
?
I
n ord
e
r t
o
com
p
u
te op
tim
al
traj
ect
ories
p
r
ed
icted
b
y
t
h
is
m
in
im
u
m
to
rqu
e
chan
ge m
o
del
,
t
h
e
dy
nam
i
cs e
qua
t
i
o
ns
o
f
t
h
e
m
u
scul
o
-
s
k
el
e
t
al
sy
st
em
m
u
s
t
fi
rst
be
s
p
eci
fi
ed
beca
use
J
d
e
pen
d
s
on
t
h
e
dy
nam
i
cs
of
t
h
e co
nt
ro
l
l
e
d
o
b
j
ect
O
p
t
i
m
i
za
t
i
o
n
Pr
i
n
ci
p
l
es
i
n
M
o
t
o
r Co
n
t
ro
l
2
6
Optimization Principles in Motor Control
?
P
roblem
: it is difficult to describe the the m
usculo-skeletal
system
exactly because it is a com
plex system
.
?
C
onsider the
f
o
llowing two-join
t sys
t
em
:
O
p
t
i
m
i
za
t
i
o
n
Pr
i
n
ci
p
l
es
i
n
M
o
t
o
r Co
n
t
ro
l
2
7
Optimization Principles in Motor Control
x
y
L
1
1
θ
2
θ
S
1
D
D
D
D
z
1
=
(
I
1
+
I
2
+
2
M
2
L
1
S
1
cos(
θ
2
)
+
M
2
L
1 2
)
×
θ
1
+
b
1
θ
1
D
D
D
D
D
+
(
I
2
+
M
2
L
1
S
2
cos(
θ
2
)
)
×
θ
2
?
M
2
L
1
S
2
(
2
θ
1
+
θ
2
)
×
θ
2
sin(
θ
2
)
D
D
D
D
D
D
z
2
=
(
I
2
+
M
2
L
1
S
2
cos(
θ
2
)
)
×
θ
1
+
I
2
θ
2
+
b
2
θ
2
D
θ
1
+
M
2
L
1
S
2
()
2
sin(
θ
2
)
O
p
t
i
m
i
za
t
i
o
n
Pr
i
n
ci
p
l
es
i
n
M
o
t
o
r Co
n
t
ro
l
2
8
Optimization Principles in Motor Control
?
S
in
ce th
e d
y
n
a
m
ics o
f
th
e m
u
lti-jo
i
n
t
system
is
no
n
l
in
ear, t
h
e
prob
lem
of
fi
n
d
i
n
g t
h
e
un
i
q
ue t
r
a
j
ec
t
o
ry
t
h
at
m
i
ni
m
i
zes
J
is
a non
lin
e
ar
o
p
tim
izatio
n pro
b
lem
.
?
C
o
n
s
equ
e
ntly,
it seem
s im
p
o
ssib
le t
o
o
b
t
ain
an
alytical
expressi
o
n
of
th
e
so
l
u
ti
on
of
th
is prob
lem
,
un
li
k
e
t
h
e case
with th
e m
in
imu
m
j
e
rk
pr
o
bl
e
m
?
P
re
di
ct
i
o
ns v
s
expe
ri
m
e
nt
–
T
raje
ctor
y
depen
d
s
on arm
pos
tur
e
and
ext
e
rnal
fo
rces
–
N
ot alway
s
str
a
ight paths
?
The m
i
nim
u
m
torque c
h
a
nge
m
odel s
u
ccee
ded i
n
re
produci
ng
o
bser
v
ed
t
r
ajector
ies und
er v
a
r
i
o
us co
nd
it
ions
O
p
t
i
m
i
za
t
i
o
n
Pr
i
n
ci
p
l
es
i
n
M
o
t
o
r Co
n
t
ro
l
2
9
Optimization Principles in Motor Control
?
P
hysiolog
i
cal advan
t
ag
e of each m
odel: W
hy would the
CNS want to m
i
nim
i
ze
–
t
or
qu
e ch
an
g
e
?
–J
e
r
k
?
O
p
t
i
m
i
za
t
i
o
n
Pr
i
n
ci
p
l
es
i
n
M
o
t
o
r Co
n
t
ro
l
3
0
Pre-Programmed Muscle Respons
e
During Dow
n
ward Jumps
?
L
it
er
at
ure r
e
vi
ew
–
E
n
gb
e
r
g and
Lu
ndb
er
g (19
69)
E
M
G
activity dur
ing walking in cat
li
m
b
s
“E
M
G
was
triggered 5 to 10
m
s
p
r
io
r to i
m
p
act”,
sor
t
of feedfor
w
ar
d activation “a centr
a
lly
pr
ogr
a
m
m
e
d
event anticipating s
t
ance”
–
Melv
ill
-Jon
es an
d W
a
tt (1
971
)
T
e
sted the
above conclusion on hu
m
a
ns dur
ing sudden falls
.
Found
consistent EM
G bur
s
t
activity
beg
i
nning 75
m
s
af
ter
d
r
op.
Concluded that “deceler
ation
r
esulted f
r
o
m
a ti
m
e
d bur
s
t
of pr
e-
p
r
ogr
a
m
m
e
d
m
u
scle
activity
”. Pr
oble
m
with this
study
: d
r
opped subjects f
r
o
m
heights
up to 20cm
!
Activ
ity
tr
igger
e
d by ves
tibular
input?
–
Gree
n
w
o
o
d
a
n
d H
o
pki
ns
(1
9
7
6
)
Studied EM
G activity dur
ing voluntary
and unexpected ju
m
p
s, heights
up to 120cm
.
Findin
g
s:
T
w
o peaks of activ
ity
:
80
m
s
after release only
in unexpected
ju
m
p
s
+ consistent
ti
m
e
befo
r
e
landing (r
elated
to
the voluntar
y control of landing)
O
p
t
i
m
i
za
t
i
o
n
Pr
i
n
ci
p
l
es
i
n
M
o
t
o
r Co
n
t
ro
l
3
1
Pre-Programmed Muscle Respons
e
During Dow
n
ward Jumps
–
D
yh
r
e
-
P
ou
lsen and
Lau
r
sen
(1
980
,
19
83
, 198
5)
Analy
z
ed landing m
e
chanis
m
s
and EM
G activity
in
m
o
nkey
s
dur
ing downwar
d ju
m
p
s. Onset
of E
M
G activity
s
t
arted occu
rr
ed with gr
eat precision 8
0
m
s befo
r
e
landin
g
.
Still an ar
gu
m
e
n
t
against p
r
e-
p
r
ogr
amm
i
ng
: visual
m
o
nitoring of distance dur
ing ju
m
p
?
L
i
ghts tur
n
ed off,
s
a
m
e
act
ivat
ion patt
e
r
n,
locked to
the t
i
m
e
of
expected
i
m
p
a
c
t
–
M
cKin
ley and
Sm
ith
(19
83)
Per
f
or
m
e
d si
m
ilar
exper
i
m
e
nts on blindfolded and laby
rinthecto
m
ized cats
.
–
W
att
et
al
. (
198
6)
p
l
u
s
nu
m
e
ro
us o
t
h
e
r
st
ud
i
e
s
It is wid
e
ly
ackn
owledged
that microgravity
expo
sure causes prof
ound chang
e
s in
hum
an balan
ce,
posture con
t
rol
a
nd locom
o
tion
.
W
a
tt e
t
a
l
.
test
ed
astronau
t
s
subjected to sudden dr
ops.
All subjects ar
e “unsteady
po
stflight”.
Reaons for
decr
e
m
ent
in
per
f
or
m
a
n
ce?
Astronauts
sta
ted
t
h
e f
loor co
m
ing up
to
m
e
et
the
m
,
and
is
the
r
e
b
ef
o
r
e
the
y
we
re
r
ead
y f
o
r
it
O
p
t
i
m
i
za
t
i
o
n
Pr
i
n
ci
p
l
es
i
n
M
o
t
o
r Co
n
t
ro
l
3
2
?
?
?
Pre-Programmed Muscle Respons
e
During Dow
n
ward Jumps
?
T
he m
i
ssi
ng
l
i
nk
:
A
p
r
o
p
o
se
d m
odel
t
o
acc
ou
nt
f
o
r t
h
e a
b
ove
o
b
s
e
r
v
at
i
o
ns
–
T
he pr
evious ex
periments suggest that th
e “fly
in
g object” has
an
estimate of th
e
tim
e of
im
pact
T
impact
wh
y?
–
P
rior to
jum
p
, a
visual
estim
ate
o
f
the
heigh
t
is pe
rform
ed
H
0
–
W
ha
t
i
s nee
d
e
d
to go from e
sti
ma
t
e
d he
i
ght t
o
e
st
ima
t
e
d ti
me
of
impa
c
t
?
A representation
of th
e gr
avity
field
in th
e sensorimotor s
y
stem, or
an
intern
al g-model
T
impact
=
CNS
g
H
0
2
?
O
p
t
i
m
i
za
t
i
o
n
Pr
i
n
ci
p
l
es
i
n
M
o
t
o
r Co
n
t
ro
l
3
3
?
Pre-Programmed Muscle Respons
e
During Dow
n
ward Jumps
?
W
hen astronauts perform postflight jum
ps, one hypotheses
regarding the perform
a
nce decrem
ent (other than m
uscle atrophy)
is th
a
t
intern
al repr
esen
ta
tion of
the g
r
av
ity f
i
e
l
d is alte
red
:
g
CNS
<
g
true
?
T
impact
>
T
true
?
H
ence the floor is “there before [they are] ready for it”!
O
p
t
i
m
i
za
t
i
o
n
Pr
i
n
ci
p
l
es
i
n
M
o
t
o
r Co
n
t
ro
l
3
4