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Motor learning through the combination of
primitives.
Mussa-Ivaldi & Bizzi
Phil.Trans. R. Soc. Lond. B 355:1755-1769.
Forward problems
a(t) = 1/m F(t) = G2(F(t))
v(t) = G1(v(t0), a(t))
x(t) = G0(x(t0), v(t))
q(t) = G(q(t0), (t))
x
v
a
F
time
Inverse problems
F(t) = G2-1(a(t))
G-1(q(t)) = (t)
x
v
a
F
time
A simple problem
D is the torque  of the previous slide
Forward and Inverse Kinematics
Kinematics
 The study of motion when only position and velocity
are considered.
 Forward Kinematics
Position is specified by setting value for each DOF
Hard to achieve world space constraints
Movement flow (relatively) easy to control
Inverse Kinematics
Specify world space constraints that one or more
parts of the skeleton must achieve
Solve for joint angles to achieve these
Good for meeting world space constraints, but
movement flow can be a problem
Most skeletons are highly redundant, so problem is
underconstrained
Solutions based on feed-back
Should be
feedback
Is
real world
Solutions based on feedforward
Not the real world,
but solving the inverse problem
x
v
a
F
time
Memory based computations
Use the real world in previous behaviour,
i.e. learn and remember
The cerebellum
The cerebellum ("little brain") has convolutions similar to those of cerebral
cortex, only the folds are much smaller. Like the cerebrum, the cerebellum has
an outer cortex, an inner white matter, and deep nuclei below the white matter.
The cerebellum
If we enlarge a single fold of
cerebellum, or a folium, we
can begin to see the
organization of cell types. The
outermost layer of the cortex
is called the molecular layer,
and is nearly cell-free. Instead
it is occupied mostly by axons
and dendrites. The layer
below that is a monolayer of
large cells called Purkinje
cells, central players in the
circuitry of the cerebellum.
Below the Purkinje cells is a
dense layer of tiny neurons
called granule cells. Finally, in
the center of each folium is
the white matter, all of the
axons traveling into and out of
the folia. These cell types are
hooked together in
stereotypical ways throughout
the cerebellum.
Equilibrium-point hypothesis
Polat & Bizzi, 1979
Isometric force fields
Bizzi et al. 1991
Fiber tracts of the spinal cord
Evidence for internal models
Forward model:
• The transformation from
a motor command to the
consequent behaviour
• Predict the expected
outcome of a command
• Estimate the current
state inthe presence of
feedback delays
Inverse model:
• The transformation of the
desired behaviour to the
corresponding motor
command
Cortical primitives
Brain
Spinal cord solving
the inverse problem
(t) = G-1(q(t)) = S ci i(q(t))
Linear interaction
Potential modules
Simulating a composite movement