2016 CoSMo - CompNeuroscicompneurosci.com/wiki/images/8/83/Francisco_2016_CoSMo.pdf · Frame...
Transcript of 2016 CoSMo - CompNeuroscicompneurosci.com/wiki/images/8/83/Francisco_2016_CoSMo.pdf · Frame...
CoSMo 2016
Francisco Valero-Cuevas, PhD Brain-Body Dynamics Laboratory
Purpose of this talk
To be informative, but also provocative, about the role of mechanics and
computational science understanding the control neuromuscular systems.
Provide several examples of how mechanics has helped us understand neuroscience.
Frame computational neuromechanics as a Big Data problem.
Hot topics of the day
Muscle redundancyMuscle Synergies
Optimization and Optimal controlMotor learningMotor policies
Bayesian motor control
My scientific approach to these problems
Neurophysiology
Sensorimotor behavior
Musculoskeletal modeling
Theories of neural control
Nylon cordsdriving tendons
Cadaverichand
Physical modeling
Clinical and rehabilitation tools
Machine learning
x
x
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+
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Some long term questions How does the neuromechanical system (or how should a versatile machine) meet the necessary
conditions for dexterous function?
and/or
What specific contributions from passive (e.g., the body) and active (e.g., the brain) components
enable dexterous function?
For example, computational modeling of neuromuscular systems is still in its infancy
Robust Control
Adaptive control
Hierarchical control
Hybrid control
Model predictive control
Reinforcementlearning
(Policy gradients,Q-learning)
*Optimal Control(LQR, LQG, iLQR, iLQG)
*Stochastic Estimation(Kalman filter, extended Kalman filter)
*Supervised learning(Backpropagation ANNs,support vector machines)
*Unsupervised learning(PCA, self-organized ANNs)
*Bayesian estimation(Particle filters, path integral estimation,
Monte Carlo simulations)
**Classical control(Root locus design,
tracking)
Minimum variance unbiased estimation (MVUE)Best linear unbiased estimator (BLUE)
etc.
Machine learning Control theory
Estimation-Detection theoryValero-Cuevas et al., IEEE RBME 2009
The problems in motor neuroscience todayWe agree on physics, mechanics, physiology, and computational principles; but the consequences of our experimental and modeling choices are hard to reconcile.
As a result, we are united by our methods but fragmented into schools of thought that tend to talk and publish past each other.
Consider the longstanding debates around topics like internal models, equilibrium point control, optimal control, synergies, etc.
Many of these debates become a matter of opinion.
The fundamental challenge for organisms:Newton and Darwin are unforgiving
Mechanics describes theundeniable physical reality.
Evolutionary biology is the response to that reality. Organisms are a result of successful
brain-body co-evolution in that context.
We must move fromComputational Neuroscience
toComputational Neuromechanics
Where is one to begin?
From The Help by Kathryn Stockett, a novel about life in the ‘60s:…get an entry-level job... When you’re not making mimeographs ... look around, Don’t waste your time on the obvious things. Write about what disturbs you, particularly if it bothers no one else.
Some mechanics-based examples of things that have been disturbing some of us
Why do we have so many muscles?
Neuromechanical Concept #0
Control of force(underdetermined—redundancy)
vs.Control of motion
(overdetermined—no redundancy)
Simplest tendon mechanics
42 4 Tendon-Driven Limbs
r =sq
r
s
q q
Fig. 4.5 Schematic representation of the calculations of the moment arm, r(q). Given tendonexcursion as a function of joint angle s(q), the moment arm is its partial derivative with respect tothe joint angle r(q) = ds
dq , Eq. 4.9 [4]
Fig. 4.6 Measurement oftendon excursion for asimple tendon path. Notethat a negative (as per theright hand rule) rotation ofthe joint −δq induces apositive (rightward) tendonexcursion δs that lengthensthe musculotendon, and viceversa. See Sect. 4.6 for adefinition of this signcovention
linear scale
angular scale+ s
- q
Fig. 4.7 A planar 1 DOFlimb driven by 6 tendons
r3
r2
r1
m1
m3
m2
m4
m6
m5
Force control vs. movement control44 4 Tendon-Driven Limbs
τ = r(q)T fm (4.14)
and
τ =
⎛
⎜⎜⎜⎝
r(q)1r(q)2...
r(q)N
⎞
⎟⎟⎟⎠·
⎛
⎜⎜⎜⎝
f1f2...
fN
⎞
⎟⎟⎟⎠=
(r(q)1, r(q)2, . . . , r(q)N
)T
⎛
⎜⎜⎜⎝
f1f2...
fN
⎞
⎟⎟⎟⎠(4.15)
This represents amapping from the vector spaceofmuscle forces (of dimensionN , or fm ∈ RN ) to the scalar value of net joint torque (of dimension 1, orτ ∈ R1),
RN → R1 (4.16)
Equations4.12–4.15 are underdetermined in that they equivalently express thata given value of the net joint torque scalar can be produced by a variety of combi-nations of muscle forces. Essentially, there are N muscle activation DOFs that canbe combined in an infinite number of ways to achieve a same goal.2 This is a formof muscle redundancy that begs the question of how the nervous system selects asolution from amongmany. This has been called the central problem ofmotor control[7]. We will discuss this in detail in Chap.5.
An equally important concept described by Fig. 4.7—which is not as well high-lighted in the literature [8]—concerns the tendon excursions of the N muscles thatcross the joint. In this case of a single DOF q
δsi = −r(q)iδq for i = 1, . . . , N (4.17)⎛
⎜⎜⎜⎝
δs1δs2...
δsN
⎞
⎟⎟⎟⎠=
⎛
⎜⎜⎜⎝
r(q)1r(q)2...
r(q)N
⎞
⎟⎟⎟⎠δq (4.18)
which in vector form is
δs = −r(q) δq (4.19)
2Note that the letter M need not stand for muscles, nor N be used only for kinematic DOFs of alimb. They are simply letters to indicate indices and dimensions. See Appendix A.
44 4 Tendon-Driven Limbs
τ = r(q)T fm (4.14)
and
τ =
⎛
⎜⎜⎜⎝
r(q)1r(q)2...
r(q)N
⎞
⎟⎟⎟⎠·
⎛
⎜⎜⎜⎝
f1f2...
fN
⎞
⎟⎟⎟⎠=
(r(q)1, r(q)2, . . . , r(q)N
)T
⎛
⎜⎜⎜⎝
f1f2...
fN
⎞
⎟⎟⎟⎠(4.15)
This represents amapping from the vector spaceofmuscle forces (of dimensionN , or fm ∈ RN ) to the scalar value of net joint torque (of dimension 1, orτ ∈ R1),
RN → R1 (4.16)
Equations4.12–4.15 are underdetermined in that they equivalently express thata given value of the net joint torque scalar can be produced by a variety of combi-nations of muscle forces. Essentially, there are N muscle activation DOFs that canbe combined in an infinite number of ways to achieve a same goal.2 This is a formof muscle redundancy that begs the question of how the nervous system selects asolution from amongmany. This has been called the central problem ofmotor control[7]. We will discuss this in detail in Chap.5.
An equally important concept described by Fig. 4.7—which is not as well high-lighted in the literature [8]—concerns the tendon excursions of the N muscles thatcross the joint. In this case of a single DOF q
δsi = −r(q)iδq for i = 1, . . . , N (4.17)⎛
⎜⎜⎜⎝
δs1δs2...
δsN
⎞
⎟⎟⎟⎠=
⎛
⎜⎜⎜⎝
r(q)1r(q)2...
r(q)N
⎞
⎟⎟⎟⎠δq (4.18)
which in vector form is
δs = −r(q) δq (4.19)
2Note that the letter M need not stand for muscles, nor N be used only for kinematic DOFs of alimb. They are simply letters to indicate indices and dimensions. See Appendix A.
r3r2r1
m1
m3
m2
m4
m6
m5
44 4 Tendon-Driven Limbs
τ = r(q)T fm (4.14)
and
τ =
⎛
⎜⎜⎜⎝
r(q)1r(q)2...
r(q)N
⎞
⎟⎟⎟⎠·
⎛
⎜⎜⎜⎝
f1f2...
fN
⎞
⎟⎟⎟⎠=
(r(q)1, r(q)2, . . . , r(q)N
)T
⎛
⎜⎜⎜⎝
f1f2...
fN
⎞
⎟⎟⎟⎠(4.15)
This represents amapping from the vector spaceofmuscle forces (of dimensionN , or fm ∈ RN ) to the scalar value of net joint torque (of dimension 1, orτ ∈ R1),
RN → R1 (4.16)
Equations4.12–4.15 are underdetermined in that they equivalently express thata given value of the net joint torque scalar can be produced by a variety of combi-nations of muscle forces. Essentially, there are N muscle activation DOFs that canbe combined in an infinite number of ways to achieve a same goal.2 This is a formof muscle redundancy that begs the question of how the nervous system selects asolution from amongmany. This has been called the central problem ofmotor control[7]. We will discuss this in detail in Chap.5.
An equally important concept described by Fig. 4.7—which is not as well high-lighted in the literature [8]—concerns the tendon excursions of the N muscles thatcross the joint. In this case of a single DOF q
δsi = −r(q)iδq for i = 1, . . . , N (4.17)⎛
⎜⎜⎜⎝
δs1δs2...
δsN
⎞
⎟⎟⎟⎠=
⎛
⎜⎜⎜⎝
r(q)1r(q)2...
r(q)N
⎞
⎟⎟⎟⎠δq (4.18)
which in vector form is
δs = −r(q) δq (4.19)
2Note that the letter M need not stand for muscles, nor N be used only for kinematic DOFs of alimb. They are simply letters to indicate indices and dimensions. See Appendix A.
underdetermined
overdetermined
Reflexes respond to tendon excursions
Motor Cotex
Motorneuron
Extensor
Flexor
MuscleSpindle
Sensory Cortex
Sensory Neuron
To lengthen a muscle….you have to silence its stretch reflex!
Neuromechanical Concept #1
Feasible functionvs.
Optimal function
for underdetermined systems
Muscle redundancy as the central problem of motor control.
Popular View:We have many more muscles than kinematic degrees of freedom.
This allows infinite solutions.
Therefore the nervous system is faced with the tough computational problem of decision-making (or optimization).
But this is paradoxical withevolutionary biology and clinical reality
Wikipedia
That is,For decades, neuroscientists and biomechanists have
been exploring how to effectively choose specific muscle actions from a set of infinite choices.
but...
If we are so redundant:Which muscle would you like to donate?
Why do people seek clinical treatment for dysfunction even after mild pathology?
Why would we evolve, encode, grow, maintain, repair, control, etc. so many muscles?
1
ISBN 978-1-4471-6746-4
Biosystems & Biorobotics
Francisco J. Valero-Cuevas
Fundamentals of NeuromechanicsFundamentals of Neuromechanics
Valero-Cuevas
Biosystems & Biorobotics
Francisco J. Valero-CuevasFundamentals of Neuromechanics
This book provides a conceptual and computational framework to study how the nervous system exploits the anatomical properties of limbs to produce mechanical function. The study of the neural control of limbs has historically emphasized the use of optimization to find solutions to the muscle redundancy problem. That is, how does the nervous system select a specific muscle coordination pattern when the many muscle of a limb allow for multiple solutions?
I revisit this problem from the emerging perspective of neuromechanics that emphasizes finding and implementing families of feasible solutions, instead of a single and unique optimal solution. Those families of feasible solutions emerge naturally from the interactions among the feasible neural commands, anatomy of the limb, and constraints of the task. Such alternative perspective to the neural control of function is not only biologically plausible, but sheds light on the most central tenets and debates in the fields of neural control, robotics, rehabilitation, and brain-body co-evolutionary adaptations. This perspective developed from courses I taught to engineers and life scientists at Cornell University and the University of Southern California, and is made possible by combining fundamental concepts from mechanics, anatomy, robotics and neuroscience with advances in the field of computational geometry.
Fundamentals of Neuromechanics is intended for neuroscientists, roboticists, engineers, physicians, evolutionary biologists, athletes, and physical and occupational therapists seeking to advance their understanding of neuromechanics. Therefore, the tone is decidedly pedagogical, engaging, integrative and practical to make it accessible to people coming from a broad spectrum of disciplines. I attempt to tread the line between making the mathematical exposition accessible to life scientists, and convey the wonder and complexity of neuroscience to engineers and computational scientists. While no one approach can hope to definitively resolve the important questions in these related fields, I hope to provide you with the fundamental background and tools to allow you to contribute to the emerging field of neuromechanics.
Engineering
Theoretical framework
http://valerolab.org/fundamentalsAll lectures are available online
Free e-Book at universities with SpringerLink
How did this paradox arise?
The brain is confronted with controlling muscles and tendons system, yet for mathematical and technical
convenience (both very good reasons!) we have historically phrased the problem as one of torque
control or of simplified musculature.
Much of what I have learned has come from using and extending techniques to study the mechanics of
tendon-driven systems.
What is a limb?What is limb function?
BrainSpinal cord
Muscles, bones and joints
Motor Cotex
Motorneuron
Extensor
Flexor
The musculo-skeletal system “filters” the propagation of neural commands
MusclesTendons
&joints
BonesMuscle
activation(activation space)
Muscleforce
(muscle force space)
Jointtorques
(torque space)
Endpointforces and torques
(wrench space)
Block diagram of transformations in a forward limb model
Input OutputMusclesTeTeT ndons
&joints
BonesMusclefofof rcrcr e
Joinini ttorqrqr ues
EnEnE dpdpd oinini tfofof rcrcr es and torqrqr ues
Outptpt ut
Motor Cotex
Motorneuron
Extensor
Flexor
The musculo-skeletal system “filters” the propagation of neural commands
MusclesTendons
&joints
BonesMuscle
activation(activation space)
Muscleforce
(muscle force space)
Jointtorques
(torque space)
Endpointforces and torques
(wrench space)
Block diagram of transformations in a forward limb model
Input OutputTeTeT ndons&
jointsBones
Joinini ttorqrqr ues
EnEnE dpdpd oinini tfofof rcrcr es and torqrqr ues
Outptpt ut
Motor Cotex
Motorneuron
Extensor
Flexor
The musculo-skeletal system “filters” the propagation of neural commands
MusclesTendons
&joints
BonesMuscle
activation(activation space)
Muscleforce
(muscle force space)
Jointtorques
(torque space)
Endpointforces and torques
(wrench space)
Block diagram of transformations in a forward limb model
Input OutputBonesEnEnE dpdpd oinini t
fofof rcrcr es and torqrqr ues
Outptpt ut
Motor Cotex
Motorneuron
Extensor
Flexor
The musculo-skeletal system “filters” the propagation of neural commands
MusclesTendons
&joints
BonesMuscle
activation(activation space)
Muscleforce
(muscle force space)
Jointtorques
(torque space)
Endpointforces and torques
(wrench space)
Block diagram of transformations in a forward limb model
Input Output
Motor Cotex
Motorneuron
Extensor
Flexor
100 7 Feasible Neural Commands and Feasible Mechanical Outputs
feasible activation set(nondimensional units)
feasible muscle force set(N)
feasible torque set(N-m)
F matrix0F matrix
R m
atrix
endpoint feasible wrench set(forces in N)
350
525
0
700
f1
f3
f2
1
1
0
1
a1
a3
a2
0 50 100
-75
-50
-25
0
25
τ2
τ3 τ1
Torque at joint 1
Torque at joint 2
J matrix-T
Force in x
Force in y
-50 -250 25
0
100
200
w1
w3
w2
(a) (b)
(c)(d)
Fig. 7.3 Sequence of linear transformations of the feasible activation set in a forward model ofstatic force production by a tendon-driven limb. Using the concepts in Figs. 7.1 and 7.2, we use asneural input a the feasible activation set (a positive unit cube) to produce b the feasible muscle, cfeasible joint torque, and d endpoint feasible wrench sets. Adapted with permission from [2]
point wrenches, respectively. Their positive linear combinations specify the set of allpossible outputs in those spaces. If we consider them as the vi vectors of Fig. 7.2,we can build the zonotopes that correspond to the feasible joint torque set and thefeasible output wrench set as shown in Fig. 7.3. In the specific simple case shownin Fig. 7.4, the endpoint wrench is a planar force vector—consisting of fx and fycomponents only. This zonotope is called the feasible force set, and not the feasiblewrench set because the endpoint output contains no torques (see Sect. 2.6).
Valero-Cuevas. Fundamentals of Neuromechanics, 2015
This propagation of neural
commands defines
feasible inputs and
outputs
Feasible actions are defined bythe anatomical routing of tendons
Spoor, An, Yoshikawa, Brand, Leijnse, Valero-Cuevas, etc.
350
700
525
0 50 100
-75
-50
-25
25
-75
-50
-25
0
25
muscle force space(N)
joint torque space(N-m)
Tendon pathsand moment arms
Force in x
Force in y
xy
0f1
Torque at joint 1
f 3
f 2
τ2
τ3 τ1
Torque at joint 2
{(f1 * r1,1), (f1*r2,1))
{(f2 * r1,2), (f2*r2,2))
{(f3 * r1,3), 0)
R matrixR matrix
-50 -25 0 25
0
100
200
0
100
200
endpointwrench space(forces in N)
J matrixJ matrix-T
w w
w
1 3
2
m3
m1m2
r1,1r1,1
r1,3r1,3
r1,2r1,2 r2,2r2,2
r2,1r2,1
ground
joint 1
joint 2
Feasible actions are defined bythe anatomical routing of tendons
Spoor, An, Yoshikawa, Brand, Leijnse, Valero-Cuevas, etc.
350
700
525
0 50 100
-75
-50
-25
25
-75
-50
-25
0
25
muscle force space(N)
joint torque space(N-m)
Tendon pathsand moment arms
Force in x
Force in y
xy
0f1
Torque at joint 1
f 3
f 2
τ2
τ3 τ1
Torque at joint 2
{(f1 * r1,1), (f1*r2,1))
{(f2 * r1,2), (f2*r2,2))
{(f3 * r1,3), 0)
R matrixR matrix
-50 -25 0 25
0
100
200
0
100
200
endpointwrench space(forces in N)
J matrixJ matrix-T
w w
w
1 3
2
m3
m1m2
r1,1r1,1
r1,3r1,3
r1,2r1,2 r2,2r2,2
r2,1r2,1
ground
joint 1
joint 2
Feasible actions are defined bythe anatomical routing of tendons
Spoor, An, Yoshikawa, Brand, Leijnse, Valero-Cuevas, etc.
350
700
525
0 50 100
-75
-50
-25
25
-75
-50
-25
0
25
muscle force space(N)
joint torque space(N-m)
Tendon pathsand moment arms
Force in x
Force in y
xy
0f1
Torque at joint 1
f 3
f 2
τ2
τ3 τ1
Torque at joint 2
{(f1 * r1,1), (f1*r2,1))
{(f2 * r1,2), (f2*r2,2))
{(f3 * r1,3), 0)
R matrixR matrix
-50 -25 0 25
0
100
200
0
100
200
endpointwrench space(forces in N)
J matrixJ matrix-T
w w
w
1 3
2
m3
m1m2
r1,1r1,1
r1,3r1,3
r1,2r1,2 r2,2r2,2
r2,1r2,1
ground
joint 1
joint 2
Feasible actions are defined bythe anatomical routing of tendons
Spoor, An, Yoshikawa, Brand, Leijnse, Valero-Cuevas, etc.
350
700
525
0 50 100
-75
-50
-25
25
-75
-50
-25
0
25
muscle force space(N)
joint torque space(N-m)
Tendon pathsand moment arms
Force in x
Force in y
xy
0f1
Torque at joint 1
f 3
f 2
τ2
τ3 τ1
Torque at joint 2
{(f1 * r1,1), (f1*r2,1))
{(f2 * r1,2), (f2*r2,2))
{(f3 * r1,3), 0)
R matrixR matrix
-50 -25 0 25
0
100
200
0
100
200
endpointwrench space(forces in N)
J matrixJ matrix-T
w w
w
1 3
2
m3
m1m2
r1,1r1,1
r1,3r1,3
r1,2r1,2 r2,2r2,2
r2,1r2,1
ground
joint 1
joint 2
Building a feasible torque set for a “complex” limb
Valero-Cuevas, 2005. J Biomech.
r1,2
r1,5 r1,3r2,5
r2,4
5 muscles
r2,1r1,4
r1,1 m1
m4
m2
m5m3
+fingertip flexors produce
positive torque
DOF 1
DOF 2
Building a feasible torque set for a “complex” limb
Valero-Cuevas, 2005. J Biomech.
r1,2
r1,5 r1,3r2,5
r2,4
5 muscles
r2,1r1,4
r1,1 m1
m4
m2
m5m3
+fingertip flexors produce
positive torque
DOF 1
DOF 2
τ2
τ1
m5
m1
m4
m2
Quad. II Quad. I
Quad. III Quad. IV
m3
r1,4r2,4
flexion torqueextension torque
flexion torque
extension torque
Building a feasible torque set for a “complex” limb
Valero-Cuevas, 2005. J Biomech.
r1,2
r1,5 r1,3r2,5
r2,4
5 muscles
r2,1r1,4
r1,1 m1
m4
m2
m5m3
+fingertip flexors produce
positive torque
DOF 1
DOF 2
τ2
τ1
m5
m1
m4
m2
m3
Building a feasible torque set for a “complex” limb
Valero-Cuevas, 2005. J Biomech.
r1,2
r1,5 r1,3r2,5
r2,4
5 muscles
r2,1r1,4
r1,1 m1
m4
m2
m5m3
+fingertip flexors produce
positive torque
DOF 1
DOF 2
τ2
τ1
m5
m1
m4
m2
m3
Building a feasible torque set for a “complex” limb
Valero-Cuevas, 2005. J Biomech.
r1,2
r1,5 r1,3r2,5
r2,4
5 muscles
r2,1r1,4
r1,1 m1
m4
m2
m5m3
+fingertip flexors produce
positive torque
DOF 1
DOF 2
τ2
τ1
m5
m1
m4
m2
m3
Building a feasible torque set for a “complex” limb
Valero-Cuevas, 2005. J Biomech.
r1,2
r1,5 r1,3r2,5
r2,4
5 muscles
r2,1r1,4
r1,1 m1
m4
m2
m5m3
+fingertip flexors produce
positive torque
DOF 1
DOF 2
τ2
τ1
m5
m1
m4
m2
m3
Building a feasible torque set for a “complex” limb
Valero-Cuevas, 2005. J Biomech.
r1,2
r1,5 r1,3r2,5
r2,4
5 muscles
r2,1r1,4
r1,1 m1
m4
m2
m5m3
+fingertip flexors produce
positive torque
DOF 1
DOF 2
τ2
τ1
m5
m1
m4
m2
m3
One analytical starting point:A working definition of “versatility”
Simply put: the ability to produce end-point force in every direction—i.e., controllability.
Not to worry, it can be extended to motion in every direction!
Lucía Valero
Valero-Cuevas. A mathematical approach to the mechanical capabilities of limbs and fingers. 2009.
Versatility ≡ feasible torque and force sets that include the origin
0 50 100
-75
-50
-25
25
-75
-50
-25
0
25
feasible torque set(N-m)
endpointfeasible force set
(forces in N)
τ 2
τ 3 τ 1
Torque at joint 1
Torque at joint 2
Force in x
Force in y
-50 -25 0 25
0
100
200
0
100
200
ww
w
1
3
2
That is, producing end-point force in every Cartesian direction requires that you produce torques in every direction in “torque space”
Valero-Cuevas 1998, 2005, 2009
convex sets remain convex under linear mapping
All’s well
m3
m1m2
ground
All’s well
m3
m1m2
ground
...but
How many muscles do you need to include the origin in torque/force space?
-100 -50 0 50 100
-100
-50
50
-
-50
0
50
0 50 100
-75
-50
-25
25
-75
-50
-25
0
25
feasible torque sets(Newton-meters)
τ 2
τ 3 τ 1
Torque at joint 1
Torque at joint 2
Torque at joint 1
Torque at joint 2
m3
m1m2
r1,1
r1,3
r1,2 r2,2
2-link limb with 3 muscles (N+1)2-link limb with 4 muscles (2*N)
r2,1
ground
m3
m1m2
r1,1
r1,3
r1,2 r2,4
r2,1
ground
m4r1,4
At least N+1 well-routed muscles
Wait a minute... but N+1 > N
Valero-Cuevas 1998, 2005, 2009
Wait a minute... but N+1 > N...so you need more muscles than degrees of freedom?
Valero-Cuevas 1998, 2005, 2009
Wait a minute... but N+1 > N...so you need more muscles than degrees of freedom?A versatile feasible torque set implies muscle redundancy for
submaximal outputs!
0 50 100
-75
-50
-25
25
-75
-50
-25
0
25
τ 2
τ 2
τ 3
τ 1
τ 1
Torque at joint 1
Torque at joint 2
Valero-Cuevas 1998, 2005, 2009
Muscle redundancy is not an accident of evolution, but rather an appropriate structural
adaptation for versatility.
(and later we will see how having more muscles allows us to meet more function constraints)
Thus, versatile tendon-driven systems require “over-actuation”
And each tendon contributes in unique waysτ
2
τ1
FTS 1,2,3,4,5
FTS 1,2,3,4
FTS 1,2,3
FTS 1,2r1,2
r1,5 r1,3r2,5
r2,4
5 muscles
r2,1r1,4
r1,1 m1
m4
m2
m5m3
+fingertip flexors produce
positive torque
DOF 1
DOF 2
τ1
FTS 1,2,3
And each tendon contributes in unique waysτ
2
τ1
FTS 1,2,3,4,5
FTS 1,2,3,4
FTS 1,2,3
FTS 1,2
Tendons define the size and shape of the feasible torque and feasible force sets
r1,2
r1,5 r1,3r2,5
r2,4
5 muscles
r2,1r1,4
r1,1 m1
m4
m2
m5m3
+fingertip flexors produce
positive torque
DOF 1
DOF 2
τ1
FTS 1,2,3
And each tendon contributes in unique waysτ
2
τ1
FTS 1,2,3,4,5
FTS 1,2,3,4
FTS 1,2,3
FTS 1,2
Tendons define the size and shape of the feasible torque and feasible force sets
So which muscle would you give up?
r1,2
r1,5 r1,3r2,5
r2,4
5 muscles
r2,1r1,4
r1,1 m1
m4
m2
m5m3
+fingertip flexors produce
positive torque
DOF 1
DOF 2
τ1
FTS 1,2,3
Redundancy does not imply robustness
With Jason KutchJ Biomech 2011
Now assistant professor at USC
MUSCLE 1Tension
MUSCLE 2Tension
MUSCLE 3Tension
Valero-Cuevas et al., 1998, 2005, 2009; Kutch & Valero-Cuevas, Inouye and Valero-Cuevas 2011
MUSCLE 1
MUSCLE 2
MUSCLE 3
Dimensionality and structure of the feasible input (solution space) for a given task
Computational GeometryVertex Enumeration (dual of Linear programming)
No cost function needed, simply description of feasible inputs and outputs!
feasible output spacefeasible input space
MUSCLE 1Tension
MUSCLE 2Tension
MUSCLE 3Tension
Max tension
Max tension
Max tension
Max tension
Valero-Cuevas et al., 1998, 2005, 2009; Kutch & Valero-Cuevas, Inouye and Valero-Cuevas 2011
MUSCLE 1
MUSCLE 2
MUSCLE 3
Dimensionality and structure of the feasible input (solution space) for a given task
Computational GeometryVertex Enumeration (dual of Linear programming)
No cost function needed, simply description of feasible inputs and outputs!
feasible output spacefeasible input space
MUSCLE 1Tension
MUSCLE 2Tension
MUSCLE 3Tension
Max tension
Max tension
Max tension
Max tension
Valero-Cuevas et al., 1998, 2005, 2009; Kutch & Valero-Cuevas, Inouye and Valero-Cuevas 2011
MUSCLE 1
MUSCLE 2
MUSCLE 3
Dimensionality and structure of the feasible input (solution space) for a given task
Computational GeometryVertex Enumeration (dual of Linear programming)
No cost function needed, simply description of feasible inputs and outputs!
feasible output spacefeasible input space
MUSCLE 1Tension
MUSCLE 2Tension
MUSCLE 3Tension
Max tension
Max tension
Max tension
Max tension
Valero-Cuevas et al., 1998, 2005, 2009; Kutch & Valero-Cuevas, Inouye and Valero-Cuevas 2011
MUSCLE 1
MUSCLE 2
MUSCLE 3
Dimensionality and structure of the feasible input (solution space) for a given task
Computational GeometryVertex Enumeration (dual of Linear programming)
No cost function needed, simply description of feasible inputs and outputs!
feasible output spacefeasible input space
MUSCLE 1Tension
MUSCLE 2Tension
MUSCLE 3Tension
Max tension
Max tension
Max tension
Max tension
Valero-Cuevas et al., 1998, 2005, 2009; Kutch & Valero-Cuevas, Inouye and Valero-Cuevas 2011
MUSCLE 1
MUSCLE 2
MUSCLE 3
Constraint 1
Dimensionality and structure of the feasible input (solution space) for a given task
Computational GeometryVertex Enumeration (dual of Linear programming)
No cost function needed, simply description of feasible inputs and outputs!
feasible output spacefeasible input space
MUSCLE 1Tension
MUSCLE 2Tension
MUSCLE 3Tension
Max tension
Max tension
Max tension
Max tension
Max tension
Constraint 1
Valero-Cuevas et al., 1998, 2005, 2009; Kutch & Valero-Cuevas, Inouye and Valero-Cuevas 2011
MUSCLE 1
MUSCLE 2
MUSCLE 3
Constraint 1
Dimensionality and structure of the feasible input (solution space) for a given task
Computational GeometryVertex Enumeration (dual of Linear programming)
No cost function needed, simply description of feasible inputs and outputs!
feasible output spacefeasible input space
MUSCLE 1Tension
MUSCLE 2Tension
MUSCLE 3Tension
Max tension
Max tension
Max tension
Max tension
Max tension
Constraint 1
Valero-Cuevas et al., 1998, 2005, 2009; Kutch & Valero-Cuevas, Inouye and Valero-Cuevas 2011
MUSCLE 1
MUSCLE 2
MUSCLE 3
Constraint 1
Constraint 2
Dimensionality and structure of the feasible input (solution space) for a given task
Computational GeometryVertex Enumeration (dual of Linear programming)
No cost function needed, simply description of feasible inputs and outputs!
feasible output spacefeasible input space
MUSCLE 1Tension
MUSCLE 2Tension
MUSCLE 3Tension
Max tension
Max tension
Max tension
Max tension
Max tension
Constraint 1
Constraint 2
Valero-Cuevas et al., 1998, 2005, 2009; Kutch & Valero-Cuevas, Inouye and Valero-Cuevas 2011
MUSCLE 1
MUSCLE 2
MUSCLE 3
Constraint 1
Constraint 2
Dimensionality and structure of the feasible input (solution space) for a given task
Computational GeometryVertex Enumeration (dual of Linear programming)
No cost function needed, simply description of feasible inputs and outputs!
feasible output spacefeasible input space
MUSCLE 1Tension
MUSCLE 2Tension
MUSCLE 3Tension
Max tension
Max tension
Max tension
Max tension
Max tension
Constraint 1
Constraint 2
Valero-Cuevas et al., 1998, 2005, 2009; Kutch & Valero-Cuevas, Inouye and Valero-Cuevas 2011
MUSCLE 1
MUSCLE 2
MUSCLE 3
Constraint 1
Constraint 2
Dimensionality and structure of the feasible input (solution space) for a given task
Computational GeometryVertex Enumeration (dual of Linear programming)
No cost function needed, simply description of feasible inputs and outputs!
feasible output spacefeasible input space
MUSCLE 1Tension
MUSCLE 2Tension
MUSCLE 3Tension
Max tension
Max tension
Max tension
Max tension
Max tension
Constraint 1
Constraint 2
Valero-Cuevas et al., 1998, 2005, 2009; Kutch & Valero-Cuevas, Inouye and Valero-Cuevas 2011
MUSCLE 1
MUSCLE 2
MUSCLE 3
Constraint 1
Constraint 2
Dimensionality and structure of the feasible input (solution space) for a given task
Computational GeometryVertex Enumeration (dual of Linear programming)
No cost function needed, simply description of feasible inputs and outputs!
feasible output spacefeasible input space
MUSCLE 1Tension
MUSCLE 2Tension
MUSCLE 3Tension
Max tension
Max tension
Max tension
Max tension
Max tension
Constraint 1
Constraint 2
Valero-Cuevas et al., 1998, 2005, 2009; Kutch & Valero-Cuevas, Inouye and Valero-Cuevas 2011
MUSCLE 1
MUSCLE 2
MUSCLE 3
Constraint 1
Constraint 2
Dimensionality and structure of the feasible input (solution space) for a given task
Computational GeometryVertex Enumeration (dual of Linear programming)
No cost function needed, simply description of feasible inputs and outputs!
feasible output spacefeasible input space
All possible ways to do
a same task
MUSCLE 1Tension
MUSCLE 2Tension
MUSCLE 3Tension
Valero-Cuevas et al., 1998Kutch & Valero-Cuevas, 2011
MUSCLE 1
MUSCLE 2
MUSCLE 3
Some muscles are more redundant than others!
MUSCLE 1Tension
MUSCLE 2Tension
MUSCLE 3Tension
Max tension
Max tension
Max tension
Max tension
Valero-Cuevas et al., 1998Kutch & Valero-Cuevas, 2011
MUSCLE 1
MUSCLE 2
MUSCLE 3
Some muscles are more redundant than others!
MUSCLE 1Tension
MUSCLE 2Tension
MUSCLE 3Tension
Max tension
Max tension
Max tension
Max tension
Valero-Cuevas et al., 1998Kutch & Valero-Cuevas, 2011
MUSCLE 1
MUSCLE 2
MUSCLE 3
Some muscles are more redundant than others!
MUSCLE 1Tension
MUSCLE 2Tension
MUSCLE 3Tension
Max tension
Max tension
Max tension
Max tension
Max tension
Constraint 1
Valero-Cuevas et al., 1998Kutch & Valero-Cuevas, 2011
MUSCLE 1
MUSCLE 2
MUSCLE 3
Constraint 1
Some muscles are more redundant than others!
MUSCLE 1Tension
MUSCLE 2Tension
MUSCLE 3Tension
Max tension
Max tension
Max tension
Max tension
Max tension
Constraint 1
Constraint 2
Valero-Cuevas et al., 1998Kutch & Valero-Cuevas, 2011
MUSCLE 1
MUSCLE 2
MUSCLE 3
Constraint 1
Constraint 2
Some muscles are more redundant than others!
MUSCLE 1Tension
MUSCLE 2Tension
MUSCLE 3Tension
Max tension
Max tension
Max tension
Max tension
Max tension
Constraint 1
Constraint 2
Valero-Cuevas et al., 1998Kutch & Valero-Cuevas, 2011
MUSCLE 1
MUSCLE 2
MUSCLE 3
Constraint 1
Constraint 2
Some muscles are more redundant than others!
MUSCLE 1Tension
MUSCLE 2Tension
MUSCLE 3Tension
Max tension
Max tension
Max tension
Max tension
Max tension
Constraint 1
Constraint 2
Valero-Cuevas et al., 1998Kutch & Valero-Cuevas, 2011
MUSCLE 1
MUSCLE 2
MUSCLE 3
Constraint 1
Constraint 2
Some muscles are more redundant than others!
MUSCLE 1Tension
MUSCLE 2Tension
MUSCLE 3Tension
Max tension
Max tension
Max tension
Max tension
Max tension
Constraint 1
Constraint 2
Valero-Cuevas et al., 1998Kutch & Valero-Cuevas, 2011
MUSCLE 1
MUSCLE 2
MUSCLE 3
Constraint 1
Constraint 2
Some muscles are more redundant than others!
MUSCLE 1Tension
MUSCLE 2Tension
MUSCLE 3Tension
Max tension
Max tension
Max tension
Max tension
Max tension
Constraint 1
Constraint 2
Valero-Cuevas et al., 1998Kutch & Valero-Cuevas, 2011
MUSCLE 1
MUSCLE 2
MUSCLE 3
Constraint 1
Constraint 2
Allo
wab
le Te
nsio
n Ra
nge
Maximal
0 MUSCLE1 2 3
Some muscles are more redundant than others!
MUSCLE 1Tension
MUSCLE 2Tension
MUSCLE 3Tension
Max tension
Max tension
Max tension
Max tension
Max tension
Constraint 1
Constraint 2
Valero-Cuevas et al., 1998Kutch & Valero-Cuevas, 2011
MUSCLE 1
MUSCLE 2
MUSCLE 3
Constraint 1
Constraint 2
Allo
wab
le Te
nsio
n Ra
nge
Maximal
0 MUSCLE1 2 3
!"#$%&'()'#$%*%'
+,*-.%*'"*'/(#'
/%-%**0&1
Some muscles are more redundant than others!
MUSCLE 1Tension
MUSCLE 2Tension
MUSCLE 3Tension
Max tension
Max tension
Max tension
Max tension
Max tension
Constraint 1
Constraint 2
Valero-Cuevas et al., 1998Kutch & Valero-Cuevas, 2011
MUSCLE 1
MUSCLE 2
MUSCLE 3
Constraint 1
Constraint 2
Allo
wab
le Te
nsio
n Ra
nge
Maximal
0 MUSCLE1 2 3
2$"*'+,*-.%'"*'/%-%**0&133
Allo
wab
le Te
nsio
n Ra
nge
!"#$%&'()'#$%*%'
+,*-.%*'"*'/(#'
/%-%**0&1
Some muscles are more redundant than others!
Can we show this for real systems?
Joining the XVI and XXI centuries
Valero-Cuevas et al, 2000-presentAnatomy Lesson of Dr.TulipRembrandt 1632
.Turn to age-old physical testing because biomechanical modeling remains a challenge
Robotic Arm
Extension Spring
Safety Shield
Fixation Device
Dynamometer
Figu
Mechanical Engineering
Mathematics
Computer ScienceHand SurgeryNeuroscience
Record the real input-output mapping
MusclesTendons
&joints
BonesMuscle
activation(activation space)
Muscleforce
(muscle force space)
Jointtorques
(torque space)
Endpointforces and torques
(wrench space)
Block diagram of transformations in a forward limb model
Input Output
Robotic Arm
Extension Spring
Safety Shield
Fixation Device
Dynamometer
Figure 2
A. Constrain only radial force
LUM FPI FDP FDI FDS EDC EIP0
100
Muscle
Allo
wab
le T
ensi
on R
ange
(% M
ax)
B. Constrain only radial and dorsal force
LUM FPI FDP FDI FDS EDC EIP0
1
Muscle
Allo
wab
le T
ensi
on R
ange
(% M
ax)
LUM FPI FDP FDI FDS EDC EIP0
1
Muscle
Allo
wab
le T
ensi
on R
ange
(% M
ax)
C. Constrain radial, dorsal, and distal force (produce a well directed force)
What is the set of all muscle actions to produce a given force?
A subset of 7-dimensional space!
Kutch & Valero-Cuevas, 2011
A. Constrain only radial force
LUM FPI FDP FDI FDS EDC EIP0
100
Muscle
Allo
wab
le T
ensi
on R
ange
(% M
ax)
B. Constrain only radial and dorsal force
LUM FPI FDP FDI FDS EDC EIP0
1
Muscle
Allo
wab
le T
ensi
on R
ange
(% M
ax)
LUM FPI FDP FDI FDS EDC EIP0
1
Muscle
Allo
wab
le T
ensi
on R
ange
(% M
ax)
C. Constrain radial, dorsal, and distal force (produce a well directed force)
What is the set of all muscle actions to produce a given force?
A subset of 7-dimensional space!
Kutch & Valero-Cuevas, 2011
A. Constrain only radial force
LUM FPI FDP FDI FDS EDC EIP0
100
Muscle
Allo
wab
le T
ensi
on R
ange
(% M
ax)
B. Constrain only radial and dorsal force
LUM FPI FDP FDI FDS EDC EIP0
1
Muscle
Allo
wab
le T
ensi
on R
ange
(% M
ax)
LUM FPI FDP FDI FDS EDC EIP0
1
Muscle
Allo
wab
le T
ensi
on R
ange
(% M
ax)
C. Constrain radial, dorsal, and distal force (produce a well directed force)
What is the set of all muscle actions to produce a given force?
A subset of 7-dimensional space!
Kutch & Valero-Cuevas, 2011
A. Constrain only radial force
LUM FPI FDP FDI FDS EDC EIP0
100
Muscle
Allo
wab
le T
ensi
on R
ange
(% M
ax)
B. Constrain only radial and dorsal force
LUM FPI FDP FDI FDS EDC EIP0
1
Muscle
Allo
wab
le T
ensi
on R
ange
(% M
ax)
LUM FPI FDP FDI FDS EDC EIP0
1
Muscle
Allo
wab
le T
ensi
on R
ange
(% M
ax)
C. Constrain radial, dorsal, and distal force (produce a well directed force)
Necessary
Necessary & Intolerant to dysfunction
What is the set of all muscle actions to produce a given force?
A subset of 7-dimensional space!
Kutch & Valero-Cuevas, 2011
Neuromechanical Concept #2
What is the nature and structure of feasible sets?
MUSCLE 1Tension
MUSCLE 2Tension
MUSCLE 3Tension
Valero-Cuevas et al., 1998Kutch & Valero-Cuevas, 2011
MUSCLE 1
MUSCLE 2
MUSCLE 3
A first approach: the bounding box
MUSCLE 1Tension
MUSCLE 2Tension
MUSCLE 3Tension
Max tension
Max tension
Max tension
Max tension
Valero-Cuevas et al., 1998Kutch & Valero-Cuevas, 2011
MUSCLE 1
MUSCLE 2
MUSCLE 3
A first approach: the bounding box
MUSCLE 1Tension
MUSCLE 2Tension
MUSCLE 3Tension
Max tension
Max tension
Max tension
Max tension
Valero-Cuevas et al., 1998Kutch & Valero-Cuevas, 2011
MUSCLE 1
MUSCLE 2
MUSCLE 3
A first approach: the bounding box
MUSCLE 1Tension
MUSCLE 2Tension
MUSCLE 3Tension
Max tension
Max tension
Max tension
Max tension
Max tension
Constraint 1
Valero-Cuevas et al., 1998Kutch & Valero-Cuevas, 2011
MUSCLE 1
MUSCLE 2
MUSCLE 3
Constraint 1
A first approach: the bounding box
MUSCLE 1Tension
MUSCLE 2Tension
MUSCLE 3Tension
Max tension
Max tension
Max tension
Max tension
Max tension
Constraint 1
Constraint 2
Valero-Cuevas et al., 1998Kutch & Valero-Cuevas, 2011
MUSCLE 1
MUSCLE 2
MUSCLE 3
Constraint 1
Constraint 2
A first approach: the bounding box
MUSCLE 1Tension
MUSCLE 2Tension
MUSCLE 3Tension
Max tension
Max tension
Max tension
Max tension
Max tension
Constraint 1
Constraint 2
Valero-Cuevas et al., 1998Kutch & Valero-Cuevas, 2011
MUSCLE 1
MUSCLE 2
MUSCLE 3
Constraint 1
Constraint 2
A first approach: the bounding box
MUSCLE 1Tension
MUSCLE 2Tension
MUSCLE 3Tension
Max tension
Max tension
Max tension
Max tension
Max tension
Constraint 1
Constraint 2
Valero-Cuevas et al., 1998Kutch & Valero-Cuevas, 2011
MUSCLE 1
MUSCLE 2
MUSCLE 3
Constraint 1
Constraint 2
A first approach: the bounding box
MUSCLE 1Tension
MUSCLE 2Tension
MUSCLE 3Tension
Max tension
Max tension
Max tension
Max tension
Max tension
Constraint 1
Constraint 2
Valero-Cuevas et al., 1998Kutch & Valero-Cuevas, 2011
MUSCLE 1
MUSCLE 2
MUSCLE 3
Constraint 1
Constraint 2
A first approach: the bounding box
MUSCLE 1Tension
MUSCLE 2Tension
MUSCLE 3Tension
Max tension
Max tension
Max tension
Max tension
Max tension
Constraint 1
Constraint 2
Valero-Cuevas et al., 1998Kutch & Valero-Cuevas, 2011
MUSCLE 1
MUSCLE 2
MUSCLE 3
Constraint 1
Constraint 2
Allo
wab
le Te
nsio
n Ra
nge
Maximal
0 MUSCLE1 2 3
A first approach: the bounding box
Chapter 9
Use computational geometry to find the structure of these high-dimensional subspaces8.3 Vertex Enumeration in Practice 121
unit N-cube convex polytopevertices of
convex polytope
Feasibleinputset
Feasibleoutput
set
(a) (b) (c)
linearmapping
Fig. 8.5 Top row Schematic representation of Algorithm 1. Note that the addition of inequalityconstraints reduces the size and distorts the shape of the original feasible input set—the positive unitN-cube—and transforms it into a new convex polytope. The more faces the convex polytope hasthe more vertices are required to define it. Columns a–c: Schematic representation of Algorithm 2.As illustrated in Fig. 7.6, the projection of the feasible input set produces a feasible output set thatis a convex hull that defines the feasible muscle force, joint torque, wrench, or output force sets,depending on the framing of the problem. Figure8.1 shows a real-world example of mapping afeasible activation set in R7 onto the feasible force set in the plane of finger flexion in R2. Note thatthe vertices that become internal points of the feasible output set hint at the shape of the feasibleinput set, which is hard to describe or visualize given its high-dimensionality. This is an area ofactive research [18]
upon the most necessary points, and the reader is encouraged to see the referencesmentioned.
Recall that in Sect. 7.6 we introduced two types of representations of a convexpolytope. In this case, we are interested in finding the vertices of a high-dimensionalconvex polytope defined by a set of linear inequality constraints [15] as in
Ax ≤ b (8.6)
where A ∈ RM×N is a matrix containing M equations in N variables, x ∈ RN , andb ∈ RM is a vector of constants.
Do you notice the similarity to the phrasing of the canonical form of the linearprogramming problem in Eq.5.9? Notably, however, there is no need to have a costfunction. That is, we are not interested in finding the maximal force in a particulardirection in the plane of finger flexion as in Eq.5.22. Instead, we are interested infinding all possible maximal forces in the plane of finger flexion.
8.3 Vertex Enumeration in Practice 121
unit N-cube convex polytopevertices of
convex polytope
Feasibleinputset
Feasibleoutput
set
(a) (b) (c)
linearmapping
Fig. 8.5 Top row Schematic representation of Algorithm 1. Note that the addition of inequalityconstraints reduces the size and distorts the shape of the original feasible input set—the positive unitN-cube—and transforms it into a new convex polytope. The more faces the convex polytope hasthe more vertices are required to define it. Columns a–c: Schematic representation of Algorithm 2.As illustrated in Fig. 7.6, the projection of the feasible input set produces a feasible output set thatis a convex hull that defines the feasible muscle force, joint torque, wrench, or output force sets,depending on the framing of the problem. Figure8.1 shows a real-world example of mapping afeasible activation set in R7 onto the feasible force set in the plane of finger flexion in R2. Note thatthe vertices that become internal points of the feasible output set hint at the shape of the feasibleinput set, which is hard to describe or visualize given its high-dimensionality. This is an area ofactive research [18]
upon the most necessary points, and the reader is encouraged to see the referencesmentioned.
Recall that in Sect. 7.6 we introduced two types of representations of a convexpolytope. In this case, we are interested in finding the vertices of a high-dimensionalconvex polytope defined by a set of linear inequality constraints [15] as in
Ax ≤ b (8.6)
where A ∈ RM×N is a matrix containing M equations in N variables, x ∈ RN , andb ∈ RM is a vector of constants.
Do you notice the similarity to the phrasing of the canonical form of the linearprogramming problem in Eq.5.9? Notably, however, there is no need to have a costfunction. That is, we are not interested in finding the maximal force in a particulardirection in the plane of finger flexion as in Eq.5.22. Instead, we are interested infinding all possible maximal forces in the plane of finger flexion.
Brian Cohn May Szedlák
with Fukuda K and Gärtner B
Valero-Cuevas et al., 1998 - 2015
The Hit-and-Run algorithm can sample such subspaces in up to 40 dimensions!
We can now sample subspaces in up to 40
dimensions to provide a statistical description of
their structure
9.5 Probabilistic Neural Control 151
a1 a3
a2
p
a1 a3
a2
p
q
a1
a1
a3
a3
a2
a2
a1 a3a2
pp1
a1 a3
a2
0
0
0
0
p1
(a) (b)
(c)
(e)
(f)
(d)
0Pro
babi
lity
dens
ityP
roba
bilit
y de
nsity
Activation Activation Activation
1 N Output
2 N Output
0
K r
s
0
0
0 018.06.02.0 4.0 018.06.02.0 4.0 6.02.0 4.0 0.8 1
0 018.06.02.0 4.0 018.06.02.0 4.0 18.06.02.0 4.0
3.04.04.0
2.0
40
20
40
10.5
25
20
2.0 1.5
1
1
11
1
1
1
1
11
1
1
Fig. 9.7 The Hit-and-Run algorithm applied to a convex body K defined as the 2D shaded polygonembedded in 3D. aYou begin by finding a valid point p0 ∈ K , and run a linear program in a randomvector direction q, (b) to find the extreme points r and s at both ends. c You then use a uniformprobability distribution to sample a point p1 along the line r s, and d use the point p1 to repeat step(a) iteratively to find the set of points pi for i = 0, . . . ,M , where M is large enough to guaranteeconvergence to a uniform sampling of K . For an example of such iterative stochastic sampling seeFig. 9.8. e The histograms of the sampled points pi are an approximation to the probability densityfunctions of the convex body K for a 1 N target output force magnitude by the limb in a particulardirection. f If we change the target magnitude of force output to 2 N, the probability distributionschange where now a2 cannot be fully activated, a1 moves toward full activation, and the plane’ssurface area is centrally distributed about a3
Submaximal force
Near maximal force
Applied to a 7-muscle finger model producing static force
Valero-Cuevas et al 1998
45'*(.,6(/'*70-%'
%+8%99%9'
"/':5
Histograms (Bayesian priors) of activation to produce static force of different magnitudes
.(;')(&-%'+0</"#,9%
+0="+0.')(&-%'+0</"#,9%
+,*-.%'> +,*-.%':?
@$(;*'$(;'
&%9,/90/-1'"*'.(*#3
Big Data approach to motor control: Parallel coordinates visualization
with associated cost for each solution
5"A%&%/#'-(*#'),/-6(/*B,*-.%*
C-6D06(/'.%D%.
E(&-%'+0</"#,9%
Summary about muscle redundancy
• Redundancy does not imply robustness
Summary about muscle redundancy
• Redundancy does not imply robustness• Every muscle contributes uniquely to function
Summary about muscle redundancy
• Redundancy does not imply robustness• Every muscle contributes uniquely to function•More muscles allow more complex tasks
Summary about muscle redundancy
• Redundancy does not imply robustness• Every muscle contributes uniquely to function•More muscles allow more complex tasks• We have barely enough muscles for real-world tasks!
Summary about muscle redundancy
The nature and structure of feasible sets
•Co-contraction is often not an option, and in fact loses meaning
The nature and structure of feasible sets
•Co-contraction is often not an option, and in fact loses meaning
•Agonist-antagonist language loses meaning
The nature and structure of feasible sets
•Co-contraction is often not an option, and in fact loses meaning
•Agonist-antagonist language loses meaning•Synergist muscles are not obvious or invariant
The nature and structure of feasible sets
•Co-contraction is often not an option, and in fact loses meaning
•Agonist-antagonist language loses meaning•Synergist muscles are not obvious or invariant•We do not have to settle for optimization approaches. We can know the entire solution space!
The nature and structure of feasible sets
•Co-contraction is often not an option, and in fact loses meaning
•Agonist-antagonist language loses meaning•Synergist muscles are not obvious or invariant•We do not have to settle for optimization approaches. We can know the entire solution space!
•Critical evaluation of synergies
The nature and structure of feasible sets
•Co-contraction is often not an option, and in fact loses meaning
•Agonist-antagonist language loses meaning•Synergist muscles are not obvious or invariant•We do not have to settle for optimization approaches. We can know the entire solution space!
•Critical evaluation of synergies•The statistical structure of the solution space shows the path to probabilistic motor control
The nature and structure of feasible sets
•Co-contraction is often not an option, and in fact loses meaning
•Agonist-antagonist language loses meaning•Synergist muscles are not obvious or invariant•We do not have to settle for optimization approaches. We can know the entire solution space!
•Critical evaluation of synergies•The statistical structure of the solution space shows the path to probabilistic motor control
•Motor control is a Big Data problem
The nature and structure of feasible sets
•Co-contraction is often not an option, and in fact loses meaning
•Agonist-antagonist language loses meaning•Synergist muscles are not obvious or invariant•We do not have to settle for optimization approaches. We can know the entire solution space!
•Critical evaluation of synergies•The statistical structure of the solution space shows the path to probabilistic motor control
•Motor control is a Big Data problem
The nature and structure of feasible sets
•Co-contraction is often not an option, and in fact loses meaning
•Agonist-antagonist language loses meaning•Synergist muscles are not obvious or invariant•We do not have to settle for optimization approaches. We can know the entire solution space!
•Critical evaluation of synergies•The statistical structure of the solution space shows the path to probabilistic motor control
•Motor control is a Big Data problem
The nature and structure of feasible sets
Conclusionsaboutdesignof tendon-drivensystems
• Thehumanhandhascri$calmorphologicalfeatureslendingitverygoodgraspcapabili6es.
• Thegraspingcapabili$esofrobo6chandscanbedras$callyimprovedbyexploringthefulldesignspace
–Non-uniformmaximaltendontension
distribu6ons
–Centerofrota6onsnotinthemiddleofthejoint
52
Roadmap to computational exercises
Brian Cohn
1
ISBN 978-1-4471-6746-4
Biosystems & Biorobotics
Francisco J. Valero-Cuevas
Fundamentals of NeuromechanicsFundamentals of Neuromechanics
Valero-Cuevas
Biosystems & Biorobotics
Francisco J. Valero-CuevasFundamentals of Neuromechanics
This book provides a conceptual and computational framework to study how the nervous system exploits the anatomical properties of limbs to produce mechanical function. The study of the neural control of limbs has historically emphasized the use of optimization to find solutions to the muscle redundancy problem. That is, how does the nervous system select a specific muscle coordination pattern when the many muscle of a limb allow for multiple solutions?
I revisit this problem from the emerging perspective of neuromechanics that emphasizes finding and implementing families of feasible solutions, instead of a single and unique optimal solution. Those families of feasible solutions emerge naturally from the interactions among the feasible neural commands, anatomy of the limb, and constraints of the task. Such alternative perspective to the neural control of function is not only biologically plausible, but sheds light on the most central tenets and debates in the fields of neural control, robotics, rehabilitation, and brain-body co-evolutionary adaptations. This perspective developed from courses I taught to engineers and life scientists at Cornell University and the University of Southern California, and is made possible by combining fundamental concepts from mechanics, anatomy, robotics and neuroscience with advances in the field of computational geometry.
Fundamentals of Neuromechanics is intended for neuroscientists, roboticists, engineers, physicians, evolutionary biologists, athletes, and physical and occupational therapists seeking to advance their understanding of neuromechanics. Therefore, the tone is decidedly pedagogical, engaging, integrative and practical to make it accessible to people coming from a broad spectrum of disciplines. I attempt to tread the line between making the mathematical exposition accessible to life scientists, and convey the wonder and complexity of neuroscience to engineers and computational scientists. While no one approach can hope to definitively resolve the important questions in these related fields, I hope to provide you with the fundamental background and tools to allow you to contribute to the emerging field of neuromechanics.
Engineering
Theoretical framework
http://valerolab.org/fundamentals
Chapters 7, 8 and 9
Define your input-output relation
MusclesTendons
&joints
BonesMuscle
activation(activation space)
Muscleforce
(muscle force space)
Jointtorques
(torque space)
Endpointforces and torques
(wrench space)
Block diagram of transformations in a forward limb model
Input OutputMusclesTeTeT ndons
&joints
BonesMusclefofof rcrcr e
Joinini ttorqrqr ues
EnEnE dpdpd oinini tfofof rcrcr es and torqrqr ues
Outptpt ut
Motor Cotex
Motorneuron
Extensor
Flexor
Define your input-output relation
MusclesTendons
&joints
BonesMuscle
activation(activation space)
Muscleforce
(muscle force space)
Jointtorques
(torque space)
Endpointforces and torques
(wrench space)
Block diagram of transformations in a forward limb model
Input OutputTeTeT ndons&
jointsBones
Joinini ttorqrqr ues
EnEnE dpdpd oinini tfofof rcrcr es and torqrqr ues
Outptpt ut
Motor Cotex
Motorneuron
Extensor
Flexor
Define your input-output relation
MusclesTendons
&joints
BonesMuscle
activation(activation space)
Muscleforce
(muscle force space)
Jointtorques
(torque space)
Endpointforces and torques
(wrench space)
Block diagram of transformations in a forward limb model
Input OutputBonesEnEnE dpdpd oinini t
fofof rcrcr es and torqrqr ues
Outptpt ut
Motor Cotex
Motorneuron
Extensor
Flexor
Define your input-output relation
MusclesTendons
&joints
BonesMuscle
activation(activation space)
Muscleforce
(muscle force space)
Jointtorques
(torque space)
Endpointforces and torques
(wrench space)
Block diagram of transformations in a forward limb model
Input Output
Motor Cotex
Motorneuron
Extensor
Flexor
Find the bounding box of the feasible static force of a 31-muscle 3D cat hindlimb
144 9 The Nature and Structure of Feasible Sets
X
Y TPPTPLPB
FDLFHLLG
SOL*MG
*PLAN
TAEDL*
PYRGMEDGMIN
*RFPSOAS
SART
GMAXQFPECADLSM*ADFBFAGRAC*BFPST*
*EDLVLVI
VM*RF
SM*LGPLAN*MG*GRAC*BFPST*
BFASM
PSOASPECADFADL
GRACSART
BFPRFSTGMAXQFGMINGMEDPYR
MGBFP
VLGRACVIRFLGEDLSMPLANVMST
FHLMG
SOLLG
PLANTP
FDL
TAPBPTEDLPL
−2.00 0.00 2.00
Moment arm (cm) Flex/Ext
−2.00 0.00 2.00
Moment arm (cm) Add/Adb
Z
Y
RFPECADLPYR
GRACBFAADFBFPQF
SARTPSOASSMSTGMAXGMEDGMIN
−2.00 0.00 2.00
Hip Int/Ext Rot (cm)
a. Sagitatal view of right hind limb b. Frontal view of right hind limb
Knee
Ankle
Hip
Fig. 9.3 Moment arms of the 7 DOF, 31 muscle cat hindlimb model [2]. Figure adapted withpermission from [70]
of a cat (Felis catus) with 31 muscles actuating 7 kinematic DOFs from the hip tothe ankle,6 Fig. 9.3.
One application of the techniques described in this book was to find feasible forcesets7 for models of the hindlimb of the cat [71]. A subsequent study explored thebounding box of the feasible activation set for force in one 3D direction [2]. Theyfound that, as the magnitude of the force in that direction increased, the feasibleranges for many muscles remain large even when approaching maximal feasibleforce.
We studied the bounding box of the feasible activation sets for force production inevery 3D direction, also at different force magnitudes [70] because, as motivated bythe previous section, understanding neural control strategies requires that we knowas much as possible about the global properties and structure of feasible activationsets.
In real life, the neural control of musculature must regulate the magnitude and thedirection of force vectors, as in locomotion [72] and manipulation [73]. Therefore,it is of interest how the structure of the feasible activation set changes as you varythe direction of force production.
In response to this need, we developed the vectormap approach as shown inFig. 9.4 [70]. This technique allows one to visualize the properties of the enclosedfeasible set more intuitively. More importantly, by mapping the properties of theenclosed feasible set onto the surface of the sphere, several calculations can be done
6The general model has 7 DOFs, but in this analysis hip ad- ab-duction was frozen so the model isreally a 6 DOF model.7That is, constraints were added to enforce that the torque elements of the output wrench be zero.
X
YTPPTPLPB
FDLFHLLG
SOL*MG
*PLAN
TAEDL*
PYRGMEDGMIN
*RFPSOAS
SART
GMAXQFPECADLSM*ADFBFAGRAC*BFPST*
*EDLVLVI
VM*RF
SM*LGPLAN*MG*GRAC*BFPST*
BFASM
PSOASPECADFADL
GRACSART
BFPRFSTGMAXQFGMINGMEDPYR
MGBFP
VLGRACVIRFLGEDLSMPLANVMST
FHLMG
SOLLG
PLANTP
FDL
TAPBPTEDLPL
��� ���� ����
Moment arm (cm) Flex/Ext
��� ���� ����
Moment arm (cm) Add/Adb
Z
Y
RFPECADLPYR
GRACBFAADFBFPQF
SARTPSOASSMSTGMAXGMEDGMIN
��� ���� ����
Hip Int/Ext Rot (cm)
D��6DJLWDWDO�YLHZ�RI�ULJKW�KLQG�OLPE E��)URQWDO�YLHZ�RI�ULJKW�KLQG�OLPE
Knee
Ankle
Hip
Implement Vectormap
9.4 Vectormap Description of Feasible Sets 145
10 20 30 40 50 60Feasible force output for a given direction (N)
2D Feasible force set 3D Feasible force set Spherical vectormap
60N{
dorsal
ventral
ante
rior
post
erio
r
(a) (b) (c)
Fig. 9.4 Vectormap visualization for a cat hindlimb in a static force production task. a Take theexample of a 2D feasible force set, where that polygon is enclosed in a circle. The thin black linesemanating from the origin are the lines of action of each of the 31 muscles. The distance from theorigin to the boundary of the feasible force set (i.e., the maximal force in that direction) is assigneda color, blue for small and yellow for large, as shown for the eight rays emanating from the origin.That color-code is vectormapped onto that point on the circle as shown for the posterior, anterior,dorsal and ventral directions, and 4 others in between. b The same can be done for a 3D feasibleforce set that is a polyhedron enclosed in a sphere (only the cross-section of the sphere is shown).c The color on the surface of the sphere now contains all the information of the enclosed feasibleforce set—but in a more intuitive way that can be compared across directions and 3D feasible sets.Adapted with permission from [70]
on that manifold. For example, one can find the difference, sum, or average of twosuch feasible sets. For example, Fig. 9.5 shows the average and standard deviationmaximal feasible force from three different cat models with different limb anatomyand moment arms. The vectormap approach is novel and useful because it allows usto generate cross-species, inter-species, and inter-task comparisons.
This same methodology can be used to represent feasible activations. Take forexample the feasible activations associated with the spherical vectormap of the fea-sible force set shown in Fig. 9.4c. For each 3D direction of the feasible force set,that muscle is associated with a particular activation level. These values are foundfrom the optimal, unique coordination patterns that produce maximal force in eachof those directions mapped onto the spherical vectormap of the feasible force set. Ifyou collect all such unique values, they are in fact a 3D object, whose vectormapcan be shown in 3D. Consider the case of one single muscle, say the vastus lateralis,as shown in the top row of Fig.9.6. The large vectormap at the far right shows the
Species average
5
10
15
20
25
30
35
40
45
50
55
dorsalposterior
Force (N)
Mean
SD
Same orientation as above
medial anteriorventral
lateral
Orientation
Implement Vectormap9.4 Vectormap Description of Feasible Sets 147
Vastus LateralisLower bounds
Muscle activation0 1.00.5
Upper bounds
α = 50% α = 60% α = 70% α = 80% α = 90% α = 100% (fmax)
Medial Gastrocnemius
Soleus
dorsal
posterior lateral
medialventral
anterior
Fig. 9.6 Vectormaps of the feasible activations sets for sub-maximal and maximal force out-put in every 3D direction. The vectormaps for 5 different sub-maximal force levels (α =50, 60, 70, 80, and 90%) for each of 3 muscles are shown, plus the larger vectormap for the maxi-mal feasible force magnitude. The top and bottom rows are, for each muscle respectively, obtainedfrom the maximal and minimal values of their bounding boxes. Figure adapted with permissionfrom [70]
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The Whitaker FoundationBiomedical Engineering Research Grant
The National Institutes of HealthNIAMS/NICHD R01-AR050520; R01-AR052345: R21-HD048566
Alexander von Humboldt FoundationMax Plank Institute for Human Brain and Cognitive
Sciences
Wenner-Gren FoundationKarolinska Institute
Department of EducationNIDRR OPTT-RERC