やわらかい知能中嶋浩平
東京大学大学院情報理工学系研究科
2022/09/09,
東京大学本郷キャンパス工学部6号館61講義室

目次
1.やわらかい身体に制御をアウトソースする
2.脳–身体–環境系に対する力学系的アプローチ

3.やわらかい身体の情報処理能力

目次
1.やわらかい身体に制御をアウトソースする→embodiment ：脳（制御機）‐身体‐環境系

2.脳–身体–環境系に対する力学系的アプローチ→非線形力学系の紹介

3.やわらかい身体の情報処理能力→physical reservoir computing

やわらかい身体に制御をアウトソースする

Embodiment(R. Pfeifer et al., Science, 2007)

The behavior of robots (or animals) is generated by the dynamic
coupling between the controller (brain), the body, and the environment!

やわらかい身体の情報処理能力

Computing with soft body dynamics?

Reservoir computing: basic settings
WReservoir
𝑓 𝑥 = tanh(𝑥)

Wout

Win

Adjust only thereadout!

readout

𝑊𝑜𝑢𝑡 = 𝑋 𝑇 𝑋
inputsH. Jaeger et al., Science, Vol.304.
no.5667, pp.78–80 (2004).

(Good points)

outputs−1𝑋𝑇𝑦

Use 𝑊𝑜𝑢𝑡 for
information processing!

𝑦ො = 𝑋𝑊𝑜𝑢𝑡(Computational power)• Learning is fast and stable!
• No local minimum problem!
• Feasible for physical platform.

• Nonlinearity• Memory

*Prerequisite: “reproducible response”(target function)𝑦𝑘+1 = 𝑔(𝑢𝑘 , 𝑢𝑘−1 , … )(reservoir computing)𝒙𝒌+𝟏 = 𝑓 𝒙𝒌 , 𝑢𝑘
෠ 𝒌+𝟏 )𝑦ො𝑘+1 = 𝜓(𝒙෠ 𝒌+𝟏 )𝑔(𝑢𝑘 , 𝑢𝑘−1 , … ) ≈ 𝜓(𝒙We want to emulate(learn) function “g”!

(prerequisite)• Reproducible response to the same input sequence!
• Reservoir states should not depend on the initial condition!

𝑓 𝒙𝒌 , 𝑢𝑘 − 𝑓 𝒙𝒌∗ , 𝑢𝑘 ≈ 0(common-signal-induced synchronization/Negative conditional Lyapunov exponents)
Z. Lu, B. R. Hunt, E. Ott, Chaos 28, 061104 (2018).

𝒙𝒌 = 𝝓(𝑢𝑘−1 , 𝑢𝑘−2 , … )(echo state property (ESP))
H. Jaeger, GMD Technical Report. 148 (2001).
I. B. Yildiz, et. al., Neural netw. 35 (2012).
G. Manjunath, et. al., Neural comp. 25 (2013).

力学系の容量(Capacity)
sequence of h previous inputs:

Dambre J., Verstraeten D., Schrauwen B.,
Massar S. Sci. Rep. 2:514, 2012.

target function:system output:

Xを使って線形にzを近似したときの最小のMSE

𝐶𝑇 [𝑋, 𝑧]=0: 全く近似できてないとき
𝐶𝑇 [𝑋, 𝑧]=1: 完全に近似できるとき

𝐶𝑇 𝑋, 𝑧 >0である場合、「力学系Xは関数zを容量𝐶[𝑋, 𝑍]で近似できる」という。

Capacityの公式
𝑚𝑖𝑛𝑊𝑖 𝑀𝑆𝐸𝑇 𝑧Ƹ
𝐶𝑇 𝑋, 𝑧 = 1 −𝑧2 𝑇=1−
𝑇𝑧 − 𝑋𝑤𝑜𝑢𝑡 2𝑧𝑇 𝑧
𝑇𝑧 𝑋 𝑋 𝑋=𝑧𝑇𝑧（今、 𝑤𝑜𝑢𝑡 =−1𝑇

𝑋 𝑧−1 𝑇
𝑇𝑋 𝑋 𝑋 𝑧）

Reservoir𝑑𝑖

Wout

𝑦1𝑑𝑖
Inputstream𝑢(𝑘)(u(k) should be i.i.d!)𝑦 2𝑑𝑖・・・𝑦 𝑀𝑑𝑖

delay (memory)
𝑃0 (𝑢 𝑡 ) 𝑃0 (𝑢 𝑡 − 1 ) … 𝑃0 (𝑢 𝑡 − 𝑖 )… 𝑃1 (𝑢 𝑡 − 𝑖 )𝑃2 (𝑢 𝑡 ) 𝑃2 (𝑢 𝑡 − 1 ) … 𝑃2 (𝑢 𝑡 − 𝑖 )𝑃3 (𝑢 𝑡 ) 𝑃3 (𝑢 𝑡 − 1 ) … 𝑃3 (𝑢 𝑡 − 𝑖 )𝑃1 (𝑢 𝑡 ) 𝑃1 (𝑢 𝑡 − 1 )…

outputs 0𝑦degree (nonlinearity)

容量の総和を定量する手順

𝑃𝑛 (𝑢 𝑡 ) 𝑃𝑛 (𝑢 𝑡 − 1 )… 𝑃𝑛 (𝑢 𝑡 − 𝑖 )Choose the polynomialfor each delay!

Total capacity

Dambre J., Verstraeten D., Schrauwen B., Massar S.
Sci. Rep. 2:514, 2012.

＊𝜖で閾値を設ける• Emulate all the combinations ofLegendre polynomial (orthogonal)!
• For each delay, one nth degree
polynomial!

linear memory

Non-linearmemory重要な性質• Reservoir Xがfading memory (echo state property)を持つとする。この時、N個の読出し口（この相関行列がフルランクとする）を用いて近似可能なすべての直交関数に対する容量の総和はNになる。

(詳細はDambre J., et. al., Sci. Rep. 2:514, 2012.のtheorem 7)

Echo state propertyを持つ力学系からN個の読出し関数があれば、全直交多項式に対する容量の総和はNとなる！（かなり強力）Physical Reservoir Computing

K. Nakajima et. al., Front. Comput. Neurosci., 7 (91), 2013.
K. Nakajima et. al., J. R. Soc. Interface 11: 20140437, 2014.
K. Nakajima et. al., Scientific Reports 5: 10487, 2015.
K. Nakajima et. al., Soft Robotics 5(3): pp.339-347, 2018.

Soft body dynamics ascomputational resource!

Arm behavior with τstate=5

K. Nakajima et. al., J. R. Soc. Interface 11: 20140437, 2014.

How does octopus arm store memory?
Only from current arm’s shape, infer previous inputs!

• Can infer previous inputs from arm’s shape!
• Rigid stick does not have this property!

ESP in the octopus arm

K. Kagaya, B. Yu, Y. Minami, K. Nakajima, inProceedings of 2022 IEEE 5th International
Conference on Soft Robotics (RoboSoft), pp. 224-230,
2022.

Injecting identical input series to different environmental conditions:air, sea water, tap water
Common-signal-induced synchronization observed in
the tap water condition but not in the air condition!

Closed loop control with τsquare =10(manual perturbation)

K. Nakajima et. al., J. R. Soc. Interface 11: 20140437, 2014.

Soft keyboard: drawing cursive letters via soft materialson“Soft keyboard”

offDraw cursive letters!

s1….

s2sN
Push soft keyboard andinduce deformation!

Deformation is monitored
as a sensory time series! time

• Push soft keyboard and inducedeformation!
• Encode your cursive letters into thedeformation dynamics!
• Control to draw with soft push!

Time series data as acomputational resource !

Another design…
(soft smartphone, etc.)

Soft modules and sensors
finger motions
uprightleft

downPress and swipe!

(“down” motion in video!)(Collaboration with Dr. Alexander Schmitz)
T.P. Tomo et al., IEEE-RAS International Conference on
Humanoid Robots (2016))• 16 force sensors (3axis) andcorrespondingvelocities.
• 16×3×2=96 nodes• Real-time monitoringof soft deformation!

s(t)Sensory response curves

up

down

leftv(t)

right

time

time

timetime

Each finger motion generates specific sensory patterns (20 trialsaveraged)!
K. Nakajima, et. al., in Proceedings of NOLTA2018, pp. 147-150, 2018.

(Process flow)

Demonstration

Training
datacollection!

Mechanicalinput••fingerPush and slide!

Encoding
the letters!

••

Draw cursive letters using a
digital pen!
Collect sensory data for thecorresponding finger
motion! (15 trials: training, 5trials: evaluation)
Using Ridge regression,tune the readout weights!
Correspond sensory data
and a cursive letter!

Soft material
readout

Drawing
the letters!

...
“b”
“a”
(Δx, Δy)“c”“d”•(overall system)“z”

Evaluate the performanceusing trained readout
weights!

•All theprocesses
take only afew minutes!

Multitasking (readout for each alphabet)
K. Nakajima, et. al., in Proceedings of NOLTA2018, pp. 147-150, 2018.

• Target cursive letters: uppercase and lowercase letters in the alphabet, numbers.
• Finger motions: “up,” “right,” “down,” “left”• For a single motion, encode multiple cursive letters by training readout weights foreach (multitasking) !

Results （using sensory dynamics of “UP” in subject 1 with multitasking）

target output
system output

Physical reservoirs …
Liquid brain

K. Nakajima, Physical reservoir computing---anintroductory perspective, Jap. J. Appl. Phys. 59:060501, 2020.

Cultured neural networks

water in a bucket
culturedneural networks

C. Fernando et. al., Lec. Comp. Sci. 2801 (2003).

Soft robots

origami robots

M. R. Dranias, et. al., J. Neurosci. 33, 1940 (2013).
T. Kubota, et. al., Lect. Comp. Sci. 11731 (2019).
Y. Yada, et. al., Appl. Phys. Lett., 119(17), 173701 (2021).

tensegrity structures

Q. Zhao et al., Proceedings of IROS, pp. 1445-1451 (2013).
K. Nakajima et al. J. R. Soc. Interface. 11: 20140437 (2014).
K. Caluwaerts et al. J. R. Soc. Interface 11:98 (2014).
K. Nakajima et al. Sci. Rep. 5: 10487 (2015).
K. Nakajima et al. Soft Robotics 5: 10487 (2018).
P. Bhovad, et. al., Sci. Rep. 11(1), 1-18 (2021).

quadruped robots

PRC for neuromorphic devices
Photonic reservoirs

In-Materia reservoirs

L. Larger, et. al., Opt. Express 20, 3241 (2012).
D. Brunner, et. al., Nat. Commun. 4, 1364 (2013).
K. Vandoorne, et. al., Nat. Commun. 5, 3541 (2014).
L. Larger, et. al., Phys. Rev. X 7, 011015 (2017).

E. C. Demis et al. Nanotechnology 26:204003 (2015).
A. Z. Stieg et al. Adv. Mater. 24:286-293 (2012).
M. Cucchi, et. al., Science Advances, 7(34), eabh0693 (2021).
Y. Usami, et. al., Adv. Mater. (2021).

Spintronics reservoirs

Quantum reservoirs

J. Torrejon et al., Nature 547, 428 (2017).
T. Furuta, et. al., Phys. Rev. Appl. 10, 034063 (2018).
S. Tsunegi, et. al., Appl. Phys. Lett. 114, 164101 (2019).
N. Akashi, et. al., Phys. Rev. Res. 2: 043303 (2020).

K. Fujii, K. Nakajima, Phys. Rev. Appl. 8: 024030 (2017).
K. Nakajima, et. al., Phys. Rev. Appl. 11: 034021 (2019).
S. Ghosh, et. al., Adv. Quantum Technol. 4: 2100053 (2021).
Q. H. Tran, K. Nakajima, Phys. Rev. Lett. 127: 260401 (2021).

Thank you!