SUGISAKAMASANORI

Journal of Robotics, Networking and Artificial Life
Vol. 9(4); March (2023), pp. 309–315
ISSN (Online):2352-6386; ISSN (Print):2405-9021
https://alife-robotics.org/jrnal.html

Ryuichi Usami, Hideyuki Tanaka
Graduate School of Humanities and Social Sciences, Hiroshima University, 1-1-1 Kagamiyama, Higashi-Hiroshima city, Hiroshima, 739-8524 Japan

Received 24 November 2021
Accepted 24 September 2022

Leap motion controller
Modeling of finger motions
State-space representation
Step response
N4SID

Modeling the motion of experts in sports is important in demonstrating their motion to beginners.
A primary approach to this objective is to study the dynamics of a transient response from one
position to another. Hence, finger-motion analysis is a primitive but crucial step to be considered.
The state-space representation with a step input is used to eliminate the personal habits of experts
from their motion and to present simple models to learners according to their learning stages.
Based on the deterministic realization and N4SID algorithms, two algorithms are developed todemonstrate simple motions according to the order of the state-space representation. They areapplied to the finger motion captured by the Leap Motion Controller.
© 2022 The Author. Published by Sugisaka Masanori at ALife Robotics Corporation Ltd.
This is an open access article distributed under the CC BY-NC 4.0 license(http://creativecommons.org/licenses/by-nc/4.0/).

expected to be effective in learning the motion. However,in sports, advanced techniques and skills, including
individual habits, are extremely complex for beginners.
Hence, we believe that a simple model that eliminates
these factors would facilitate motion learning by
beginners. We therefore introduce the different levels of
simplification by means of the order of the system.

1. Introduction
In sports, player movements have been measured using
sensors. For example, motion sensors were used tocapture the movements of the golf swing [1], andpressure sensor systems were presented to monitor the
ankle supination torque during sport motions [2]. A
wearable inertial sensor network and its associatedactivity recognition algorithm were proposed toaccurately recognize human activities in daily life andsports [3]. Motion capture has also been used to analyze
sports performance; see a systematic review on the recent
developments of motion capture systems for the analysisof sport performance [4]. Moreover, a system applicableto the marker-less sport-movement analysis has beenpresented [5].

In this study, we develop mathematical models to provide
a simplified model for beginners and a complicated one
for advanced learners. We introduce the state-space
representation driven by a step input, often used formodeling in control engineering [7]. We use the statespace representation, because it can handle dynamics ofmultiple inputs and multiple outputs, and a simplified
model can be obtained by reducing the order of the statespace representation. Moreover, using a step input
response of the state-space representation, it will bepossible to develop models of a transient response fromone position to another such as a crouching start of a

The demonstration of models for learners has provenparticularly effective in enhancing motor learning [6].
Presenting a model of an expert to beginners is therefore
Corresponding author’s E-mail: tanakalpha@hiroshima-u.ac.jp309

where 𝑝𝑥 (𝑡) , 𝑝𝑦 (𝑡) , and 𝑝𝑧 (𝑡) ∈ 𝐑21 denote thepositions in the X-Y-Z coordinates of the 21 joints of thefingers, respectively. The positions of finger joints aremeasured over the same sampling period between the
movement from the open hand to the grasped hand.

short-distance race. As a simple example of motion fromone position to another, we will model a single grasping
motion of a hand, because the motions of finger joints are
related to each other and so are the motions of humanbody joints in the transient response of sports movements.
We will apply the deterministic realization [8] andN4SID algorithms [9] to the problem. They are suitable
for this study, because the order of the model can besystematically determined by computing the singularvalue decomposition (SVD) [7]. Since the deterministic
realization algorithm was developed to determine a statespace model from an impulse response, we modify thealgorithm to obtain a state-space model of finger motion.
Since N4SID is applicable only to data with persistently
excited inputs, we reconfigure the input and output data.
We compare these algorithms by applying them to fingermotion data.

(a) start of measurement

We utilize the Leap Motion Controller (LMC), an optical
hand tracking module that captures hand movements[10]. The accuracy and robustness of the LMC have been
analyzed [11], and its performance was evaluated using aprofessional, high-precision, fast motion tracking system[12].

(c) measuring points(d) coordinate system
Fig. 1 Finger motion and measurement system
We measure the finger positions during the grasping
motion, using Processing [13] [14]. The time intervalfrom the start to end is 1(s), and 100 data points are
sampled during 1(s). The countdown from 5(s) before thestart time is displayed on the console.

The remainder of this paper is organized as follows:
Section 2 describes how the finger motions are measured,and Section 3 formulates the problem setting. Sections 4
and 5 solve the problems using deterministic realization
and N4SID, respectively. Section 6 presents
experimental results, and Section 7 concludes the paper.

3. Problem Setting
We formulate the problem settings. Suppose that the
sampling time is ℎ and let 𝑦𝑘 be the coordinate 𝑦(𝑡) at
𝑡 = 𝑘ℎ:

2. Measuring Data𝑦𝑘 = 𝑦(𝑘ℎ).

We describe how the finger motions are measured. Fig.
1(a) and Fig. 1(b) illustrate the start (open hand) and end(grasped hand) of transient motion, respectively. The
LMC can measure 21 finger joints in a three-dimensional
Cartesian coordinate system, which implies that the LMC
obtains 63 elements for each sample. Fig. 1(c) and Fig.
1(d) present the measuring points and the coordinate
system, respectively. We define the output 𝑦(𝑡) as𝑦(𝑡) = [𝑝𝑥 (𝑡)⊤𝑝𝑦 (𝑡)⊤(b) end of measurement(1)

We model 𝑦𝑘 by means of the discrete-time state-space
representation:
𝑥𝑘+1 = 𝐴𝑥𝑘 + 𝐵𝑢𝑘 ,𝑦𝑘 = 𝐶𝑥𝑘 + ζ,

(2𝑎)
(2𝑏)

where 𝑥𝑘 is a state of the system, 𝑥0 = 0, ζ is a constantvector, and 𝑢𝑘 is as follows:⊤ ⊤𝑝𝑧 (𝑡) ] ∈ 𝐑63 ,𝑢𝑘 = {
310

0 (𝑘 = −1, −2, … )1 (𝑘 = 0, 1, 2, … )(3)

We suppose that 𝑘 = 0 when the transient motion starts
and that there is no explicit delay in starting the finger
movement after the start time is displayed. We consider
the following problems:

For positive integers 𝜏 > 𝑛 and 𝑁 > 𝑛 , define the
Hankel matrix 𝐻 ∈ R𝜏𝑝×𝑁 from 𝑣𝑘 as follows:
𝑣1
𝑣2
𝐻=[⋮
𝑣𝜏

Problem 1: Find (𝐴 , 𝐵 , 𝐶 , ζ) within the degrees of
freedom of similarity transformations, given 𝑦𝑘 in (2)and (3).

⋯⋯⋮⋯𝑣𝑁+1−𝜏
𝑣𝑁+2−𝜏
⋮ ].
𝑣𝑁

(7)

We also define the extended observability andreachability matrices as follows:

Problem 2: Suppose that the positions of the finger joints
are measured as presented in (1). Find a mathematicalmodel for 𝑦𝑘 in (2) and (3).

𝒪𝜏 = [𝐶 ⊤ (𝐶𝐴)⊤ ⋯ (𝐶𝐴𝜏−1 )⊤ ]⊤ ,𝒞𝒩 = [𝐵 𝐴𝐵 ⋯ 𝐴𝑁−1 𝐵 ].

The complexity of the finger-motion model in (2) isdetermined by the size of the matrix 𝐴 or the order of thesystem. The lower the order, the simpler the model is.
Conversely, the higher the order, the more accurate thefinger motion is, and the closer the motion is to theoriginal one.

(8𝑎)
(8𝑏)

From (6), 𝐻 satisfies the following equation:(9)

𝐻 = 𝒪𝜏 𝒞𝒩 .

We compute the singular value decomposition (SVD):
𝑈2 ] [𝛴10⊤= 𝑈1 𝛴1 𝑉1 ,𝐻 = 𝑈𝛴𝑉 ⊤ ≈ [𝑈1

By selecting the order of the model (2), we can generatesamples of the motion of experts for different levels oflearners. It is expected that beginners will learn the
motions by simpler dynamical models, whereas
advanced learners will learn the motions by more
complex models. It should be noted that the dynamicalsystem (2) can model the transition from one point toanother, although we model finger motions in this study.
4.

𝑣2
𝑣3
⋮
𝑣𝜏+1

0 𝑉1⊤
][ ]
0 𝑉2⊤
(10)

where 𝛴 is a diagonal matrix, and 𝑈 and 𝑉 are
orthogonal matrices satisfying 𝑈 ⊤ 𝑈 = 𝐼 and 𝑉 ⊤ 𝑉 = 𝐼 .
From (9), 𝒪𝜏 and 𝒞𝒩 are expressed as
1𝒪𝜏 = 𝑈1 𝛴12 ,1

𝒞𝒩 = 𝛴12 𝑉1⊤ .

(11)

Solution via Deterministic Realization
From (8), 𝐶 and 𝐵 can be obtained as follows:

We solve Problem 1 using the deterministic realization
algorithm [8] and apply the algorithm to Problem 2. We
consider Problem 1. Let us describe 𝑦𝑘 for (2) as

𝐶 = 𝒪𝜏 (1: 𝑝, : ),

(4)

𝑗=1

𝒪𝜏↓ = 𝒪𝜏 (1: 𝑝(𝜏 − 1), : ),𝒪𝜏↑ = 𝒪𝜏 (𝜏 + 1: 𝑝𝜏, : ).

and let 𝑣𝑘 be as follows (𝑘 ≥ 1):
𝑣𝑘 = 𝑦𝑘 − 𝑦𝑘−1 .

(13𝑎)
(13𝑏)

From (8a), we have(5)𝒪𝜏↓ = [𝐶 ⊤ (𝐶𝐴⊤ ) ⋯ (𝐶𝐴𝜏−2 )⊤ ]⊤ ,𝒪𝜏↑ = [(𝐶𝐴)⊤ (𝐶𝐴2 )⊤ ⋯ (𝐶𝐴𝜏−1 )⊤ ]⊤ ,The signal 𝑣𝑘 is the difference of the output 𝑦𝑘 . From (4),𝑣𝑘 satisfies the following equation:
𝑣𝑘 = 𝐶𝐴𝑘−1 𝐵.

(12)where we use the colon notation [15]. Let us define 𝒪𝜏↓and 𝒪𝜏↑ as follows:𝑘𝑦𝑘 = ∑ 𝐶𝐴 𝑗−1 𝐵 + ζ,

𝐵 = 𝒞𝒩 (: ,1),and hence

(6)
311

(14𝑎)
(14𝑏)

𝒪𝜏↓ 𝐴 = 𝒪𝜏↑ .

number of the non-zero diagonal elements of 𝐻 in (10).
Thus, we simplify the state-space model (3) and obtain asimplified transient movement from the open hand to thegrasped hand. The lower the order of the state-spacemodel is, the simpler the transient motion becomes.

(15)

We can thus obtain 𝐴 by solving the least-squares
method. Let the estimates of 𝐴, 𝐵, 𝐶 and ζ be denoted as
𝐴̂, 𝐵̂ , 𝐶̂ , and ζ̂, respectively. By setting the initial state as𝑥̂0 = 0, we compute 𝜂𝑘 for 𝑘 = 0, ⋯ , 𝑁 as follows:𝑥̂𝑘+1 = 𝐴̂𝑥̂𝑘 + 𝐵̂𝑢𝑘 ,𝜂𝑘 = 𝐶̂ 𝑥̂𝑘 ,5.

We solve Problem 1 by using the N4SID algorithm [9].
The N4SID algorithm can also systematically determine
the order of the model by computing the SVD. However,
N4SID cannot be directly applied to the system in (2).
Moreover, the input data in (3) do not satisfy theassumptions of N4SID. We therefore demonstrate how toapply the input-output data to N4SID.

(16𝑎)
(16𝑏)

where 𝑢𝑘 is the step input in (3). We then have 𝜂𝑘 =∑𝑘𝑗=1 𝐶̂ 𝐴̂ 𝑗−1 𝐵̂ ≈ ∑𝑘𝑗=1 𝐶𝐴 𝑗−1 𝐵 and hence from (4):𝑘

ζ = 𝑦𝑘 − ∑ 𝐶𝐴 𝑗−1 𝐵 ≈ 𝑦𝑘 − 𝜂𝑘Let the number 𝑁 of given data be as follows:𝑗=1

𝑁 = 2𝜏 + 𝑙 − 1

We therefore estimate ζ, by averaging 𝑦𝑘 − 𝜂𝑘 :1∑(𝑦𝑘 − 𝜂𝑘 ) .
𝑁+1

(17)

𝑢0 𝑢1 ⋯ 𝑢𝑙−1
𝑢1 𝑢2 ⋯𝑢𝑙
𝑈𝑝 = [ ⋮⋮⋮⋮ ],𝑢𝜏−1 𝑢𝜏 ⋯ 𝑢𝜏+𝑙−2
𝑢𝜏𝑢𝜏+1 ⋯ 𝑢𝜏+𝑙−1
𝑢𝜏+1 𝑢𝜏+2 ⋯𝑢𝜏+𝑙
𝑈𝑓 = [ ⋮⋮⋮⋮ ].
𝑢2𝜏−1 𝑢2𝜏 ⋯ 𝑢2𝜏+𝑙−2

𝑘=0

Thus, estimates of (𝐴, 𝐵, 𝐶, ζ) for Problem 1 are obtainedby (𝐴̂, 𝐵̂ , 𝐶̂ , ζ̂), and those of 𝑦̂𝑘 in (2) are given by𝑥̂𝑘+1 = 𝐴̂𝑥̂𝑘 + 𝐵̂𝑢𝑘 ,𝑦̂𝑘 = 𝐶̂ 𝑥̂𝑘 + 𝜁̂.

(𝜏 > 𝑛).

(19)

Using the input 𝑢𝑡 and output 𝑦𝑡 , define

𝑁

ζ̂ =

Solution via N4SID

(18𝑎)
(18𝑏)

(20𝑎)

(20𝑏)

In applying N4SID, the following assumption is made:We summarize the above procedure for Problem 1 as thefollowing algorithm:

𝑈𝑝rank ([ ]) = 2𝜏𝑚,
𝑈𝑓

Algorithm based on deterministic realization:
Step 1: Calculate 𝑣𝑘 in (5).
Step 2: Construct the block Hankel matrix 𝐻 in (7).
Step 3: Compute the SVD in (10).
Step 4: Determine 𝒪𝜏 and 𝒞𝒩 as in (11).
Step 5: Compute the estimate (𝐴̂, 𝐵̂ , 𝐶̂ ) of (𝐴, 𝐵, 𝐶) from
(12) and (15).
Step 6: Obtain an estimate ζ̂ of ζ as (17) and calculate 𝑦̂𝑘in (18).

(21)

where 𝑚 = 1 is the size of input. However, the input data𝑢𝑡 = 1 in (3) satisfy
rank ([

𝑈𝑝]) = 1.
𝑈𝑓

We reconfigure the data to satisfy (21). Defining
𝑢𝑘 = 0 (𝑘 = −1, ⋯ , −𝑀),
𝑦𝑘 = 𝑦0 (𝑘 = −1, ⋯ , −𝑀),

By applying this algorithm to Problem 2, we havemathematical models for finger motions. The model issimplified by the SVD of the Hankel matrix in (10), andthe order is determined by the number of the dominant
singular values. We select the order by choosing the

we determine new input and output (𝑘 = 0, 1, . . . , 𝑁 +𝑀 − 1), respectively:𝑢̅𝑘 = 𝑢𝑘−𝑀 − 𝜇𝑢 ,
312

(22𝑎)

(22𝑏)𝑦̅𝑘 = 𝑦𝑘−𝑀 − 𝜇𝑦 ,𝑉⊤𝑈2 ] [𝛴1 0] [ 1⊤ ]0 0 𝑉2= 𝑈1 𝛴1 𝑉1⊤ .

𝑂𝑋̅ ≈ [𝑈1where 𝜇𝑢 and 𝜇𝑦 are computed as follows:
𝑁−1

𝜇𝑢 =

𝑁−1

1
1
∑ 𝑢𝑡 , 𝜇 𝑦 =∑ 𝑦𝑡 ,
𝑁+𝑀
𝑁+𝑀
𝑡=0

Approximating 𝒪𝜏 𝑋̅𝑓 ≈ 𝒪̂𝜏 𝑋̂𝑓 , 𝒪̂𝜏 and 𝑋̂𝑓 are as follows:𝑡=01𝒪̂𝜏 ≈ 𝑈1 𝛴12 ,since N4SID is applied to the input-output data withmean zero. Thus, we define the following matrices:𝑢̅0 𝑢̅1 ⋯ 𝑢̅𝑙−̅ 1
𝑢̅
𝑢̅2 ⋯
𝑢̅𝑙 ̅
𝑈𝑝 = [ 1],⋮⋮⋮⋮𝑢̅𝜏−1 𝑢̅𝜏 ⋯ 𝑢̅𝜏+𝑙 −̅ 2𝑢̅𝜏𝑢̅𝜏+1 ⋯ 𝑢̅𝜏+𝑙−̅ 1𝑢̅𝑢̅𝜏+2 ⋯𝑢̅𝜏+𝑙 ̅𝜏+1
̅𝑓 = [
𝑈],⋮⋮⋮⋮𝑢̅2𝜏−1 𝑢̅2𝜏 ⋯ 𝑢̅2𝜏+𝑙 −̅ 2

̅𝑝𝑈]) = 2𝜏,̅𝑓
𝑈𝑋̃𝜏 = 𝑋̂𝑓 (: ,1: 𝑙 ̅ − 1),̅𝑋̃𝜏+1 = 𝑋̂𝑓 (: ,2: 𝑙 ).

(23𝑎)

[𝐴̂

(24)

].

(25)

(31)

By applying this algorithm to Problem 2, we also havemodels for finger motions. The model is simplified bycomputing the SVD of the Hankel matrix in (28). It
should be noted that both algorithms provide simplified
model based on the SVDs; however, the models obtained
from them are different, because different matrices areused in (10) and (28). Thus, in Section 6, we compare
different simplifications, using both the algorithms basedon deterministic realization and N4SID.

We have the following equations:
(26𝑎)
(26𝑏)

where 𝑋̅𝑝 and 𝑋̅𝑓 are as follows:𝑋̅𝑝 = [𝑥̅0 𝑥̅1 ⋯ 𝑥̅𝑙 −̅ 1 ],𝑋̅𝑓 = [𝑥̅𝜏 𝑥̅𝜏+1 ⋯ 𝑥̅𝜏+𝑙−̅ 1 ],𝐶̂ = 𝑌̅𝜏 𝑋̃𝜏† ,Algorithm based on N4SID:
Step 1: Calculate {𝑢̅𝑘 , 𝑦̅𝑘 } as (22).
̅𝑝 , 𝑈̅𝑓 , 𝑌̅𝑝 and 𝑌̅𝑓 as (23).
Step 2: Construct 𝑈
Step 3: Compute the SVD in (28).
Step 4: Find an estimate of the state as (29).
Step 5: Compute the estimate (𝐴̂, 𝐵̂ , 𝐶̂ ) from (31).
Step 6: Obtain an estimate ζ̂ of ζ as (17).

𝐷̅𝑝,𝑌̅𝑝 = 𝒪𝜏 𝑋̅𝑝 + 𝒯𝜏 𝑈
̅𝑓 ,𝑌̅𝑓 = 𝒪𝜏 𝑋̅𝑓 + 𝒯𝜏 𝑈

†
𝑋̂𝜏
̃
]
̂
=
𝑋
[
]
,
𝐵
𝜏+1 ̅
𝑈𝜏

We summarize the above procedure for Problem 1:

0
⋱
𝐶𝐵

(30𝑎)
(30𝑏)

where (⋅)† expresses the pseudo-inverse. We compute ζ̂from (17) using (𝐴̂, 𝐵̂ , 𝐶̂ ).

̅𝑝 ,𝑈̅𝑓 , 𝑌̅𝑝
We apply N4SID to the input and output data 𝑈
and 𝑌̅𝑓 and explain how to select the order. We define theblock Toeplitz matrix as follows:
𝐷
⋱
⋯

(29)

The matrices 𝐴, 𝐵 and 𝐶 are estimated as follows.

(23𝑏)

which satisfies the assumption of N4SID in (21).

𝐷
𝐶𝐵
𝒯𝜏 = [⋮
𝐶𝐴𝜏−2 𝐵

1𝑋̂𝑓 ≈ 𝛴12 𝑉1⊤ .

Using 𝑋̂𝑓 , define the followings:

where 𝑙 ̅ = 𝑁 + 𝑀 − 2𝜏 + 1. We construct 𝑌̅𝑝 and 𝑌̅𝑓 in
the same way, using 𝑦̅𝑡 . Then, we have:rank ([

(28)

(27𝑎)
(27𝑏)

6. Experimental Results

and where 𝑥̅𝑘 = 0 (𝑘 = 0, . . . , 𝑀 − 1) and 𝑥̅𝑘 = 𝑥𝑘−𝑀
(𝑘 = 𝑀, … , 𝑁 + 𝑀 − 1) . We estimate 𝒪𝜏 𝑋̅𝑓 based on
N4SID. Describing the estimate of 𝒪𝜏 𝑋̅𝑓 as 𝑂𝑋̅ , wecalculate the SVD as follows:

We demonstrate the algorithms by experimental
results.We present 𝑝𝑦 (𝑡) at the measuring point 13,
which is the tip of the middle finger with one of thelargest motions among the measuring points of fingers.
313

contrast, the high-order models (𝑛 = 12, 18) given by
the deterministic realization provide the motion close to
the original one, although the motion given by the
reduced-order models (𝑛 = 4, 7) is far away from theoriginal one. From this experiment, we infer that the
N4SID-based algorithm gives simple models that moreaccurately capture the motion of fingers than thedeterministic-realization-based algorithm.

In Fig. 2 and Fig. 3, the horizontal and vertical axes
represent the time (s) and position (mm) of 𝑝𝑦 (𝑡) ,respectively. The red lines in Fig. 2 and Fig. 3 indicate
the position of the original motion, whereas the blue ones
show models with the order of 4, 7, 12, and 18. Fig. 2 andFig. 3 demonstrate that the output of models (blue lines)
becomes closer to the original motion (red lines) as theorder is higher. By comparing blue lines for 𝑛 = 4 and𝑛 = 18, it is observed that the motion drawn by the blueline for 𝑛 = 4 is much simpler than that for 𝑛 = 18 inboth Fig. 2 and Fig. 3.

We compute the SVDs to obtain simplified models andselect the order. We show singular values of the matricesin (10) and (28) in Fig. 4 and Fig. 5, respectively.

Fig. 4 Singular values (deterministic realization)
Fig. 2 Modeling based on deterministic realization forthe different sizes of 𝛴1 ∈ 𝐑𝑛×𝑛 in (10) (𝑛 =4, 7, 12, and 18).

Fig. 5 Singular values (N4SID)

7. Conclusions
We developed mathematical models of finger motions tosimplify the motion of experts in sports. We solved the
problem of modeling finger motions based on the step
response of a state-space representation, by modifying
the deterministic realization and N4SID algorithms. The
experimental results demonstrated that the modelingmethods satisfied the purpose of simplifying complex
finger motions by reducing the order of the state-space
representation. By selecting the order, we could select theFig. 3 Modeling based on N4SID for the different sizesof 𝛴1 ∈ 𝐑𝑛×𝑛 in (28) (𝑛 = 4, 7, 12, and 18).
We compare the motions provided by the algorithms
based on deterministic realization and N4SID. The
reduced-order models (𝑛 = 4, 7) computed by N4SID
capture the motion of the finger almost exactly. In
314

14. ultraleap, “API Overview”, https://developer-archive.
leapmotion.com/documentation/java/devguide/Leap_Ove
rview.html.
15. G. H. Golub and C. F. Van Loan, “Matrix computations,
4th Edition”, The Johns Hopkins University Press, 2013.

model of finger motion at different levels including
simple and accurate motions.
The N4SID-basedalgorithm provided simple models that more accurately
capture the motion of fingers than the deterministicrealization-based algorithm.
Acknowledgment: This work is partly supported by JSPS
KAKENHI Grant Number 20K04535.

Authors Introduction
Mr. Ryuichi Usami
He received his bachelor's degree fromthe School of Education, Hiroshima
University, Japan in 2021. He iscurrently a Master’s Program studentof Graduate School of Humanities andSocial Sciences in Hiroshima
University, Japan.

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Dr. Hideyuki Tanaka
He graduated master course at
graduate school of engineering in
Kyoto University and received Dr.
(Eng.) from Kyoto University. He is amember of Division of Educational
Sciences, Graduate School of
Humanities and Social Sciences in
Hiroshima University. He is a memberof IEEE, SICE, and ISCIE.

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