ICROS-SICE International Joint Conference 2009
August 18-21, 2009, Fukuoka International Congress Center, Japan

Observation of Human Multi-Joint Arm Movementfrom the Viewpoint of a Riemannian Distance
Masahiro Sekimoto1 , Suguru Arimoto1,2 , Boris I. Prilutsky3 , Tadao Isaka1,4 , and Sadao Kawamura1,4
Research Organization of Science and Engineering, Ritsumeikan University, Kusatsu, Japan(Tel: +81-77-561-4910; E-mail: sekimoto@fc.ritsumei.ac.jp)
2
RIKEN-TRI Collaboration Center, RIKEN, Nagoya, Japan
3
Center for Human Movement Studies, Georgia Institute of Technology, Atlanta, Georgia, USA
4
Department of Robotics, Ritsumeikan University, Kusatsu, Japan
1

Abstract: This paper aims at analyzing dynamic characteristics of human arm movements from the viewpoint of Riemannian distance. In order to evaluate the amount of inertia-induced movement of a multi-joint arm, a measure calledinertia-induced measure is developed. By applying the measure to actual human reaching, it is shown that the smoothreaching tends to be closer to the inertia-induced movement than the clumsy reaching. From this observation, it is suggested that humans use their own inertia properties efﬁciently in smooth reaching.
Keywords: Inertia-induced movement, Motion analysis, Human reaching, Riemannian distance, Dynamics

1. INTRODUCTION
1

Humans and robots constitute a common structure:a series of links that are connected through rotational
joints. During fast movements of such a multi-body system, inertia-induced effects (i.e., inertia, centrifugal, andCoriolis forces) are dominant except for robots with highgear-ratio joints, and Lagrange’s equation of motion forthis multi-body system becomes intrinsically nonlinear.
A particle moves straight with no acceleration when there
is no external force or the sum of forces acting on theparticle is zero; this is generally known as the law of inertia. In contrast, physical characteristics of motion ofa multi-body system in such a case have not yet beenshown explicitly. Very recently, it has been demonstrated
that inertia-induced movement of a multi-body system ischaracterized by a geodesic curve corresponding to a Riemannian distance in a Riemannian manifold [1].
A Riemannian distance is a technical term used in Riemannian geometry which is an important branch of differential geometry. The traditional Riemannian geometry focuses on a broad range of geometries whose metric properties vary from point to point. Three decadesago (1978), Arnold [2] noticed the importance of Riemannian geometry in modeling, analysis, and control ofmechanical systems. Recently, Bullo and Lewis [3] haveshown that an equation of motion of a multi-body system can be treated using Riemannian geometry. Thus,
Riemannian geometry can also deal with movement of amulti-body system. Arimoto [4], introducing the ideasof Riemannian geometry in Robotics independently of
Bullo and Lewis’s work, has shown that given a robot
as a multi-body mechanism with n degrees-of-freedom(DOFs) and the free endpoint, the set of its all postures
can be regarded as a Riemannian manifold (M, g) associated with the Riemannian metric g that constitutes
the robot’s inertia matrix. On the basis of this fact, Arimoto et al. [5] have suggested the new movement stability analysis (the nonlinear closed-loop dynamics analysis) of a multi-joint robot arm with joint redundancy whenxd = (xd, yd)T
2

8

10cm
3 y

7
4

6

x = (x, y)T

q2

5

q1 x
O
Fig. 1 Planar movement of a human upper arm with two
jointsthe passivity based control was applied to the system.
Then, a geodesic connecting any two postures with the
Riemannian distance corresponds to an orbit expressedin a local coordinate chart and generated by a solution
to the Euler-Lagrange equation of robot motion originating only from inertia-induced force [1]. Furthermore,
the Riemannian-geometry viewpoint has been extended
to an important class of dynamics of multi-body systems
physically interacting with an object or with environment
through holonomic or/and nonholonomic (but Pfafﬁan)constraints [5].
Aside from movements of a multi-joint robot, humanmulti-body movements have attracted a lot of attention
and various studies have been conducted. The widely
known features of human skilled reaching by the arm area straight endpoint trajectory and a bell-shaped endpoint
velocity proﬁle [6]. A lot of studies of human movement including the above observations have focused on
the kinematic properties. However, fewer studies havedealt with dynamics due to lack of mathematical tools foranalyzing dynamic characteristics of human movements.

- 2664 -PR0002/09/0000-2664 ¥400 © 2009 SICE

and v = v i (∂/∂qi ) ∈ Tp M , gp (u, v) is a symmetric positive deﬁnite quadratic form:

This paper aims at analyzing dynamic characteristics
of human arm reaching movements from the viewpoint of
Riemannian distance. In particular, this paper focuses on
how humans cope with dynamics of their own arms during reaching. In order to analyze the role of arm inertial
properties in human movement, a measure for evaluating
the amount of inertia-induced movement of a multi-joint
arm, called an inertia-induced measure, is proposed. The
inertia-induced measure was used to analyze human planar reaching. It is shown that in smooth reaching, the calculated inertia-induced measure proﬁle has peaks at theinitial and ﬁnal stages of motion and the measure values
are small during other stages of motion. However, thevalues of the inertia-induced measure are relatively large
in general in clumsy reaching. Based on these observations, it is suggested that humans use their own inertia
properties efﬁciently in smooth reaching.

gp (u, v) =gij (p)ui v j

(2)

i,j=1

Then, the manifold (M, p) can be regarded as a connected
Riemannian manifold and becomes complete as a metricspace. Hence, it is possible to deﬁne a length of c(t) ∈ M
to be
� b
� b�
L(c(t)) =�ċ(t)�dt =gc(t) (ċ(t), ċ(t)) dt (3)aaThen, since any connected Riemannian manifold becomes a metric space whose induced topology is coincident with the given manifold topology, it is also possible
to deﬁne a Riemannian distance d(p, p� ) for any pair ofpoints p, p� ∈ M as the inﬁmum of all admissible curves
from p to p� as follows:

2. RIEMANNIAN DISTANCE:INERTIA-INDUCED MOVEMENT OF
A MULTI-JOINT SYSTEM

d(p, p� ) = inf L(c(t))(4)

As described in the textbooks on Riemannian geometry[8, 9], an admissible curve c in a Riemannian manifold
is said to be minimizing if L(c) ≤ L(c̃) for any otheradmissible curve c̃ with the same endpoints. It follows
immediately from the deﬁnition of distance that c is minimizing if and only if L(c) is equal to the distance between
the endpoints. Furthermore, it is known that if the Riemannian manifold (M, g) is complete then for any pairof points p and p� there exists at least a minimizing curvec(t), t ∈ [a, b], with c(a) = p and c(b) = p� . If sucha minimizing curve c(t) is described with the aid of coordinate chart (U, φ) as φ(c(t)) = (q1 (t), · · · , qn (t))(=q(t)) then it satisﬁes the 2nd-order differential equation:

Let us consider a multi-joint system with n rigid bodies connected in series through rotational joints as shownin Fig.1. If a posture of the system is represented byp, the set of all such possible postures M and a family of subsets of M can be determined naturally. Since
the set M becomes Hausdorff and compact as far as theobject is conﬁned to multi-joint systems existing in thereal world, the set M with such a family of open subsets can be regarded as a differentiable manifold of class
C ∞ [7]. Furthermore, every point p of M has a neighborhood U that is homeomorphic to an open subset Ωof n-dimensional numerical space Rn constructed by local coordinates (q1 , . . . , qn ), and such a homeomorphism(φ : U → Ω) is called a coordinate chart. As the local
coordinates, it is reasonable to choose generalized coordinates of the system, that is, the joint angles given inqi ∈ (−π, π] (i = 1, . . . , n) as deﬁned in Fig.1.
Next, let I be an interval (−ε, ε) and deﬁne a curvec(t) by a mapping c : I → M such that c(0) = p. If forany given curve c̄ a relation (∼) is deﬁned byd(φ ◦ c)d(φ ◦ c̄)c ∼ c̄ ⇐⇒
(0) =
(0)
dt
dt

n
�

q̈k (t) +n�Γkij (q(t))q̇i (t)q̇j (t) = 0

i,j=1(k = 1, · · · , n)where Γkij denotes Christoffel’s symbol deﬁned by��m1 � kh ∂gih
∂gjh
∂gij
k
Γij =
g
+
−
2
∂qj
∂qi
∂qh

(5)

(6)

h=1

(1)

and (g kh ) denotes the inverse of matrix (gkh ). A curve
q(t) : I → U satisfying eq.(5) together with φ−1 (q(t))
is called a geodesic and eq.(5) itself is called a geodesic
equation. Furthermore, the geodesic equation of eq.(5)can be transformed, by multiplying eq.(5) from the lefthand by (gki ), into

in a coordinate chart (U, φ) around p, then it becomes theequivalence relation, and the equivalence class of curvesc : I → M is called a tangent vector to M at p. Thisdeﬁnition of tangent vectors to M at p does not dependon choice of the coordinate chart at p, as discussed in the
textbooks [8, 9]. Furthermore, the set of all tangent vectors to M at p is denoted by Tp M and called the tangent
space at p ∈ M . It has an n-dimensional linear spacestructure like Rn .
Now, for a given n × n symmetric positive deﬁnite
matrix G(c(t)) whose (i, j)-entry is denoted by gij , deﬁne a Riemannian metric in the differentiable manifold(M, p) as a mapping gp : Tp M × Tp M → R such thatp → gp is of class C ∞ and, for u = ui (∂/∂qi ) ∈ Tp M

gki q̈i (t) +n�Γikj (q(t))q̇i (t)q̇j (t) = 0

i,j=1(k = 1, · · · , n) (7)where Γikj denotes another Christoffel’s symbol deﬁnedby��
1 ∂gik∂gjk∂gij
Γikj =
(8)
+
−
2 ∂qj
∂qi
∂qk

- 2665 -Meanwhile, in mechanics and robotics, Lagrange’s
equation of motion of the multi-joint arm [10] is described by:
H(q(t))q̈(t) +��1Ḣ(q(t)) + S(q(t), q̇(t)) q̇(t) = u(t)2of eq.(3) together with a Riemannian distance of eq.(4)based on a system inertia matrix is available to evaluate
the amount of inertia-induced movement of a multi-joint
system.
In general, dynamics of an actual human or robot differ from the ideal dynamics shown by eq.(9) and containviscoelastic, Coulomb friction, and gravity forces. An
equation of motion of such complicated system can be
described by��
1
H(q)q̈ +Ḣ(q) + S(q, q̇) q̇ + C q̇2
(11)
+KΔq + f c + g(q) = u(9)where q(t) = (q1 (t), . . . , qn (t))T denotes the vector of joint angles, H(q(t)) the inertia matrix,
S(q(t), q̇(t))q̇(t) the gyroscopic force term including
centrifugal and Coriolis forces, S(q(t), q̇(t)) the skewsymmetric matrix, u the control input torque at joints.
The inertia matrix H(q(t)) is symmetric and positive definite. Furthermore, each entry hij of H(q(t)) is a constant or a sinusoidal function of components of joint angle vector q(t). Hence, H(q(t)) is symmetric, positivedeﬁnite and differentiable of class C ∞ .
Now, suppose that the inertia matrix hij of the system is chosen as gij in the Riemannian metric deﬁned byeq.(2), i.e., gij = hij for i, j = 1, . . . , n. Then, the lefthand side of the geodesic equation of eq.(7) is coincident
with that of the system dynamics of eq.(9) because byreferring to the book [10] and the paper [7] it satisﬁes
�
�
n
�
1
Γikj (q)q̇i q̇j(10)Ḣ(q) + S(q, q̇) q̇ =2i,j=1

where C q̇ denotes joint viscosities, KΔq denotes joint
elastics, f c denotes a Coulomb friction, g(q) denotes thegravity effect. Even in the case of such complicated dynamics it is possible to evaluate the amount of inertiainduced movement of the system. Now, we focus onsome interval t ∈ [a, b] during the motion governed by
the dynamics of eq.(11) with u = 0. For a measured orbit of state variables of a system, a length corresponding
to the motion within [a, b] can be calculated by applying
the motion data to eq.(3) or the following expression
� b�L=q̇(t)T H(q(t))q̇(t) dt(12)a

Hence, the system dynamics with u = 0 is coincident
with the geodesic equation [7]. This means that if an inertia matrix of system is chosen as gij in the Riemannian metric of eq.(2) then a geodesic q(t) represents amotion from q(a) = p to q(b) = p� over an intervalt ∈ [a, b] without input torques in the physical sense,that is, inertia-induced movement (pure inertia, Coriolis,
and centrifugal forces movement) of a multi-joint system.
Furthermore, the Riemannian distance itself, which is theminimum in lengths determined by eq.(3) of all possible
motions from a posture p(= q(a)) to another p� (= q(b)),
corresponds to inertia-induced movement of a multi-joint
system. Note that a Riemannian distance itself is invariant even if for given positive constants α and β the subsidiary variable t is replaced with s = αt + β, that is, it
depends only on the endpoints p and p� in the Riemannian
manifold as discussed in the textbooks [8, 9].

On the other hand, the corresponding Riemannian distance between the postures q(a) and q(b) is available
from eq.(4). In more detail, the Riemannian distancecan be obtained by solving the two-point boundary value
problem under the geodesic equation of eq.(5) or (7) together with the two boundary conditions q(a) and q(b)and applying the obtained geodesic orbit to eq.(12). Thus,
it is possible to evaluate the inertia-induced amount of thetarget motion by comparing the obtained length with thecorresponding Riemannian distance. However, the evaluated motion must be far from the inertia-induced movement because the other forces except the inertia-induced
force are exerted on the system. As another situation, we
consider the case that a control input torqueu = C q̇ + KΔq + f c + g(q)(13)

is applied to the system so as to cancel the forces except the inertia-induced force. In this case, the calculated length must be close to the Riemannian distance,
hence, the evaluated motion should be close to the inertiainduced movement. Thus, even if other forces except theinertia-induced force are exerted on the system, it is possible to evaluate the amount of inertia-induced movementof a multi-joint system. This is physically reasonablegiven an analogy with a particle motion under action of
external forces.
Now, we suggest a measure to evaluate the amount ofinertia-induced movement of a multi-joint system on thebasis of physical properties of the Riemannian distance,
called inertia-induced measure. For an interval t ∈ [a, b]
arbitrarily set within movement, the measure is deﬁnedby

3. INERTIA-INDUCED MEASURE
As described in the previous section, a length in eq.(3)based on a Riemannian metric with a system inertia matrix represents the amount of inertia-induced movementof a multi-joint system. For some multi-joint movementfrom a posture p(= q(a)) to another posture p� (= q(b)),
if the motion is far from the inertia-induced movement(the geodesic orbit obtained by eq.(5) or (7)), the trajectory length is longer than the Riemannian distance between the points p and p� . On the other hand, if the motion is similar to the geodesic orbit, the trajectory lengthis close to the Riemannian distance. Note that a length of
any admissible curve c between p and p� in a Riemannian
manifold is more than or equal to a corresponding Riemannian distance, i.e., L(c) ≥ d(p, p� ). Hence, a length

R = L − d(q(a), q(b))- 2666 -(14)

y-component [cm]

60

55

50

45

40

35
0

5

10

15

20

25

30

x-component [cm]Fig. 3 Endpoint trajectories of 20 trials

Fig. 2 An experimental setup

Table 1 The arm parameters of the subject
where L denotes the length deﬁned by eq.(12) andd(q(a), q(b)) denotes the Riemannian distance deﬁned
by eq.(4) between the postures q(a) and q(b). Whenthe motion is exactly equal to the inertia-induced movement, the measure shows R = 0, while the measure has
the large values when the target motion is far from theinertia-induced movement. Furthermore, we suggest thenormalized measure deﬁned by
RN = (L − d(q(a), q(b))) / d(q(a), q(b))

Link number
Length [m]Center of mass [m]Mass [kg]Inertia moment [kgm2 ]1
0.27800.14982.001
1.023 × 10−220.40800.14561.621
1.951 × 10−2

narm is capable of generating a force ﬁeld imposed on
the arm through the motor-actuated torques applied at theshoulder and elbow joints, but this function was not usedin the experiment. Figure 3 depicts endpoint trajectoriesof all 20 trials of one subject. The joint angles and angular velocities of the arm were measured by the encoders
built in the robotic arm of Kinarm. The endpoint position
shown in Fig.3 was calculated from the kinematic information of the arm: subject’s segment lengths and the joint
angles.
The evaluation of the inertia-induced measure wasconducted by using the obtained data. First, each movement was divided into 30 segments according to movement time. Next, for each interval the length of eq.(12)and the Riemannian distance were calculated from thedata according to the procedure described in Section 3.
Finally, the values of the inertia-induced measure as described by eqs.(14) and (15) were calculated by using the
calculated length and the Riemannian distance. These
steps were repeated for all 30 segments to obtain the proﬁles of the inertia-induced measure. The inertia matrix ofhuman arm required for the calculation of the length andthe Riemannian distance, i.e., the link inertias and linkmass, was estimated by referring to the book [12]. Table
1 shows the estimated arm parameters of the subject.
Figures 4 and 5 depict two typical results observed in
the experiment. Figure 4 shows mechanical characteristics of reaching to target 2 in trial 13; Figure 5 shows thesame characteristics for trial 18. Each ﬁgure presents theendpoint trajectory, the estimated kinetic energy proﬁle,the joint angle proﬁle, the angular velocity proﬁle, theproﬁle of the inertia-induced measure deﬁned by eq.(14),and the proﬁle of the normalized measure deﬁned byeq.(15). The kinetic energy was calculated on the basisof the estimated inertia matrix. It can be seen that the
motion in Fig.4 becomes clumsy around the target (i.e.,
the path curvature sharply changed). On the other hand,a smooth reaching movement is observed in Fig.5, al-(15)

It is possible to set the interval t ∈ [a, b] according touser’s demands. If transient effects are of interest, a smallinterval should be set. However, if the inertia-induced
amount of motion is investigated over an entire movement, the movement onset and offset times should be set
as the interval.

4. OBSERVATION OF REACHING FROM
THE VIEWPOINT OF RIEMANNIANDISTANCE
In order to investigate dynamic properties in humanreaching from the viewpoint of the inertia-induced measure, human planar reaching experiments were conducted
as shown in Fig.1. Figure 2 demonstrates an experimental setup for reaching experiments using a robotic system Kinarm (BKIN Technologies, Canada). The subject
was sitting in a Kinarm chair with the arms resting on
two-joint robotic arms in the horizontal plane (see the paper by Scott [11] for Kinarm description). The chair andthe robotic arm were adjusted so that the subject could
move the arms in the horizontal plane at a chest level.
The subject was instructed to reach as fast and accurately
as possible one of 8 targets, which appeared randomly on
the horizontal screen in front of the subject, with the index ﬁnger tip of right hand. The targets were circles of 1
[cm] in diameter arranged radially around the start position 10 [cm] away from it. When the target was reached,
the color of target was changed so that the subject could
assess if the task was successfully accomplished. It disappeared within 1 [s] of ”touch” and the subject moved
the arm back to the initial (center) position and waited fora next target to appear. After reaching all 8 targets in random order (one trial), next trials of reaching movements
were performed until 20 trials were completed. The Ki-- 2667 -

Goal

Goal

0.025

51
50
49

0.01
0.008
0.006
0.004
0.002

Start
14

15

16

17

18

19

20

21

0.2

X-component [cm](a) Endpoint trajectory
Solution dq [deg/s]

Joint angles q [deg]

q2

70
60

q1

50
40
30
0

0.2

0.4

0.6

50
49

14

0.8

.

.
q

-30

2

-40

0

0.2

0.4

0.6

0.8

18

19

20

0
0

21

60

q1

40

0.4

0.6

1

0.8

R

RN

0.2

0.04

0.1

0.02

0
0

0
0

0.2

0.4

0.6

0.8

1

-70

0

0.2

0.8

0
0

1

0.4

0.6

0.8

0.1
0.08
0.06
0.04
0.02

0.1

0.6

-60

0.12

0.2

0.4

q2

-50

0.14

0.5

0.3

0.2

.

-40

0.16

0.4

0.06

0.3

-30

(d) Angular velocities

0.6

0.1
0.08

0.4

-20

0.18

0.7

0.12

0.5

q1

0

-5
x 10

0.8

0.14

0.6

.

10

-10

Time [s]

0.9

0.16

0.7

0.8

20

(c) Angles(d) Angular velocities

0.6(b) Estimated kinetic energy

q2

0.2

0.4

Time [s]

70

50

0.2

Time [s]

0.18

0.8

17

80

30
0

1

-5
x 10

0.9

16

x-component [cm]Time [s](c) Angles

0.01

0.005

(a) Endpoint trajectory

-10

-50

1

15

90

-20

0.02

0.015

Start

48

1

q1

0

Time [s]

R

0.8

51

(b) Estimated kinetic energy
10

80

0.6

52

Time [s]

90

1

0.4

53

RN

13

0
0

Joint angles q [deg]48

Kinetic energy K [J]52

Angular velocities dq [deg/s]

53

54

0.012

y-component [cm]Kinetic energy K [J]y-component [cm]

0.014
54

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0
0

0.2

0.4

0.6

0.8

Time [s]Time [s]Time [s]Time [s](e) The measure(f) The normalized measure(e) The measure(f) The normalized measure

Fig. 5 Experimental results in the case of smooth reaching

Fig. 4 Experimental results in the case of clumsy reachingproperties between the two reaching movements. As discussed in Section 3, the length of eq.(12) depends not
only on the initial and ﬁnal postures of the reaching motion within a predeﬁned time interval, but also on movement time. In contract, the Riemannian distance depends
only on the initial and ﬁnal postures, regardless of movement time. Hence, the normalized R measure can be useful for comparing movements with different speed anddifferent Riemannian distance.

though the end-point did not reach the target and passed
very close to it. The fact that the target was missed intrial 18 does not present a problem since our concern inthe paper is the application of the inertia-induced measure analysis to human reaching rather than the reaching performance. The proﬁles of the non-normalized andnormalized inertia-induced measure show that the values tend to be large around the initial and ﬁnal stagesof movement. As discussed in Section 3, the quantity
R becomes small when the observed movement is close
to the inertia-induced movement and large when it is far
from it. Hence, it appears that the large R values coincidewith subject’s attempts to initiate or stop the arm at these
movement phases. In particular, during the clumsy reaching, the subject attempted to move the arm and correct its
trajectory so as to reach the target at the ﬁnal stage of
motion. The inertia-induced measure represents this behavior by large R values in Fig.4(e). On the other hand,
the results show that the R values tend to be relatively
small during other movement stages (e.g., middle stage)
especially if reaching is smooth (Figs.4 and 5). Based onthese results it can be suggested that humans take into account their own inertia properties and use them efﬁciently
during fast reaching motion.

5. CONCLUSIONS
The inertia-induced measure was proposed to evaluate the amount of inertia-induced movement of a multijoint system, and dynamic properties of human reachingwere analyzed by using this measure. An important beneﬁt of the measure is that it allows for analysis of inertiainduced movement of a multi-joint system. The inertiainduced measure can be used to evaluate the amount ofinertia-induced motion for variety of multi-joint movements. The analysis can be used when a motion is subject
to gravity and external forces as long as the motion is primarily governed by an inertia-induced force. Hence, weplan to apply the inertia-induced measure to analyses ofother movements (for instance, human walking) in futurework.

The inertia-induced measure can also provide theamount of inertia-induced movement over an entire
reaching motion by setting boundary conditions at thestart and end of motion. The overall amount of inertiainduced movement was RN = 10.1[%] in the clumsy
reaching, and RN = 1.07[%] in the smooth reaching.
These results clearly illustrate the difference in dynamic

ACKNOWLEDGMENTS
This work was partially supported by KAKENHI(No.20760175): Grant-in-Aid for Young Scientists (B) of
the MEXT, Japan. The authors thank Ms. Linda Harleyand Mr. Gregory Philips of Georgia Institute of Tech-- 2668 -nology, USA, for their help in data collection. Also, theauthors thank Dr. Takahiro Wada of Kagawa University,
Japan, for valuable discussions.

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