Adaptive Internal Model of Intrinsic Kinematics

Adaptive Internal Model of Intrinsic Kinematics
Involved in Learning an Aiming Task

Hiroshi Imamizu,
Yoji Uno,and
Mitsuo Kawato

ATR Human Information Processing
Research Laboratories

Address correspondence to:Hiroshi Imamizu
Kawato Dynamic Brain ProjectJapan Science and Technology Corporation
2-2, Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-02, Japan
Telephone:+81-774-95-1220
Facsimile:+81-774-95-3001
E-mail:imamizu@erato.atr.co.jp

Running head: Adaptive Internal Model of Intrinsic Kinematics-1-Adaptive Internal Model of Intrinsic Kinematics-2-

Abstract

We multiplied the elbow joint angle and the shoulder joint angle of participants
aiming at targets in an experiment using a position recording system and a CRT screen.
The linear transformation in joint angles (intrinsic coordinates) involved in the experimentcorresponded to a nonlinear transformation between the hand coordinates and the screen
coordinates (extrinsic coordinates). The present study examined whether the participants
could learn this transformation in the intrinsic coordinates or in the extrinsic coordinates,by investigating intermanual (between-hands) transfer under an intrinsically-consistentcondition and under an extrinsically-consistent condition. The participants learned toadjust for the transformation in the first stage for both conditions. In the second stage
under the intrinsically-consistent condition, the participants learned to adjust for the sametransformation in the intrinsic coordinates as that in the first stage. Likewise, in thesecond stage under the extrinsically-consistent condition, they learned to adjust for thesame transformation in the extrinsic coordinates as that in the first stage. Positive
intermanual transfer was observed under the intrinsically-consistent condition but notunder the extrinsically-consistent condition. Results suggest that participants can learn thelinear transformation in joint angles in the intrinsic coordinates and that the central
nervous system adaptively represents the intrinsic kinematics.

Adaptive Internal Model of Intrinsic Kinematics-3-

Introduction

In everyday life, people interact continuously with objects in extra-personal space.
Reaching movements— e.g., bringing a hand to the location of an object—are
fundamental behaviors by means of which we interact with the external world. Because ofthis, the study of reaching movements has received a great deal of interest. Recent
advances in computational models of the complex mechanisms underlying reaching
movements have provided a comprehensive account of how a purposeful act may beplanned and executed from sensory input to motor output in the central nervous system(CNS). Some computational studies have suggested that the motor system contains aninternal representation of the geometry of the limbs, such as muscle lengths or joint angles,
to control movement. Whether the geometric aspects of motion (kinematics) are
adaptively represented in the CNS in intrinsic space is examined in the present study, by
an investigation of intermanual transfer in the learning of an aiming task under different
conditions.

A Hierarchical and a Non-Hierarchical Model of Visually-Guided Movements
Many computational models have been proposed for visually-guided reaching
movements. These models can be divided into two classes depending on their approach.
One approach is to divide the problem of "control of visually-guided movements" into
several sub-processes and solve each of the sub-processes in a hierarchical manner
analogous to that used in the control of robot arms (Hollerbach, 1982; Saltzman, 1979;
Saltzman & Kelso, 1987). The other approach is based on a non-hierarchical method thattranslates a sensory stimulus directly into time-varying patterns of muscular activation(Massone & Bizzi, 1989).

Adaptive Internal Model of Intrinsic Kinematics-4-

To provide a comprehensive example of the hierarchical model, the upper row in
Figure 1 schematically illustrates a computational model based on the former approach,and the lower rows in the figure show hierarchical levels of coordinate reference frames
used in the control of a robot arm. During reaching tasks, the target position is usually
identified visually, resulting in a representation in the CNS in terms of visual space(extrinsic space). In the present study, Cartesian coordinates were assumed, in which theorigin is at the center of the shoulder, consistent with assumptions of previous
computational studies (Atkeson, 1989; Hollerbach, 1990). The position information
represented in such a space, however, is not directly linked to parameters of the motor
apparatus (i.e., joint angle and torque) and must be translated into these coordinates(intrinsic space).
The first problem is translation from extrinsic space to intrinsic space, and isreferred to as a coordinate transformation. In the model illustrated in Figure 1, the target
location is translated into joint angle space. The second problem is determination of a
trajectory from the current position to the target and is termed trajectory planning. The
last problem is computation of the joint torque necessary to obtain the planned trajectory,
that is, geometric properties of motion (kinematics). The planned trajectory must betranslated into dynamic properties of the motor apparatus related to the force required tocomplete the motion (dynamics). This is called the control problem.
------------------------------------------Insert Figure 1 about here------------------------------------------Although the reference frames of motion mentioned above (i.e., Cartesian space,joint angle space, and joint torque space) are often used in the control of a robot arm, it isunlikely that these frames are used in the CNS. Nevertheless, there are parameters that are
known to be perceived in the CNS that are functionally equivalent to these artificial

Adaptive Internal Model of Intrinsic Kinematics-5-

frames. For example, the parameters of joint angle and torque are closely related tomuscle length and tension, respectively.
A non-hierarchical approach to modeling visually-guided reaching movements was
proposed by Massone and Bizzi (1989). They proposed a neural network model of asensory-motor transformation that translates a sensory stimulus directly into time-varying
patterns of muscular activation. Figure 2 shows a schematic diagram of their model and asimplified architecture of the proposed network. The input for the network is sensory
stimulation representing the target location in extrinsic space and the output is a timevarying trajectory in muscle space (i.e., motor commands in intrinsic space). This model
does not require intermediate representations (i.e., the target location in intrinsic space or
the trajectory in intrinsic space), which are required in the hierarchical model.
------------------------------------------Insert Figure 2 about here------------------------------------------The classes of models underscore the current controversy in neuroscience
regarding how movement is controlled in the CNS—whether in a hierarchical or nonhierarchical manner. Although a hierarchical approach provides an exact andcomprehensive account of actual problems associated with motor control, several
neurophysiologists (e.g., Alexander, DeLong, & Crutcher, 1992; Kalaska & Crammond,
1992) have pointed out that such sequential and analytic approaches give implausible
accounts of how movement is controlled in the CNS.
Three problems—coordinate transformation, trajectory planning, and control—
must be solved to bring the hand from the starting position to the target for the following
reasons. First, the coordinate system used in the visual system is different from that usedin the motor system. Thus, a coordinate transformation is necessary to link sensory input
to motor output. Second, it is well-known that the hand’s trajectory between any twoAdaptive Internal Model of Intrinsic Kinematics-6-

points has some invariant properties—the hand’s path is roughly straight and the hand’s
speed-profiles are bell-shaped (Abend, Bizzi, & Morasso, 1982; Morasso, 1981). These
invariant properties suggest that the trajectories are planned in the CNS according to some
optimization principles (Flash & Hogan, 1985; Uno, Kawato, & Suzuki, 1989). Third, thefact that the hand rapidly and smoothly moves to the target suggests that the arm issuccessfully controlled taking its dynamic properties into account. It is not clear, however,whether these problems are sequentially — hierarchically — solved using several modules
or simultaneously — non-hierarchically — solved using one module.

Internal Representation of Intrinsic Kinematics
The goal of the present study is to investigate whether a representation of intrinsic
kinematics is used during the control of visually-guided movements in the CNS andwhether this representation changes during the learning of an aiming task in which thevisual feedback has been altered with a transform function. The hierarchical model ofvisually-guided movements predicts that an intermediate representation of intrinsic
kinematics is formed between the computational modules (Figure 1), whereas the nonhierarchical model does not require such a representation (Figure 2). Thus, the presence ofa representation of intrinsic kinematics in the CNS would support the hierarchical model
rather than the non-hierarchical model of visually-guided reaching movements.
It is worth noting that the presence of such representation does not reject the
network model proposed by Massone and Bizzi (1989) because there is a possibility thatsome neurons (computational elements) in the model might come to represent the intrinsic
kinematics by chance after the training of the network. However, the presence of suchneurons would indicate that the coordinates transformation is solved before these neurons
are activated and that the control problem is solved afterwards. This suggests that theAdaptive Internal Model of Intrinsic Kinematics-7-

model is hierarchical. In other words, the representation of the intrinsic kinematics makes
the model hierarchical.
Some sources of evidence supporting the idea that the CNS contains
representations of intrinsic kinematics have emerged through recent neurophysiologicalstudies on multi-joint behavioral tasks. Kalaska and his co-workers showed that theactivity of the proximal-arm-related neurons in the superior parietal lobule (Brodmannarea 5) of monkeys continuously changes depending on the direction of arm movements(Kalaska, Caminiti, & Georgopoulos, 1983), and this activity was not affected by an
external load on the arm in various directions (Kalaska, Cohen, Prud'homme, & Hyde,1990). The authors concluded that area 5 neurons encode movement kinematics, notmovement dynamics. Lacquaniti, Guigon, Bianchi, Ferraina, and Caminiti (in press)
reported that an intrinsic coordinate system centered at the shoulder and defined by theelevation and azimuth angles of the proximal arm (i.e., shoulder-joint configuration) andby the angle of extension at the elbow, provides a better account of neural modulation inarea 5 than an extrinsic coordinate system does (Cartesian coordinate system parallel to thelaboratory frame). These data suggest that proximal-arm-related neurons encode theintrinsic kinematics of arm movements and that area 5 is involved in the integration of thissensory and motor information.
In this research we investigated whether a representation of intrinsic kinematics isused in the control of reaching movements using a behavioral paradigm in which visualfeedback of the hand position of participants was altered with a transform function basedon multiplication of the shoulder-joint and elbow-joint angles by constant values during anaiming task. If a representation of intrinsic kinematics in the CNS is used in the control ofarm movements, then an adjustment to this transform would be learned in terms of the
intrinsic space. However, if a representation of intrinsic kinematics is not used, then theadjustment would not be learned or would be learned using another mechanism, perhaps inAdaptive Internal Model of Intrinsic Kinematics-8-

terms of extrinsic space. As mentioned above, it is unlikely that joint angles are
represented in the CNS. However, parameters that are functionally equivalent to joint
angles might be represented using information from sensory receptors within the muscles,
tendons, joints, and skin (i.e., proprioception; Rosenbaum, 1991).

Visual-Motor Learning Under Transformed Visual Feedback
Many investigators have studied the learning of motor tasks (e.g., reaching, aiming,and drawing) under transformed visual feedback to investigate how the relation between
visual information and motor commands is learned. In some studies, the visual feedback
was transformed using an optical device such as a prism or a mirror. Experimental
paradigms using such devices are called prism adaptation (Held & Gottlieb, 1958) or
mirror drawing (Cook, 1933). Other studies have used a position recording system, suchas a digitizing tablet and CRT screen controlled by a computer (Fig. 3), to process therecorded position through a transform function before display (Cunningham, 1989;
Imamizu & Shimojo, 1995).
------------------------------------------Insert Figure 3 about here------------------------------------------The majority of these studies have reported that participants who were initially
exposed to transformed visual feedback made large errors in the motor task and that theerrors decreased as the number of trials increased. Some of the studies revealed important
facts concerning the nature of the adaptation to (or the adjustment to) the transformed
visual feedback: 1) adaptation to the transformed visual feedback causes subsequenterrors under normal visual feedback. This phenomenon, called an aftereffect, can beobserved even when participants are informed of the return to normal visual feedback,suggesting that the adaptation is not entirely under conscious control (Welch, 1978). 2)

Adaptive Internal Model of Intrinsic Kinematics-9-

Milner, Corkin, and Teuber (1968) reported that a patient who had a deficit in verbal andcognitive memory because of brain damage could learn a mirror drawing task, and that thelearning was stable for a long period. 3) Held and Hein (1958) compared two conditionsof prism adaptation: with active or passive arm movement. The participants either swung
the arm back and forth in the frontal plane (active movement) or had it transported in thesame manner by means of a moving "cradle" to which it was strapped (passive
movement). They found that the active movement produced adaptation, whereas thepassive movement did not.
These features of visual-motor learning under transformed visual feedback (i.e.,some degree of independence from conscious control and cognitive memory and therequirement of active movements during the training period) suggest that it is like otherkinds of motor learning, such as skill acquisition in the playing of sports or musicalinstruments. The advantage of experimental paradigms using transformed visual feedback
is that the learning process can be investigated using reaching or aiming tasks that have aclear objective (i.e., to bring the hand to the exact location of an object) and is well-suited
for studying the translation of sensory input to motor output in the CNS.
Although a similar experimental paradigm of visual-motor learning undertransformed visual feedback was used in the present study, there is a critical difference
between previous research and the present study. In previous research, the positions of theparticipant's hand and objects in the visual field were altered by a transform function
expressed simply in the task-oriented extrinsic coordinates (denoted by f in Fig. 4). For
example, in prism adaptation, the hand and object positions were “translated” and in
mirror drawing, the positions were “reflected.” In the present study, however, a transformfunction that can be expressed simply in intrinsic coordinates was used (denoted by g in
Fig. 4); a “magnification” of the joint angles.
-------------------------------------------

Adaptive Internal Model of Intrinsic Kinematics-10-

Insert Figure 4 about here------------------------------------------A transform function represented in extrinsic coordinates can be translated into one
that is represented in intrinsic coordinates, or vice versa; however, a transform functionthat is simple (e.g., translation or reflection) in extrinsic coordinates becomes complicatedin intrinsic coordinates, and vice versa. This is because the relationship between extrinsic
and intrinsic coordinates is complex and nonlinear. This relation is discussed in detail atthe end of the next section.

Linear Transformation of Joint Angles
The kinematics of a two-link arm moving in a horizontal plane, as illustrated in
Figure 3, is usually expressed as follows:p = l1 cosθ1 + l2 cos(θ1 + θ 2 )q = l1 sin θ1 + l2 sin(θ 1 +θ 2 ) ,(1)

where (p, q) represents the hand position in the horizontal plane (i.e., the surface of aboard above which the hand moves) and θ 1 and θ 2 denote the shoulder-joint angle andelbow-joint angle (Fig. 3) respectively. l1 and l2 are the upper arm and forearm length
respectively.
Figure 5 illustrates the linear transformation of joint angles used in the present
study. P( p, q ) represents the hand position in the horizontal plane (i.e., surface of theboard above which the hand moves), while X ( x, y ) represents the cursor position on the
CRT screen. The cursor position was determined so that a mapping in joint angles isrepresented asθ 1* = 1.25( θ1 − h1 ) + h1
θ 2* = 0.5( θ 2 − h2 ) + h2 ,(2)Adaptive Internal Model of Intrinsic Kinematics-11-

where (h1, h2 ) is a fixed point for this mapping. θ 1 and θ 2 denote the shoulder-joint**

angle and elbow-joint angles of a transformed arm configuration (not visible in the display
shown to participants) shown in the top portion of Figure 5; that is,

x = l1 cosθ1* + l2 cos(θ 1* + θ 2* )y = l1 sin θ1* + l2 sin(θ1* + θ 2* ) .

(3)

Thus, Equation 3 represents the magnification of the shoulder-joint angle andminimization of the elbow-joint angle.
------------------------------------------Insert Figure 5 about here------------------------------------------It should be noted that this linear transform of intrinsic coordinates corresponds toa nonlinear transform function of extrinsic coordinates. The relation between the hand
position P( p, q ) and the cursor position X ( x, y ) on the CRT screen was obtained byrearranging Equations (1), (2), and (3) so as to eliminate the intrinsic parameters (i.e., θ 1 ,

θ 2 , θ 1* and θ 2* ), that is,
X = ϕ(P) .

(4)

The introduced transform function ϕ is nonlinear and highly complex. Figure 6A
illustrates the effect of the transform ϕ on hand position P( p, q ) in the map of the screen
coordinates X ( x, y ) . As a participant moves a hand along the straight grid lines (thebottom of Fig. 6A), the cursor trajectories on the CRT screen become curved (the top of
Fig. 6A).
------------------------------------------Insert Figure 6 about here

Adaptive Internal Model of Intrinsic Kinematics-12--------------------------------------------

Testing Whether Participants Learn to Adjust for the Transform in terms of the Intrinsic
Space
Because the purpose of the present study was to determine whether a model
corresponding to the artificial transformation of joint angles is represented in the CNS interms of the internal kinematics, it was critical that participants learn to adjust for thetransformation either in terms of joint angles (i.e., intrinsic coordinates) or in terms ofhand and screen coordinates (i.e., extrinsic coordinates). If there is a representation of the
intrinsic kinematics in the CNS and this representation is adaptive, then the participants
should learn to adjust for the transformation in terms of the intrinsic coordinates.
Alternatively, if there is no internal representation of the intrinsic kinematics or if therepresentation is not plastic, the participants should learn the nonlinear and complicatedmapping of ϕ using the hand p − q coordinates and the screen x − y coordinates.
We examined whether participants learned to adjust for a transformation of joint
angles in intrinsic coordinates or in extrinsic coordinates using two experimental
conditions, i.e., an extrinsically-consistent condition and an intrinsically-consistentcondition. Each of the conditions consisted of two successive stages of learning. The
participants executed an aiming task using different arms in the two stages. The joint
angles were transformed by Equation 2 in the first stage for both conditions.
Under the extrinsically-consistent condition (Figures 6A and 6B) in the second
stage, the participants executed the aiming task under the same mapping between thescreen coordinates and the hand coordinates as that in the first stage using their other arm(compare the top of Figure 6A to that of Figure 6B). Although the mapping in the second
stage was the same in the extrinsic coordinates as that in the first stage, it was quite
different in the intrinsic coordinates (in terms of joint angles) because the arm used by theparticipants in the first stage was different from the arm used in the second stage.

Adaptive Internal Model of Intrinsic Kinematics-13-

The bottom of Figure 6B illustrates the meaning of the transformation used in thesecond stage under the extrinsically-consistent condition in the intrinsic coordinates.
When the hand and the elbow are located in the horizontal plane at the level of theshoulder, the other geometrically possible arm configuration appears, whose forearm
length, upper arm length and positions of the shoulder and hand are the same as those of
the original configuration, as indicated by the broken lines in the figure; in general,
however, it is biologically impossible because of the limitations of the joint angles. The
transformation in the second stage corresponded to that of the joint angles ( ρ1 and ρ2 in
the figure) of the biologically impossible arm configuration instead of the actual arm
configuration.
Under the intrinsically-consistent condition, the participants executed the aiming
task under a different mapping between the screen and the hand coordinates in the second
stage from that in the first stage. The mapping in the second stage was symmetric to thatin the first stage with respect to the y-axis (compare the top of Figure 6A to that of Figure
6C). However, the two mappings corresponded to the same transformation of joint angles
in the intrinsic coordinates, because the arm used by the participants in the first stage wasdifferent from the arm used in the second stage (compare the bottom of Figure 6A to thatof Figure 6C).
Therefore, we can predict the two simplest results depending on whether theparticipants learn to adjust for the transformation of the joint angles in extrinsic
coordinates, i.e., the nonlinear mapping between the hand coordinates and the screen
coordinates, or in intrinsic coordinates in the first stage. On the one hand, if theparticipants learn to adjust for the transformation in extrinsic coordinates in the first stage,then a larger positive intermanual transfer of the learning effect would be observed underthe extrinsically-consistent condition than under the intrinsically-consistent condition,
because the participants would learn the same mapping under the extrinsically-consistentAdaptive Internal Model of Intrinsic Kinematics-14-

condition while they would learn mappings that differed under the intrinsically-consistentcondition (Fig. 7A). On the other hand, if the participants learn to adjust for thetransformation in intrinsic coordinates in the first stages, then a larger positive intermanualtransfer would be observed under the intrinsically-consistent condition than under the
extrinsically-consistent condition, because they would learn the same intrinsic
transformation under the intrinsically-consistent condition while they would learntransformations that differed under the extrinsically-consistent condition (Fig. 7B).
The relation between the transformation in the first stage and that in the second
stage under the extrinsically-consistent condition is clear in terms of the intrinsic
kinematics when they are represented as transformations of joint angles. The
transformation in the first stage is represented as Equation 2 while that in the second stage
is represented as

 l sinθ2 θ 1* = 1.25θ1 + (1 −1.25)π − h1 + 2 ⋅1.25arctan  2
 l1 + l2 cosθ 2 
 l sin(0.5θ2 + h2 ) −2arctan  2
 l1 + l2 cos(0.5θ 2 + h2 ) θ 2* = 0.5θ2 + h2

(5)using the joint angles of the real arm configuration ( θ 1 , θ 2 ) instead of ρ1 and ρ2 (for thefull account of the difference in the intrinsic coordinates, see Appendix A).
-----------------------------------------Insert Figure 7 about here-------------------------------------------

Methods

ParticipantsAdaptive Internal Model of Intrinsic Kinematics-15-

Eight undergraduate students (six women and two men, ranging in age from 17 to
23 years) volunteered to participate. All participants were naive as to the purpose of thisexperiment and self-reported to be right-handed. They were paid for their participation
and were treated in accordance with the "Ethical Principles of Psychologists and Code of
Conduct" (American Psychological Association, 1992).

Apparatus
The participants were seated in a dentist's chair (Nagashima SN-OP) facing a 33inch CRT screen controlled by a computer (TOSHIBA J3100ZX with a 33 MHz mainprocessor; Fig. 3). The distance between the participants’ forehead and the screen wasapproximately 1.0 m. The participants wore a custom molded cast that immobilized thewrist and a shoulder harness attached to the chair so that the shoulder position was fixed.
They moved the arm above a large board (1 m long, 2 m wide, 2 cm thickness) placed
horizontally in front of them. The height of the chair was adjusted so that the participants’shoulders were approximately at the level of the board. The participants gripped a verticalrod (10 cm long, 2 cm in diameter) in their palm. One end of the rod was firmly attachedto the center of a small light-weight board (10 cm long and wide, 0.5 cm thickness) so thatit always stood vertically on the large horizontal board. The participants were instructed tomove the rod so that the underside of the small board attached to the rod slid over the top
surface of the large board. The large board was covered with a sheet of Teflon to reduce
friction. A marker (an infrared light emitting diode) of the position recording system(Northern Digital OPTOTRAK) was attached to the top of the hand-held rod. Its position
was sampled at 100 Hz and stored in the computer. The position of the marker was
displayed as a movable cursor, i.e., a filled circle 0.9 mm in diameter. The ratio of themarker movement on the board to the cursor movement was approximately 1:1 in all
directions when the visual feedback was not altered. 1 A shield was placed above theAdaptive Internal Model of Intrinsic Kinematics-16-

board and the participants’ hand to eliminate direct vision of the hand and arm. The room
was almost completely dark.

Tasks
Each trial followed a 4-s preparatory phase: a target (a filled circle 5 mm indiameter) and a starting zone (an open circle 5 mm in diameter) appeared on the screen,
and the participant moved the cursor into the starting zone. All participants could obtainthe target easily within the 4-s interval even under transformed visual feedback. After theend of the preparatory phase, two successive auditory cues (i.e., clicks) were generated by
the computer to signal the beginning and the end of each trial. The time interval between
the first cue and the second cue was 900 ms. The participants were required to move thecursor as close to the target as possible within this time limit. They were instructed tofreeze the movement for a moment after the second cue, and then to move the cursor to thetarget at any speed. The hand trajectory was analyzed after the onset of the first auditory
cue and before the participants momentarily froze the movement.
The test area was determined for each participant considering the physicalconstraints of the arm configurations as follows: the far border of the test area was a semicircle centered at the shoulder and with a diameter equal to 85% of each participant's arm
length to ensure comfortable and stable arm movements (Fig. 8A). The near border was astraight line 30% of the arm length in front of the shoulder so that the body trunk would
not be an obstacle to hand and arm movements. The targets and starting zones were
pseudo-randomly located in the test area under the following constraints: 1) the distance
between the center of the target and that of the starting zone was less than 65% and morethan 32% of each participant's arm length and 2) those positions resulting in cursor
positions outside the border of the CRT screen after the transform were excluded. Most
locations in the right and the left corners of the test area were excluded for this reason.

Adaptive Internal Model of Intrinsic Kinematics-17-

Figures 8B and C show the pairs of targets and starting positions in one of the trial blocks
in the screen coordinates and in the hand coordinates, respectively.
-----------------------------------------Insert Figure 8 about here------------------------------------------While the participant was executing the aiming task, the visual feedback (thecursor position) was altered by a real-time transform function using a microcomputer
operation as follows. First, the joint angles (θ 1 , θ 2 ) were calculated from the position of
the marker (p, q) attached to the participant's hand according to the inverse kinematics:

 p 2 + q 2 − l12 − l22 θ 2 = arccos
2l 1l 2
 l sin θ2 
 q
θ 1 = arctan   − arctan 1.
 p l1 + l2 cosθ 2 (6)Thus, the joint angles were not directly measured, but estimated from the marker position(p, q) and the length of each participant's upper arm ( l1 ) and forearm ( l2 ). Second, thejoint angles were transformed according to Equation 2. Third, the cursor position (x, y)
corresponding to the transformed joint angles (θ 1 , θ 2 ) was calculated according to**Equation 3. It is difficult to measure the precise length of time required for the computer
to complete this operation, but it is estimated to be less than 10 ms, and the participants
did not detect a substantial time difference between their hand movement and the cursor
movement.
As indicated by Equation 2, deviations of joint angles from constant values(h1, h2 ) were multiplied by 1.25 or 0.5. These constant values were determined for eachparticipant so that the hand position (p, q) corresponding to the constant joint angles(h1, h2 ) was at approximately the center of the test area, that is, p was 0.0 (straight aheadof the shoulder position), and q was 64% of the arm length. The average of h1 wasAdaptive Internal Model of Intrinsic Kinematics-18-

37.69˚, ranging from 36.72˚ to 40.54˚, that of h2 was 100.08˚, ranging from 100.00˚ to
100.15˚, and that of q was 40.74 cm, ranging from 38.57 cm to 43.07 cm.
The above procedure for the cursor alteration is for the transformation of the joint
angles in the first stage when the participants used one arm, and it generates the mapping
between the hand and the screen coordinates shown in Figure 6A. The transformation in
the second stage under the extrinsically-consistent condition when the participants usedtheir other arm is generated by a similar procedure but we use ρ1 and ρ2 (the shoulderjoint angle and the elbow-joint angle of the other arm configuration whose forearm length,
upper arm length and positions of the shoulder and hand are the same as those of the actual
configuration: Fig. 6B) instead of θ 1 and θ 2 , respectively, after calculation of joint angles
from the position of the marker. We calculate ρ1 and ρ2 using the relations: l sinθ 2 
ρ1 = π − 2arctan 2
 − θ1
 l1 + l2 cosθ 2 
ρ2 = θ2 .

(7)(for a full account of these relations, see Appendix A).

Procedure and Experimental Design
At the beginning of the experiment, the length of the upper arm ( l1 ) and forearm( l2 ) of each participant were measured. The upper arm length was measured from theshoulder (the tip of the acromial process) to the elbow (the lateral condyle of the humerus),
and the forearm length was measured from the elbow to the center of the marker attachedto the top end of the rod held in the participant's hand. The average of l1 was 30.6 cm,
ranging from 29.4 cm to 32.3 cm, while that of l2 was 32.8 cm, ranging from 31.2 cm to35.5 cm. The transformation of the visual feedback and determination of the positions ofAdaptive Internal Model of Intrinsic Kinematics-19-

the targets and starting zones were modified according to the arm length of eachparticipant.
After measurement of the arm length, the participants were instructed about theprocedure of the task as follows (scripts have been translated to English from Japanese forthis report):
A movable cursor represented by a small filled circle, a target represented by a filled
circle, and a starting zone represented by an open circle will appear on the screen.
Move the cursor into the starting zone, and wait until the computer generates the first
auditory cue. After hearing the first cue, please move the cursor as close to thetarget as possible. Your time limit is 900 ms. At the end of this limit, the computer
will generate the second cue. You must immediately freeze your movement for a
moment. Then, you can move the cursor to the target at any speed you like.
The participants performed sixteen practice trials under normal conditions (i.e., themapping between the hand coordinates and the screen coordinates was the identity map):

x = p,y =q.

(8)

The experimenter instructed them to remember the time interval of 900 ms during thispractice period.
Following the practice trials, the transformation was described to the participants
as follows:
The cursor position on the screen will be transformed as if your shoulder-joint angle
is multiplied by 1.25 and your elbow-joint angle is multiplied by 0.5 after the
beginning of the next trial. It may be difficult to move the cursor to the target, but
you will find that the task will become easier with practice.

Adaptive Internal Model of Intrinsic Kinematics-20-

The participants were not allowed to experience any trials under transformed visualfeedback for practice.
Then, each participant began to learn the aiming task under transformed visualfeedback according to the learning schedules shown in Table I. As described earlier, theparticipants were randomly divided into extrinsically-consistent and intrinsicallyconsistent groups. All of the participants learned the aiming task using one arm in the firststage while the cursor position was altered according to Equation 2 (Fig. 6A). The
participants changed the performing arm in the second stage. The rules on how to alter the
cursor position in the second stage was different between the two groups. The cursor
position on the screen was altered so that the mapping between the screen coordinates andthe hand coordinates would be identical to that in the first stage in the extrinsicallyconsistent group (Fig. 6B), whereas it was altered so that the transformation of joint angles
would be identical to that in the first stage in the intrinsically-consistent group (Fig. 6C).
Furthermore, the participants of each group were divided into subgroups: the participantsof one subgroup used the right (i.e., preferred) arm in the first stage and the left (i.e., nonpreferred) arm in the second stage, while those of the other subgroup used the left arm in
the first stage and the right arm in the second stage.
-----------------------------------------Insert Table I about here------------------------------------------The test area in the second stage was shifted to the left or right direction so that thecenter of the test area was directly in front of the shoulder point under both conditions.
The shift in the position of the test area may have had some effect on the initial
performance of the participants in the second stage, and may have impeded thegeneralization of the learning effect to the performance in the second stage. The effect
was assumed to be minimal because Bedford (1989) reported that respondents wearingAdaptive Internal Model of Intrinsic Kinematics-21-

prism spectacles can easily generalize the effect of learning when aiming targets are
shifted in the left or right direction.
Each of the stages consisted of 12 trial blocks and lasted about 3 h. Each trial
block consisted of 50 trials of the aiming task and lasted about 12 min. A 3 min rest
period was allowed between blocks, and a 30 min rest period was allowed between thestages.
The experiment described above takes from seven to eight hours including themeasurement of arm lengths and rest time, and was conducted within a single day for eachparticipant. The participant was required to return on an additional day within one week
after the experiment to execute 14 trial blocks of the aiming task under normal conditions.
They used the right arm in seven trial blocks and the left arm in the remaining seven trial
blocks, successively. The data obtained on the second day was used in a linear regression
analysis of the aiming error.

Data analysis
The position of the marker attached to the hand-held rod was sampled at 100 Hz
and stored in the computer. Although the position recording system was carefully adjusted
before the experiment, it did not detect the position of the marker in some trials. Suchtrials were called missing trials, and were not included in the analysis described below.
The maximum number of missing trials for each stage (600 trials) was eight for any
participant.

Aiming error. After the end of the experiment, the end point of the first ballistic
movement 2 in each trial of the aiming task was determined using the curvature methods
developed by Pollick and Ishimura (1996). Figure 9 illustrates how the end point of thefirst ballistic movement was detected. The end point of the ballistic movement wasAdaptive Internal Model of Intrinsic Kinematics-22-

defined as the point where the movement was corrected for the first time. At that point,
the tangential velocity is close to zero and the curvature increases markedly. The

p˙ 2 + q˙ 2 , where p˙ and q˙ are the instantaneous

tangential velocity is defined as

velocities in the p- and q-coordinates. Figures 9A and B show a typical trajectory in one
trial. The curvature (C) at each point in the trajectory was calculated using the equationC=p˙ ˙q˙ + ˙p˙q˙,( p˙ + q˙ 2 )3 / 2(9)2

˙˙ are the accelerations at that point. The velocity was computed by a fifthwhere ˙p˙ and q
order central difference based on the interpolation polynomial and is defined asp˙ i =

p i − 2 − 8 pi −1 + 8p i +1 − pi+ 2
,
12∆t

(10)

˙ i is the i-th data value (position of the p-coordinate of the marker), and ∆t is thewhere ptime interval between data values. ∆t was 10 ms because the position was sampled at
100 Hz in this experiment. The acceleration was computed using a similar method and isdefined as

˙p˙i =

2 p˙ i− 2 − p˙ i−1 − 2 p˙ i− 1 − p˙ i +1 + 2 p˙ i +2.
7(∆t) 2

(11)

The velocity and acceleration of the q coordinates were computed in the same manner as
those of the p coordinates. In Figure 9C, the velocity and the curvature are plotted asfunctions of time from the onset of the first auditory cue signalling the beginning of thetrial. The end point of the first ballistic movement was defined as the point at which thecurvature exceeded 0.1 mm -1 for the first time after the velocity was maximum. The
distance between the center of the target and the end point of the first ballistic movement

Adaptive Internal Model of Intrinsic Kinematics-23-

was called the aiming error and used as an indicator of learning and the difficulty of theaiming task. The aiming error is expected to decrease with learning and increase with taskdifficulty.
-----------------------------------------Insert Figure 9 about here-------------------------------------------The effect of distortion caused by the transformation on the aiming error. The
aiming error may be affected by factors other than learning and the difficulty of the task.
For example, the aiming error increases as the average velocity of the movement increases,
according to Schmidt's law (Schmidt, Zelaznik, Hawkins, Frank, & Quinn, 1979). In thecurrent experiment, the distance between the target and the starting zone was randomly
determined for each trial, whereas the movement time was relatively constant, because theparticipants were required to move the cursor to the target within 900 ms. Thus, theaverage velocity of the ballistic movement was markedly different from trial to trial.
To exclude the effect of velocity, a linear model of the aiming error was assumed(E):

E = aV + b

L
+K
T

(12)

and the values of a, b, and K were estimated using linear regression analysis. Here, L isthe distance between the center of the starting zone and that of the target, and T is the
movement time (i.e., from the onset of the movement to the time when the first ballistic
movement terminates) in each trial. The onset of the movement was detected using amethod similar to that used for detecting the end point of the first ballistic movement.
This was defined as the point where the curvature last dropped below 0.1 mm -1 before theAdaptive Internal Model of Intrinsic Kinematics-24-

velocity reached maximum (Fig. 9C). Note that L/T equals the average velocity, and thesecond and third terms on the right side of the equation correspond to Schmidt's law (K isthe constant error). The first term on the right side is specific to this experiment. V
represents the magnitude of distortion caused by the transform (Fig. 10), and is defined as

r

r

r

the subtraction of one vector, S , from the other vector, G . G represents therdisplacement from the target on the hand plane to that on the screen, while S represents
the displacement from the starting zone on the hand plane to that on the screen. It is
assumed that distortion affects the aiming error (i.e., the error becomes larger as thedegree of distortion (V) becomes larger) before learning of the aiming task. Afterlearning, however, the error is small, and may become somewhat independent of thedegree of distortion. Thus, the value of a reflects learning and the difficulty of the aiming
task independent of the average velocity. The values of b and K were estimated using
linear regression analysis of the aiming error when the participants aimed at the targets
under normal conditions (data obtained on the second day).
------------------------------------------Insert Figure 10 about here------------------------------------------A simple method to investigate the effect of intermanual transfer of learning is toinvestigate the significant difference between the aiming error in the last block of the firststage and that in the first block of the last stage. However, we compared the values of aestimated for each of these blocks because the aiming error was possibly affected by thefactors described above.

Results

Trajectories and Velocity Profiles

Adaptive Internal Model of Intrinsic Kinematics-25-

Figures 11A and D show the trajectories of the hand coordinates of one participant(K.S.) in the first and the last blocks, respectively, of the first stage under the extrinsicallyconsistent condition. Figure 11G shows the trajectories in the first block of the secondstage under the extrinsically-consistent condition in the same manner as those in Figures
11A and D. Figures 11B, E and H show the tangential velocity profiles of the movements
recorded in the three blocks illustrated in Figures 11A, D and G. Although one block
consisted of 50 trials, only every other (25) trajectory and velocity profile are shown forclarity. The terminal points of these trajectories and velocity profiles correspond to theend of the first ballistic movement. Therefore, neither trajectories nor velocity profiles of
movements occurring after the end of the ballistic movement are shown. The trajectories
are approximately straight, and the velocity profiles are single-peaked and bell-shaped.
These kinematic features coincide with those of ballistic and pre-planned
movements described by other investigators (Abend, Bizzi, & Morasso, 1982; Atkeson &Hollerbach, 1985; Flash & Hogan, 1985; Morasso, 1981; Uno, Kawato, & Suzuki, 1989).
Thus, the fundamental kinematic features of the movements in the first block and those in
the last block are similar to each other. The movements in the last block, however,
become more accurate than those in the first block as shown in Figures 11C and F, which
illustrate the original trajectories (Figs. 11A and D) after normalization such that the centerof the starting zone and the target of each trial become (0, 0) and (0, 100), respectively.
The end points of the trajectories were distributed over a wide region and most ofthem undershot the target in the first block, while they concentrated near the target (Figs.
11C and F) in the last block of the first stage. However, the end points were also
distributed over a wide region in the first block of the second stage (Fig. 11I ), and thissuggested that intermanual transfer did not occur under the extrinsically-consistentcondition.
------------------------------------------

Adaptive Internal Model of Intrinsic Kinematics-26-

Insert Figure 11 about here------------------------------------------

Aiming Error
In Figure 12, the mean and standard error of the aiming error in each block of trials
are plotted separately for one of the participants in the extrinsically-consistent group(M.N.) and one in the intrinsically-consistent group (M.Y.). The mean aiming error (y)decreased with increasing number of trial blocks (x) in both the first and second stages,
and was well-fit (r>.85) by the exponential decay function:y = k 0 + k1 ⋅e − k ⋅x
2

(13)by adjusting the parameters k0, k1, and k2 to minimize the χ2 value (Fig. 12).
Furthermore, a one-way analysis of variance (ANOVA) of the mean aiming error with thenumber of trial blocks as a factor revealed that the number of trial blocks was a significant
factor (the first stage of participant M.N.: F (11, 586) = 7.38, p < .001; the second stage ofparticipant M.N.: F (11, 586) = 7.38, p < .001; the first stage of participant M.Y.: F (11,588) = 10.28, p < .001; the second stage of participant M.Y.: F (11, 587) = 12.07, p <.001). Post hoc comparisons (Tukey HSD multiple comparisons) between the trial blocks
revealed a significant difference in the mean aiming error in the first block versus the last
block in all cases (p<.001 level). These results suggest that learning occurred in all stages.
-----------------------------------------Insert Figure 12 about here------------------------------------------The mean aiming error in the first stage was compared to that in the second stage(Fig. 12) to investigate the difficulty of the aiming task in each of the stages. For theparticipant (M.N.) in the extrinsically-consistent group, the mean error in the second stage(M = 63.11, SD = 43.16) was significantly larger than that in the first stage (M = 45.05,

Adaptive Internal Model of Intrinsic Kinematics-27-

SD = 38.09; t (1197) = 7.68, p < .0001). This result suggests that the aiming task in thesecond stage was more difficult than that in the first stage. Furthermore, for the participant
in the intrinsically-consistent group (M.Y.), the error in the second stage (M = 38.70, SD =30.83) was significantly smaller than that of the first stage (M = 54.04, SD = 36.89; t'(1161) = 7.68 3, p < .0001). Thus, it is unlikely that the reason for the larger aiming error
in the second stage was that M.N. was fatigued.
The results of the remaining six participants were similar to those of M.N. or M.Y.
(Fig. 13). The mean aiming error in the second stage was significantly larger than that in
the first stage for the extrinsically-consistent group, whereas it was significantly smaller
for the intrinsically-consistent group. Because M.N. used the right (preferred) arm in thefirst stage and the left (non-preferred) arm in the second stage, the reason for the larger
error in the second stage may be that M.N (Fig. 12A) used the non-preferred arm. Notethat the results of S.M. and H.W., however, who used the left (non-preferred) arm in thefirst stage and the right (preferred) arm in the second stage, were similar to that of M.N.
Therefore, the larger error in the second stage for M.N. did not appear to be due to arm
preference.

-----------------------------------------Insert Figure 13 about here-------------------------------------------The Effect of Distortion Caused by the Transformation
Table II shows the values of b (the effect of average velocity on the aiming error)
and K (constant error) estimated from linear regression analysis of the aiming error undernormal conditions. The values of b are significant (p<.05) in the majority of cases (14 of
16 cases) and support Schmidt's law.

Adaptive Internal Model of Intrinsic Kinematics-28------------------------------------------Insert Table II about here------------------------------------------The values of the distortion effect (a) were estimated from linear regression
analysis of the aiming error pooled for each trial block (n = 50 except for some blocks
containing the missing trials) using the values of b and K shown in Table II. Figure 14
shows the estimated values of a plotted as a function of the number of trial blocks
separately for each of the participants. The majority of values were significant (p < .05).
For the extrinsically-consistent group, the 99% confidence interval 4 of the estimated value
in the last block of the first stage did not overlap that in the first block of the second stage,suggesting that the learning curves of the two stages were discontinuous. For the
intrinsically-consistent group, however, the former confidence interval overlapped thelatter except in one case (participant Y.F.), suggesting that the learning curves were
continuous. Thus, positive intermanual transfer in learning was observed in the
intrinsically-consistent group, but no positive transfer was observed in the extrinsicallyconsistent group.
-----------------------------------------Insert Figure 14 about here-------------------------------------------

Discussion

To investigate whether the intrinsic kinematics is adaptively represented in the
CNS in the control of arm movements, visual feedback during an aiming task was alteredby a real-time transform function based on the magnification of the elbow- and shoulder-Adaptive Internal Model of Intrinsic Kinematics-29-

joint angles of the participants. A linear transformation of joint angles (intrinsic
coordinates) corresponded to a nonlinear transformation of hand and screen coordinates(extrinsic coordinates).
The present study examined whether the participants learned to adjust for atransformation of joint angles in intrinsic coordinates or in extrinsic coordinates, byinvestigating whether positive intermanual transfer occurred under the extrinsicallyconsistent condition or intrinsically-consistent condition. The participants practiced theaiming task under a linear transformation of the joint angles in the first stage of bothconditions. On the one hand, when the participants executed the aiming task under thesame mapping between the screen and the hand coordinates using the different arm in the
succeeding stage (extrinsically-consistent condition), no positive intermanual transfer wasobserved, that is, 1) the aiming error in the second stage was larger than that in the firststage, and 2) the learning curves of the two stages were discontinuous. On the other hand,when the participants executed the aiming task under the same transformation of the joint
angles using the different arm in the succeeding stage (intrinsically-consistent condition),
positive intermanual transfer was observed, that is, 1) the aiming error in the second stage
was smaller than that in the first stage, and 2) the learning curves of the two stages were
continuous. The results suggest that the participants learned to adjust for thetransformation in the first stage in intrinsic coordinates.
The fact that positive intermanual transfer of learning was observed under the
intrinsically-consistent condition suggests that adjustments were made to a central
representation not specific to either arm but common to both arms. This result is at least
partly consistent with previous physiological findings. Tanji, Okano, and Sato (1987)
reported that the firing rates of primary motor cortex neurons are directly correlated with
contralateral muscle activity. The majority of non-primary motor cortices, such as the
supplementary motor cortex and the pre-motor cortex, however, do not appear to code theAdaptive Internal Model of Intrinsic Kinematics-30-

activity of particular muscles, because their discharge rates are independent of the handused (i.e., their firing rates increased when the monkey used the right hand or the lefthand). Rather, these neurons appear to encode the particular motor task (in this case"pressing a button"). These physiological findings, indicate that there are two functional
levels in motor control systems, one level is hand- (or muscle-) specific and the other is
non-specific. Presently, however, there are little data concerning the neurological locus orcorrelates of motor learning specific to either arm or both arms.
One possible reason why the aiming error was larger in the second stage under the
extrinsically-consistent condition compared to that in the first stage is as follows. The
mapping between the hand and screen coordinates used in the second stage under the
extrinsically-consistent condition was very complicated and nonlinear as represented by
Equation 5 while the transform in the first stage was simply represented as Equation 2.
Although there is no assurance that the mathematical complexity and nonlinearity
correspond to the difficulty of learning for a biological system, it appears that thetransformation by Equation 5 is more difficult in the CNS than the simple magnificationsof joint angles represented by Equation 2. However, it is unclear whether the participantslearned the transformation used in the second stage under the extrinsically-consistentcondition in intrinsic coordinates. There is a possibility that the participants adjusted tothe transformation in the first stage in terms of intrinsic coordinates and adjusted to thetransformation in the second stage in terms of extrinsic coordinates. One participant of theextrinsically-consistent group (M.N.) reported that he tried to remember the relation
between the hand and the screen planes in the later blocks in the second stage.
In any case, the fact that positive intermanual transfer could not be observed underthe extrinsically-consistent condition implies that the participants tried to adjust for thedifferent transformations in the intrinsic coordinates or that they tried to adjust for thetransformations in coordinates that differed in the first stage from the second stage.

Adaptive Internal Model of Intrinsic Kinematics-31-

It is still unclear whether the participants learned to adjust to the transformation of
joint angles in terms of kinematic parameters or dynamic parameters (e.g., joint torque andmuscle tension). It appears unlikely, however, that the participants learned it in terms ofdynamic parameters, because the relation between the intrinsic kinematic coordinate
variables and the intrinsic dynamic coordinate variables is highly nonlinear. A linear
transform of joint angles becomes nonlinear and complex when represented in dynamic
parameters. In other words, when the shoulder- and elbow-joint angles are multiplied by α
and β, respectively, the inertial and viscous forces of the shoulder joint increase
approximately in proportion to α, the centripetal and Coriolis forces increase
approximately in proportion to α2 and αβ, respectively, the elastic force is constant, andthe resultant joint torque is the sum of these forces. Furthermore, this nonlinear andcomplex transform of joint torques is unlikely to become linear and simple when it isrepresented as other dynamic parameters that the biological motor system may use (e.g.,muscle tension and motor neuron activity). Thus, it appears reasonable to assume that theparticipants learned to adjust to the transformation of joint angles in terms of kinematicparameters.
Recent behavioral and neurophysiological studies have suggested how the intrinsic
kinematics is used in the CNS to control arm movements. Some of them indicate that the
intrinsic kinematics is used to represent the target's locations and others indicate that it isused to represent the motion trajectories. Soechting and Flanders (1989a, 1989b)
investigated pointing errors in a three-dimensional space and showed that they can accountfor the observed errors on assumption that the CNS does not directly compute the muscle
tensions from the visual information of the target's locations relative to the shoulder butthat it computes the shoulder and the elbow joint angles from the target's locations.
Graziano, Yap, and Gross (1994) found that the position of the visual receptive field of
neurons in the premotor cortex of monkeys changes as the intrinsic kinematic information

Adaptive Internal Model of Intrinsic Kinematics-32-

changes (i.e., hand position represented by joint angle or muscle length). They showedthat these neurons respond to visual stimuli and that the visual receptive field includes
areas that surround sections of the surface of the body, such as the hand or arm.
Furthermore, the visual receptive field moves with the arm, even when the arm is
passively moved without visual feedback. These studies suggest that the intrinsic
kinematics is closely related to the visual information of the target locations.
Kalaska and his colleagues suggested that a trajectory is represented in terms of
intrinsic kinematics by neuronal activity in area 5 in monkeys. As mentioned in the
Introduction, area 5 neurons encode the intrinsic kinematics of arm movements (Kalaska,
Cohen, Prud'homme, & Hyde 1990; Lacquaniti, Guigon, Bianchi, Ferraina, & Caminiti, inpress). Kalaska et al. (1990) reported that some of the directionally-tuned cells in area 5
exhibit marked changes in discharge prior to reaching movements. The time function of
their discharge rate closely resembles that of the movement velocity (i.e., a bell-shaped
function). This suggests that the activity of these phasic cells encodes a time sequence of
movement velocities along the path. Rosenbaum and his colleagues (e.g., Rosenbaum,
Loukopoulos, Meulenbroek, Vaughan & Engelbrecht, 1995) proposed a kinematic modelof motion control (Knowledge model) in which postures are stored as vectors in joint
space and used to select reaching movements evaluating accuracy costs and travel costs.
In conclusion, the results of the present study suggest that a representation of
intrinsic kinematics is formed in the CNS and used in the control of reaching movements
during the learning of an aiming task. Moreover, this representation appears to becontinuously adaptive, able to adjust in response to novel circumstances. The presence ofan adaptive representation corresponding to the intrinsic kinematics in the CNS supports
the hierarchical model rather than the non-hierarchical model of visually-guided reaching
movements.

Adaptive Internal Model of Intrinsic Kinematics -Adaptive Internal Model of Intrinsic Kinematics-34-

where, θ 1 and θ 2 are the joint angles after the transformation, k1 and k 2 are**

magnification or minimization factors, and h1 and h2 are constant values. The
transformation of joint angles of the other arm configuration ( ρ1 , ρ2 ) in the same manneras Equation 15 givesρ1* = k1 ρ1 + h1
ρ*2 = k2 ρ 2 + h2 .

(

(16)

)

The transformed joint angles ρ1 , ρ2 can be represented by those of the actual arm

(

)

*

*

configuration θ1 ,θ 2 using the same relation as Equation 7:**

 l 2 sin ρ *2 *
θ = π − 2arctan
*  − ρ1
 l1 + l2 cos ρ 2 *1

θ 2* = ρ2 .

(17)

Substituting Equation 7 into Equation 16 and then substituting Equation 16 into Equation
17 gives l sinθ 2 
θ 1* = k1θ 1 + (1 − k1 )π − h1 + 2 ⋅k1 arctan 2
 l1 + l2 cosθ 2 
 l sin(k2θ 2 + h2 ) −2arctan  2l1 + l2 cos(k2θ 2 + h2 ) θ 2* = k 2θ 2 + h2 .

(18)Thus, Equation 5 is obtained by replacing k1 and k 2 with 1.25 and 0.5, respectively.

Adaptive Internal Model of Intrinsic Kinematics-35-

Appendix B
Additional Experiment to Test the Effect of a Slightly Different Ratio of the Marker
Movement to the Cursor Movement

An experiment was conducted to test the effect of different ratios of the marker
movement to the cursor movement in the x and y directions. The aiming error wascompared when the ratios in the x and y directions were identical (1.0:1.0 in both
directions) to that when the ratios were different (1.0:1.0 in the x direction and 1.0:0.8 inthe y direction).

Methods
Participants. Three volunteers who did not participate in the main experimentparticipated in this experiment. All participants were naive as to the purpose of thisexperiment and self-reported to be right-handed.

Apparatus and procedure. The apparatus was the same as that used in the main
experiment, and the procedure was similar. Visual feedback was not altered in thisexperiment. There were two types of trials: identical and different trials. In the identical
trials, the ratios in the x and y directions of the marker movement to the cursor movement
were identical (1.0:1.0 in both directions). In the different trials, they were different(1.0:1.0 in the x direction and 1.0:0.8 in the y direction).
The participants completed two blocks of 50 trials. The two types of trials wererandomly mixed within a block so that the total number of each type was 25. A 3 min rest
period was allowed between blocks. The participants were not informed of the two typesof trials nor of the ratio of the marker movement to the cursor movement.

Adaptive Internal Model of Intrinsic Kinematics-36-

Results and Discussion
Table B1 shows the mean and standard deviation of the aiming error in theidentical trials and different trials. A t-test with 98 df was performed separately for thedata from each participant. There was no significant difference between the two types oftrials. Additionally, none of the participants realized that different types of trials existed.
The results suggest that the slight difference of the ratio did not significantly affect theaiming error. Therefore, the data of the participants (H.W., M.N., Y.F., and M.Y.), who
conducted the aiming task with different ratios in the main experiment, was analyzed in
the same manner as that of the other participants.
-----------------------------------------Insert Table B1 about here-------------------------------------------

Adaptive Internal Model of Intrinsic Kinematics-37-

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Author Notes

Hiroshi Imamizu, Yoji Uno, and Mitsuo Kawato, Advanced Telecommunications
Research (ATR) Human Information Processing Research Laboratories, Kyoto, Japan.
Hiroshi Imamizu is now a member of the Kawato Dynamic Brain Project, Japan Scienceand Technology Corporation, Kyoto, Japan. Yoji Uno is now at the Department ofInformation and Computer Sciences, Toyohashi University of Technology, Aichi, Japan.
We are grateful to Frank E. Pollick for developing the method of detecting the end
point of the first ballistic movement. We thank Carol A. Fowler and Helen Cunningham
for their helpful suggestions.
Correspondence concerning this article should be addressed to Hiroshi Imamizu,
Kawato Dynamic Brain Project, Japan Science and Technology Corporation, 2-2
Hikaridai, Seika-cho, Soraku-gun, Kyoto, 619-02, Japan. Electronic mail may be sent via
the Internet to imamizu@erato.atr.co.jp.

Adaptive Internal Model of Intrinsic Kinematics-42-

Footnotes

1

For four participants (H.W., M.N., Y.F., and M.Y.), the ratio in the x direction (1.0

: 1.0) was slightly different from that in the y direction (1.0 : 0.8, see Figure 5 for thedefinitions of the directions) due to an incorrect setting of the CRT monitor. The ratio in
the x direction was equal to that in the y direction (1.0 : 1.0) for the other participants.
Therefore, an additional experiment was conducted to test the effect of the different ratios(see Appendix B). The results of this experiment clearly showed that the ratio difference
did not significantly affect the aiming error. Consequently, the data of the four
participants were analyzed using the same methods as that for the other participants.

2

Woodworth (1899) made a distinction between the components for voluntary

movements in the aiming task: an initial impulse phase and a series of secondary
adjustments made subsequently to attain the final target position. The first component is afast, preprogrammed ballistic movement that brings the hand into the general area of thetarget. The second component comprises a number of adjustments. In this latter phase,
movements are continuously monitored and adjusted in relation to sensory information.
The first component, the initial impulse phase in Woodworth’s terminology, can be called
a ballistic movement and the second current control component can be called a corrective
movement (Flowers, 1975).

3

A Welch test was used instead of a t-test when the variances of the two groups

could not be considered to be equal to each other, that is, the F ratio of one group to theother was significantly large (p < .05).

Adaptive Internal Model of Intrinsic Kinematics

4

-43-

In the linear regression analysis, the 100(1-α)% confidence interval of theestimated value ( βˆ ) is defined asβˆ − t(α /2,n − p)⋅ SE ≤ β ≤ βˆ + t(α /2, n − p) ⋅SE ,where n is the number of data, p is the number of parameters in the assumed linear model,t(α /2,n − p) is the upper 100(1-α/2)% point of the t-distribution, and SE is the standard
error of the estimated value.

Adaptive Internal Model of Intrinsic Kinematics-44-

Table I Learning schedule of each participantGroup

Participants

Arm used

Arm usedin the first stage

in the second stage

Extrinsically-

H.W. and S.M.

Left

Right

consistent

M.N. and K.S.

Right

Left

Intrinsically-

Y.F. and Y.H.

Left

Right

consistent

M.Y. and K.A

Right

Left

Adaptive Internal Model of Intrinsic Kinematics

Table II.

-45-

Estimated values of b and K from the aiming error under normal conditions.

Right Arm

Left Arm

Participantb (mm)K (mm)b (mm)K (mm)

H.W.

12.73n.s.

16.64***

25.80***

10.10**

S.M.

12.56*

12.92***

37.84***

2.27 n.s.

M.N.

55.65***-3.57n.s.

55.65***

10.66***

K.S.

47.08***-3.57n.s.

73.99***-10.12*

Y.F.

45.31***

3.66n.s.

13.93n.s.

19.71***

Y.H.

18.84*

16.62***

62.54***

0.29n.s.

M.Y.

44.31***

7.75n.s.

27.35**

14.50**

K.A.

28.21***

10.08**

18.87***

10.09**

Note. n.s. Not significant; * p <.05; ** p <.01; *** p <.001;
b and K are defined in Equation 12.

Adaptive Internal Model of Intrinsic Kinematics

Table B1.

Means and standard deviations of the aiming error for identical anddifferent trials for each participant.

Identical

Different

Participant

M

SD

M

SD

t-valueprobability

R.O.

24.53

12.12

23.97

11.29

0.04

.83

T.F.

34.09

18.16

28.90

17.74

2.09

.15

T.Y.

36.98

21.62

37.72

20.95

0.03

.86

Note. Units of the mean (M) and the standard deviation (SD) are in mm.

-46-Adaptive Internal Model of Intrinsic Kinematics-47-

Figure Captions

Figure 1

Schematic diagram of visual-motor control based on a hierarchicalapproach. The top row illustrates the computational modules and the intermediate
representations connecting each module. The next two rows illustrate examples ofrepresentations in robot control. (x, y) denotes the Cartesian coordinates of the object's
position. θ 1 and θ 2 denote the shoulder-joint angle and elbow-joint angle, and τ 1 and τ 2
denote the shoulder-joint torque and elbow-joint torque. Paths in the joint space and in thetorque space are arbitrary examples in point-to-point reaching movements.

Figure 2

Schematic diagram of visual-motor control based on a non-hierarchicalapproach. The computational model is at the top with the neural network model
underneath (Massone and Bizzi, 1989).

Figure 3

Overview of the experimental setup illustrated from behind the rightshoulder of the participant.

Figure 4

An illustration of the transform in extrinsic coordinates (f) in the top figure

and that in intrinsic coordinates (g) in the bottom figure. (x, y) denotes the Cartesian
coordinates of the hand position with the origin at the shoulder. θ 1 and θ 2 denote theshoulder-joint angle and elbow-joint angle. The filled circles indicate the hand positions
before the transformation and the open circles indicate those after the transformation.

Figure 5

An illustration of joint angle transformation. The top portion is a front

view of the CRT screen, and the bottom is a top view of the participant. The shoulder- andAdaptive Internal Model of Intrinsic Kinematics-48-

elbow-joint angles (θ 1 and θ 2 , respectively) when the hand is located at a fixed point forthis transformation (indicated by a cross), are h1 and h2 , respectively.

Figure 6

Illustrations of the experimental design. The grids in each of the panels

show the mapping between the screen coordinates (x-y: the upper panels) and the handcoordinates (p-q: the lower panels). As the participant moved the hand along the straight
grid lines (the lower panels), the cursor trajectories on the screen became curved lines (theupper panels). The participants learned the same mapping between the screen coordinatesand hand coordinates in the first and the second stages under the extrinsically-consistentcondition (A and B) while they learned the same transformation of the joint angles in bothstages under the intrinsically-consistent condition (A and C). θ 1 and θ 2 denote theshoulder-joint angle and elbow-joint angle of the actual arm configuration (solid lines) of
the participants. ρ1 and ρ2 denote the joint angles of the other geometrically possible
arm configuration (broken lines) whose forearm length, upper arm length and positions of
the shoulder and hand were the same as those of the actual arm configuration.

Figure 7

Predictions of the results under the extrinsically-consistent condition andthe intrinsically-consistent condition. A Illustration of the prediction of the results on theassumption that the participant learns to adjust for the transformation in the first stage as anonlinear one in terms of the screen and hand coordinates (i.e., extrinsic coordinates). B
Illustration of the prediction of the results on the assumption that the participant learns toadjust for the transformation in the first stage as a linear one in terms of the joint angles(i.e., intrinsic coordinates).

Figure 8

Determination of test area and starting and target positions. A Definitionof the test area (hatched area); top view of the hand plane. B Targets (filled circles) andAdaptive Internal Model of Intrinsic Kinematics-49-

starting zones (open circles) in a trial block on the screen (in the screen coordinates). Eachpair connects a target and a starting zone (in a trial) by a solid line. The broken lines
indicate the border of the screen. These fifty pairs were used in one of the trial blocks forparticipant K.A. C Targets and starting zones on the board (in the hand coordinates)
corresponding to those shown in Figure B. The broken lines show the border of the testarea.

Figure 9

Detection of the first ballistic movement. A and B A typical trajectory:

panel A illustrates the entire trajectory, and panel B focuses on the part of it near thetarget. The open circle and the filled circle indicate the starting zone and the target,
respectively. The square indicates the end point of the first ballistic movement. C
Tangential velocity (solid line) and curvature (gray bars) are plotted against time from theonset of the first auditory cue signaling the beginning of the trial.

Figure 10

Definition of V indicates the degree of distortion caused by thetransformation. A and B illustrate trajectories of the cursor on the CRT screen and thehand on the board, respectively. These two figures are overlapped in C. The vector S
represents the displacement from the starting point on the board to that on the screen, andthe vector G represents the displacement from the target on the board to that on the screen.
A scalar V is defined as the absolute value of subtraction of vector S from vector G.

Figure 11

Trajectory and velocity profiles. A, D and G Trajectories of the handcoordinates of one participant (K.S.) in the first and the last trial blocks of the first stage,and in the first trial block of the second stage, respectively. The dots are 10 ms apart (i.e.,sampled at 100 Hz). (0, 0) is a fixed point for the transformation (position of the shoulderof the participant is (0, -384)). Every other trajectory (25 each) of the trial blocks isAdaptive Internal Model of Intrinsic Kinematics-50-

superimposed on each graph. The terminal points of the trajectories correspond to the endof the first ballistic movement. B, E and H Tangential velocity profiles of the trajectories
shown in A, D and G, respectively. C, F and I Trajectories that were normalized so thatthe center of the starting zone and the target of each trial became (0, 0) and (100, 0),
respectively. The filled circles indicate the normalized position of the target.

Figure 12

Aiming error as a function of the number of trial blocks. A shows a typicalresult of a participant in the extrinsically-consistent group (M.N.) while B shows that of aparticipant in the intrinsically-consistent group (M.Y.). The error bars represent thestandard error of the mean, and the gray bars indicate the mean aiming error of each stage.
The solid lines and equations show the best fitting lines using the least squares method. r
is the correlation coefficient between the mean aiming error and the predicted best fit
value.

Figure 13

Aiming error as a function of the number of trial blocks for the participants

other than M.N. and M.Y., whose results are shown in Figure 12. The conventions are thesame as for Figure 12. (Right) and (Left) indicate the arm that the participant used in eachof the stages.

Figure 14

Effect of distortion caused by the transform on the aiming error as afunction of the number of trial blocks. The errors of each trial block were analyzed using
linear regression analysis. The open circles, gray filled circles, and black filled circles
indicate that the estimated values are significant (p < .05), marginal (.05 ≤ p ≤ .1), andnot significant (p > .1), respectively. The error bars represent 99% confidence intervals,
and the gray rectangles highlight the confidence intervals of the last block of the first stageAdaptive Internal Model of Intrinsic Kinematics-51-

and the first block of the second stage to show continuity or discontinuity of the learning
curves. (Right) and (Left) indicate the arm that the participant used in each of the stages.

Coordinate
Controltransformation

VisualsystemTrajectory
planning

Control

Target locationin extrinsic space

Target location
in intrinsic space

Trajectory
in intrinsic space

y

θ2

θ2

TargetTargetx
Extrinsic kinematics

θ1

Current
positionmusculoskeletal
systemMotor commandin intrinsic space

τ2
Targetτ1

θ1Intrinsic kinematics

Intrinsic dynamics

θ2

y

τ2
x

θ1

τ1
Fig. 1

VisualsystemCoordinate transformation
Trajectory planning
Control

Target locationin extrinsic spacemusculoskeletal
systemMotor commandin intrinsic space

State
units

Output units
Hidden units
Plan units

Fig. 2

33-inch CRT screen
Cursor
Occluder1.0 m

θ2MarkerBoard (desktop)

Extrinsic coordinates(e.g., Task-oriented
y(x, y)

visual coordinates)(x * , y * )(x * , y * ) = f (x, y)xIntrinsic coordinates(e.g., Joint angles)

θ2(θ1* ,θ 2* ) = g(θ1,θ 2 )
θ 2*

θ1

θ1*

Fig.4

ScreencoordinatesyX

Cursor

θ 2*+
CRT screen

x

θ 1*

θ 1* = 1.25(θ1 − h1 ) + h1
 *
θ 2 = 0.5(θ 2 − h2 ) + h2
Hand
coordinates

q

P
Board
(Desktop)

+

θ2

θ1

p

Fig. 5

A First stage of any condition

B Second stage of

extrisically-consistent condition

C Second stage of

intrisically-consistent condition

Extrinsically-consistent
Screencoordinates

Screencoordinatesy

Screencoordinates

y

x

x

θ 1* = 1.25(θ 1 − h1 ) + h1
 *
θ 2 = 0.5(θ 2 − h2 ) + h2
Hand
coordinatesHand ρ1* = 1.25(ρ1 − h1 ) + h1
 *
 ρ 2 = 0.5(ρ2 − h2 ) + h2
Hand
coordinates

q

y

θ2

x

θ 1* = 1.25(θ 1 − h1 ) + h1
 *
θ 2 = 0.5(θ 2 − h2 ) + h2

qHand
coordinates

θ2

q

θ2

ρ2
Elbow

θ1

θ1
Shoulder

θ1

ρ1
p

p

p

Intrinsically-consistentFig. 6

Extrinsically-consistent condition
First Stage

Intrinsically-consistent condition

Second Stage

First Stage

Second Stage

screenMapping between
the screen and thehand coordinates

First Stage

Second Stage

Aiming Error

A

Learning
at the extrinsic
coordinate levelin the first stage

Aiming Error

hand# of trials

Aiming Error

Second Stage

Aiming Error

B

First Stage

Second Stage# of trials# of trials

Learning
at the intrinsic
coordinate levelin the first stage

First Stage

First Stage

Second Stage# of trials

Fig.7

q

A
Arm length

Far border
85% of thearm lengthp
Near border 30% of the arm lengthy (cm)

Bx (cm)q (cm)Cp (cm)

Fig.8

End of the firstballistic movement

A
160

Starting zone

q (mm)

120
80

Target

40
-200

-100

0

100

p (mm)B

Aiming Error

Movement
onset

C

End of the firstballistic movement
Corrective
movement
0.2

600
500
400

0.1

300
200

Curvature (mm-1)

Tangential Velocity (mm/s)

The first ballistic
movement

100
0

0.0
0

Onset of thefirst auditory cue

4080

Movement
time (T)120

Time (ms)Fig. 9

A Screen coordinatesy

C Overlapped

r
G

x

r
S

B Hand coordinatesqr r
V = G−Sp

Fig. 10q (mm)q (mm)q (mm)A

D

G
p (mm)

Tangential Velocity (mm/s) Tangential Velocity (mm/s)

p (mm)

p (mm)

Tangential Velocity (mm/s)

B

E
Time (10 ms)

Time (10 ms)

HTime (10 ms)

q' (relative scale)q' (relative scale)q' (relative scale)

C

F

I

p' (relative scale)p' (relative scale)p' (relative scale)Fig. 11

B Intrinsically-consistent GroupA Extrinsically-consistent Group(Participant: M.Y.)(Participant: M.N.)
1st Stage
2nd Stage(right arm used) (left arm used)

Aiming Error (mm)y = 49.4

y = 35.7

100+ 51.3 e-0.370x(r = .879)+ 40.7 e-0.508x(r = .882)

80

1st Stage
2nd Stage(right arm used) (left arm used)
y = 27.8 + 55.1 e-0.162x(r = .960)
y = -1.66 + 61.9 e-0.0881x(r = .960)

60
40
20

p <.0001(t-test)p <.0001(Welch-test)

0
1

12

1

12

1

12

1

12

# of Trial Blocks (1 Block=50 Trials)

Fig. 12Extrinsically-consistent Group

Intrinsically-consistent Group
140

100

K.S.
(Right )(Left)80

K.A.
(Right )120(Left)

100

60

80

40

60

20

p<.0001 (Welch test)

40

0

20

p<.0001 (Welch test)

0

Aiming Error (mm)

100

S.M.
(Left)

80(Right)100

Y.F.
(Left)(Right)

80

60

60

40

40

20

p<.0001 (t test)

20

p<.0001 (t test)

0

0
120

100

H.W.
(Left)(Right)100(Right)

80

80

60

60

40

40

20

Y.H.
(Left)

20

p<.0001 (t test)0

p<.0001 (Welch test)

0
1

12

1st stage

112

2nd stage

1

1st stage

12

1

12

2nd stage# of Trial Blocks (1 Block=50 Trials)
Fig.13Extrinsically-consistent Group
H.W.
Stage 1
(Left)

0.5

Effect of Distortion on Aiming Error

0.4

0.3

0.2

0.2

0.1

0.1

0.0

0.0

0.6

S.M.
Stage 1
(Left)

Stage 2 0.5(Right)
0.4

0.3

M.N.
Stage 1
(Right)Intrinsically-consistent Group
Stage 2(Right)K.S.
Stage 1
(Right)Stage 2
(Left)

12

Stage 2(Right)K.A.
Stage 1
(Right)Stage 2
(Left)

0.2
0.0
M.Y.
Stage 1
(Right)0.5
0.4

Stage 2
(Left)

0.8
0.6

0.3

0.4

0.2

0.0
12 1

Y.H.
Stage 1
(Left)

0.4

0.0

0.1

1

0.6

0.1

0.2

0.0

0.4

Stage 2(Right)

0.2

0.3

0.2

0.5

0.3

Stage 2 0.5(Left)
0.4

0.4

Y.F.
Stage 1
(Left)

1# of Trial Blocks

12 1

12

0.1

0.2

0.0

0.0
1

12 1

12

1

12 1

12

# of Trial Blocks

Fig. 14