Modeling Kinematic and Dynamic of Human Arm Movement

Background

Study of human arm movement is still an emerging research until now. For years,

much research has been conducted in this field which varies in specific movement

tasks and describes performance of those movements. As other gait analysis, human

arm movement can be related to many applications in real world such as in sport,

entertainment, and clinical rehabilitation. The measurement and analysis can be

different for each task which depends on its purpose. The observation and assessment

for clinical application will be different from those of sport performance.

There are several steps performing by human to do arm motion. Human motion

trajectory is firstly planned in central nervous system (CNS). It requires the help of

visual system to provide the spatial location of the final target to decide the desired

trajectory. Then, the visual coordinate is transformed to body coordinate such as joint

angles or muscle lengths followed by generation of force by muscles to realize the

desired trajectory. These steps may be done simultaneously.

Problems in modelling human Arm

Many attempts have been done to model human arm movement. One of the

difficulties of modelling human arm is that it is task-dependent. Different from

walking which involves only repeating movement, human arm movement can be

differentiated into several movement depend on its task (non-cyclic movement). In

this case, mechanical characteristics of the arm will change based on the arm

configuration. A model can only give a good performance for a specific task based on

the context and its instruction. No standard protocol and adopted system are available

(Rau et.al, 2001). It even shows in existing research that none of model could give a

good prediction of data.

Recently, a number of models of human arm movement with different complexity

have already been developed for a better understanding of its biomechanics. Many

existing research tried to adopt robotic manipulator principle to form a model of

human arm movement. By this, they stated that the end-effector or in this case hand

trajectory is firstly planned at the object level (Flash, 1998).However, Admiral et.al

(2004) argues that movements actually are planned at kinematic levels such as in joint

level and the muscle will generate forces to execute the planned movements.

In doing experiment of arm movement, several restrictions have to be applied such

that all task have to be simple, easy to be applied, reliable, not time consuming and

clear in result. Therefore, in many arm movement experiments mostly focusing on

two different tasks with visually guided which are pointing and reaching movement.

These two movements represent arm movement with fully extended arm and

changing between fully extended and unextended arm. The task involves two joints

which are shoulder and elbow which then can be assumed as movement in 2

dimensions.

Most of the arm movement were tested in planar horizontal plane. However, they

could not represent the real movement which mostly performed in vertical plane. By

testing in vertical plane, the influence of gravity in this case should not be neglected.

In ….(..), it showed that gravity has a insignificant influence in the arm trajectory

formation. Kodek and Munih(2003) argued with this result as in their experiment, it

shows that arm dynamic especially at low velocities is only influenced by the sum of

gravitational effects, passive moment and velocity dependent. Yet, their study only

used programmed trajectories which allowed a very good motion repeatability which

is not possible in normal unconstrained movements. Moreover, they only inspected

the trajectories of elbow joint by claiming that the other joint do not have any

dynamic contributions to the dynamic of motion.

One of the commonly problems in arm movement is the flexibility of arm to execute

either a simple or complex movement by using various posture. The difficulties of

accessing human arm movement trajectory is a very large number of possible

trajectory that which could be performed by arm to move from a point to another

point in unconstrained environment which determined either by the path or the time

required to reach each point within a path(Sanger, 2000). This raises a question of

how human can select a particular trajectory to position their arm into a specific

target….

One of the confirmed results from existing research is that human arm is consistent

and reproducible. By using several model to predict the trajectory of human arm it

showed that there is a tendency for human to move their arm in straight trajectory

with almost a bell-shaped velocity profile. However, experiment on human arm

movement showed that the trajectory of arm mostly formed a curved profile. This

could also support the argument that most of the models do not accurately describe

the movement.

As mentioned above, the arm trajectory is firstly planned in CNS. Therefore, it can be

assumed that the straight trajectory is also planned before do the motion. In fact, most

experiment demonstrated that hand trajectory commonly in curved profile. This case

happened in spontaneous hand movement. Nevertheless, after repeating the same

motion several times, hand tends to form almost straight trajectory with bell-shaped

velocity profile (……).

Moreover, stability is other very important factor in human movement. The major aim

in controlling human movement is to make a response or movement trajectory (after

giving small perturbation or noise) remains close to the undisturbed

trajectory/response. However, it is very difficult to obtain stability in human

movement as there is large variability in consecutive performances of the same action.

The environment interaction also adds to the variability of the movement. The human

motor control as with any physical system produces movement in finite duration of

time.

In examining human arm stability, there are two factors that have to be considered in

the model. Firstly, joint stiffness which is defined as loss range of motion of human

arm. Secondly, mechanical impedance is dynamic mapping of motion to force.

Impedance of human arm depends on position (Mussa-Ivaldi et.al 1985), force (Gomi

and Osu 1998; Perreault et.al 2001) and instability (Burdet et.al 2001). Therefore, to

obtain impedance of human arm, many movements have to be done in one

measurement. However, it can be either done in static position (Mussa-Ivaldi et.al

1985) or during arm movement (Bennet et.al 1992; Bennet 1993; Milner 1993; Gomi

and Kawato 1997; Burdet et.al 2000). To answer the question, Tee et.al (2004)

introduced a simple model of joint torque and impedance of human arm movement by

considering every dynamic interaction between arm and environment. However, their

model cannot be adapted in all environment quite well as they only design the model

on joint-based.

Moreover, several research use joint space model to describe the movement of arm.

Burdet et.al. (2005) consider the muscle force which produced in joint limb in

modelling the movement. The model is build based on inverse dynamic model of the

task which is arm movement in horizontal plane. Arm is described as two links

(shoulder and elbow) which move ahead of the body in x-y plane to reach one target

point. The result shows that human arm movement is generally stable as muscle

elasticity and reflex compensate the perturbation which adding to arm motion.

In addition, external forces in arm movement experiment are difficult to access in real

application. In many experiment, robot manipulator have been used in assisting

human arm to ensure that the arm will follow specific trajectory. It could also help in

measuring external forces which could be given to the arm.

By looking all problems that related in human arm movement, it could be questioned

how the kinematic of arm affect the formation of hand trajectory in moving arm

between points.

Posture-Based Principle

Donder’s law

This model is derived based on the movement of eyes. The method to model

movement of eyes in 3D can also be applied to arm movement. It states that

Equlibrium trajectory hypothesis.

Study of human arm movement is mostly focusing on the concepts of equilibrium and

virtual trajectories as a means of executing movement of the arm. Equilibrium point

hypotheses states that muscle-generated points as a natural description of equilibrium

or attractor point of muscle activation have sufficient influence to generate two-joint

arm movements. Most of existing research show that this attractor point exist in the

virtual path or trajectories of human arm movement but it just has small influence

compared to the dynamics of human movement. Won and Hogan (1995) argued this

statement by proving that a moving attractor point has significant influence in the

dynamic of human arm movement. Their results show that this attractor point is

bounded closely to the actual trajectory of human arm movement. Therefore, this

point can be used in planning and coordinating multijoint movement which is aligned

to the equilibrium hypotheses.

However, the model based on this principle has a weakness that it has too many

version about the formation of hand trajectories. The formation can be performed

either in single joint level, single muscle level or in single effector level. By this

reason, the general prediction based on this model with respect to arm postures would

be impossible to be performed.

Principle of minimum cost

This model is conducted based on principle of minimum work. The common principle

in this model is that human tends to do work that consumes the minimum energy. As

mentioned above, in moving their arms, they tend to follow straight trajectory with

bell-shaped speed profile(….).As it is counted as the path that requires minimum

work to move arm from one point to another point, the tendency could be assumed as

one of the possible human strategies to save energy.

However, in other study of arm movement (Atkenson et.al, 1985; Isenberg and

Conrad,1994) shows that the trajectories of human arm in planar horizontal as well as

unrestrained vertical arm movements are not ideally straight but show some amount

of curvature profile depending on the workspace.

.

Proposed Method

Spherical Pendulum Model

Pendulum model can be used to quantitatively describe the position of a body or

segment. Commonly used tools is in Cartesian coordinate.

Within the available research, the study of human arm which use robot manipulator

theory to model human arm contributes a significant number. The researcher used the

system that resembles the actual system of human arm. This can be done by assuming

that for both robot and human arm, mechanical objectives of movement and

manipulation are identical and by applying the same law of physics. However, this

system cannot be used to represent the “real” human arm system as its model involves

many constraints that cannot be applied to human model.

The position of arm should only depend on joint angle but not on other property of

arm geometry such as length of the limb (Cruse,1993). However, as we seen on

mathematical model of human arm……………….

Mathematical Modeling

The human arm considering two limbs, upper arm and forearm, with two joint

(shoulder and elbow). In this case, wrist is considered as a part of forearm. and

represents the length of upper arm and forearm respectively. Link masses and inertia

are denoted by and where . All links are considered rigid and frictions at

each joint are neglected.

Picture 1. View from sagittal plane

z

x

y

elbow

wrist

shoulder

Picture 2. View from medial plane (x-y plane)

= rotation angle about the vertical Z-axis = rotation angle about an axis perpendicular to the plane of the arm = rotation about humeral axis = angle of flexion of the forearm

The location of the elbow is given by :

Where = length of the upper arm

The location of the wrist is given by :

Where = length of the forearm

The three separate components of the angular-velocity vector of the upper arm :

x

y

As the velocity of the centre of mass is expressed as : , then the velocity for the centre of mass of the upper arm :

The velocity of the centre of mass for the forearm :

In this case there are 4 angles that should be computed which are .The amount of work needed to move arm:

Where : = the moment of inertia for upper arm and forearm respectively

Therefore taking all above equations and put them into equation of amount of work :

In this equation, upper arm and forearm are assumed to be solid cylindrical shape, so that :

where r = distance from joint to centre of mass.

The arm trajectory is derived by using principle of minimum cost which in this case

could be obtained by minimizing the amount of work subject to the position of wrist

in the workspace.