D evelop m en t of op tim ization -b ased sim u lation too...
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Abstract² Rehabilitation of people with upper extremity motor deficits is typically focused on relearning of motor abilities and functionalities requiring interaction with physiotherapists and/or rehabilitation robots. In a point-to- point movement training, the trajectories are usually arbitrarily determined without considering the motor impairment of the individual. In this paper, we report on development of an optimal control model based on arm dynamics enabling also incorporation of muscle functioning constraints (i.e. simulation of muscle tightness) to study optimal trajectories for planar arm reaching movements. In preliminary tests of the developed model we first tested the ability of the minimum joint torque cost function to replicate trajectories obtained in previously published experimental trials done by neurologically intact subjects, and second, we explored optimal trajectories when muscle constraints were modeled. The trajectories generated by the model without implemented muscle constraints show considerable similarity as compared to the experimental data, while muscle constraints considerably changed optimal trajectories. I. INTRODUCTION PPER extremity motor deficits are prevalent post stroke requiring efficient motor rehabilitation, which is typically focused on relearning of motor abilities and functionalities, often necessitating one-on-one manual interaction with physiotherapists and/or rehabilitation robots. In recent years, rehabilitation robots made their way to clinical practice as they can apply high-intensity, repetitive, task-specific, interactive treatment with an objective and reliable means of monitoring patient progress. Robot-aided therapy can also evaluate patient's movements and assist them in moving the upper extremity through a predetermined trajectory during a given motor task. In current rehabilitation robot assisted arm training predominantly straight line trajectories connecting the starting and ending points of upper extremity movement are being used. This might be to a large extent based on predictions of the minimum jerk trajectory formation model, proposed by [1]-[3], which is however valid only under assumptions that no constraints either in movement space, i.e. boundaries of range of motion (ROM), or musculo- skeletal system are present. In contrast, some experimental [4], [5] and theoretical [5]-[7] results suggest curved paths M. Zadravec is with University Rehabilitation Institute, Republic of Slovenia, Linhartova 51, 1000 Ljubljana, Slovenia (phone: +386 1 4758 206; fax: +386 1 4372 070; e-mail: [email protected]). = 0DWMDLü LV ZLWK 8QLYHUVLW\ 5HKDELOLWDWLRQ ,QVWLWXWH 5HSXEOLF RI Slovenia, Linhartova 51, 1000 Ljubljana, Slovenia (e-mail: [email protected]). when either of constraints are invoked, especially when the target point is at the boundary of arm's ROM or in the case if the path is long-distanced. Furthermore, one may expect entirely different arm trajectories when muscle spasticity or any kind of arm weakness is considered. More natural cost function in rehabilitation robotics supported movement training, where muscle weakness and muscle tightness pose considerable constraints, would be one related to required joint torques, since the rehabilitation robot needs to supply a ‡PLVVLQJ· MRLQW WRUTXHV In this paper we report on development on an optimal control model based on arm dynamics enabling also incorporation of altered muscle functioning constraints, which can simulate arm tightness. With the developed model we first compared the experimental data of the arm reaching movements performed by neurologically intact population reported in previous study [4] to test the ability of the selected cost function minimizing a sum of squared torques in the shoulder and elbow, to replicate experimentally obtained trajectories. We then incorporated stiffness-based muscle tightness within the model to investigate its potential influence on optimal reaching trajectories. II. METHODS A. Human arm model We modeled a human arm that consisted of two links, where the first link is upper arm, while the second link is consisted of forearm and hand. This 2 DOF human arm model has two rotational joints representing shoulder and elbow joints. The model is simplified for planar arm reaching and does not contain the gravitational vector. A schematic model of the human arm is shown in Fig. 1(a), where all the variables and most of the arm parameters are indicated. The shoulder joint is located at the position :rÆr; of Cartesian coordinate system as shown in Fig. 1. This model also contains six muscles attaching to the arm links as shown in Fig. 1(b). Two monoarticular muscles causing torque in the shoulder joint (1-pectoralis major and 2- posterior deltoid), two monoarticular muscles around elbow joint (3-brachialis and 4-lateral head of triceps brachii), and two biarticular muscles (5-biceps brachii and 6-long head of triceps) are shown. Development of optimization-based simulation tool for trajectory planning in planar arm reaching after stroke 0DWMD =DGUDYHF DQG =ODWNR 0DWMDLü U The Fourth IEEE RAS/EMBS International Conference on Biomedical Robotics and Biomechatronics Roma, Italy. June 24-27, 2012 978-1-4577-1198-5/12/$26.00 ©2012 IEEE 1446
Transcript of D evelop m en t of op tim ization -b ased sim u lation too...
�
Abstract² Rehabilitation of people with upper extremity
motor deficits is typically focused on relearning of motor
abilities and functionalities requiring interaction with
physiotherapists and/or rehabilitation robots. In a point-to-
point movement training, the trajectories are usually
arbitrarily determined without considering the motor
impairment of the individual. In this paper, we report on
development of an optimal control model based on arm
dynamics enabling also incorporation of muscle functioning
constraints (i.e. simulation of muscle tightness) to study optimal
trajectories for planar arm reaching movements. In
preliminary tests of the developed model we first tested the
ability of the minimum joint torque cost function to replicate
trajectories obtained in previously published experimental
trials done by neurologically intact subjects, and second, we
explored optimal trajectories when muscle constraints were
modeled. The trajectories generated by the model without
implemented muscle constraints show considerable similarity
as compared to the experimental data, while muscle constraints
considerably changed optimal trajectories.
I. INTRODUCTION
PPER extremity motor deficits are prevalent post stroke
requiring efficient motor rehabilitation, which is
typically focused on relearning of motor abilities and
functionalities, often necessitating one-on-one manual
interaction with physiotherapists and/or rehabilitation robots.
In recent years, rehabilitation robots made their way to
clinical practice as they can apply high-intensity, repetitive,
task-specific, interactive treatment with an objective and
reliable means of monitoring patient progress. Robot-aided
therapy can also evaluate patient's movements and assist
them in moving the upper extremity through a
predetermined trajectory during a given motor task. In
current rehabilitation robot assisted arm training
predominantly straight line trajectories connecting the
starting and ending points of upper extremity movement are
being used. This might be to a large extent based on
predictions of the minimum jerk trajectory formation model,
proposed by [1]-[3], which is however valid only under
assumptions that no constraints either in movement space,
i.e. boundaries of range of motion (ROM), or musculo-
skeletal system are present. In contrast, some experimental
[4], [5] and theoretical [5]-[7] results suggest curved paths
M. Zadravec is with University Rehabilitation Institute, Republic of
Slovenia, Linhartova 51, 1000 Ljubljana, Slovenia (phone: +386 1 4758
206; fax: +386 1 4372 070; e-mail: [email protected]).
=�� 0DWMDþLü� LV� ZLWK� 8QLYHUVLW\� 5HKDELOLWDWLRQ� ,QVWLWXWH�� 5HSXEOLF� RI�
Slovenia, Linhartova 51, 1000 Ljubljana, Slovenia (e-mail:
when either of constraints are invoked, especially when the
target point is at the boundary of arm's ROM or in the case if
the path is long-distanced. Furthermore, one may expect
entirely different arm trajectories when muscle spasticity or
any kind of arm weakness is considered. More natural cost
function in rehabilitation robotics supported movement
training, where muscle weakness and muscle tightness pose
considerable constraints, would be one related to required
joint torques, since the rehabilitation robot needs to supply a
³PLVVLQJ´�MRLQW�WRUTXHV��
In this paper we report on development on an optimal
control model based on arm dynamics enabling also
incorporation of altered muscle functioning constraints,
which can simulate arm tightness. With the developed model
we first compared the experimental data of the arm reaching
movements performed by neurologically intact population
reported in previous study [4] to test the ability of the
selected cost function minimizing a sum of squared torques
in the shoulder and elbow, to replicate experimentally
obtained trajectories. We then incorporated stiffness-based
muscle tightness within the model to investigate its potential
influence on optimal reaching trajectories.
II. METHODS
A. Human arm model
We modeled a human arm that consisted of two links,
where the first link is upper arm, while the second link is
consisted of forearm and hand. This 2 DOF human arm
model has two rotational joints representing shoulder and
elbow joints. The model is simplified for planar arm
reaching and does not contain the gravitational vector. A
schematic model of the human arm is shown in Fig. 1(a),
where all the variables and most of the arm parameters are
indicated. The shoulder joint is located at the position :rár; of Cartesian coordinate system as shown in Fig. 1. This
model also contains six muscles attaching to the arm links as
shown in Fig. 1(b). Two monoarticular muscles causing
torque in the shoulder joint (1-pectoralis major and 2-
posterior deltoid), two monoarticular muscles around elbow
joint (3-brachialis and 4-lateral head of triceps brachii), and
two biarticular muscles (5-biceps brachii and 6-long head of
triceps) are shown.
Development of optimization-based simulation tool for trajectory
planning in planar arm reaching after stroke
0DWMDå�=DGUDYHF�DQG�=ODWNR�0DWMDþLü
U
The Fourth IEEE RAS/EMBS International Conferenceon Biomedical Robotics and BiomechatronicsRoma, Italy. June 24-27, 2012
978-1-4577-1198-5/12/$26.00 ©2012 IEEE 1446
B. The kinematics of links
The hand position of the two-link arm, which is defined in
Cartesian coordinate system as the coordinates :TáU;, can be
expressed by joint angles :à5áà6;, yielding forward
kinematics:
BTUC L d.5?KO�à5 E .6 ���:à5 E à6;.5OEJ�à5 E .6 ���:à5 E à6;h. (1)
The inverse kinematics relations can be written as
dà5à6h L f������t:UáT; F ������@å.>Å-.?Å..6�Å-å
ANF ������@P-.>P..?p.
6Å-Å.A j, (2)
where N L ¥T6 E U6 and ������t:UáT; L ������@ìëAE
���:U;�ksF ���:T;o ��6 .
C. Arm dynamics
Model dynamics were modeled with the Lagrangian
formulation without using potential energy. The manipulator
dynamics are specified by (3), where Î L >ì5 ì6?Í are
torques around shoulder and elbow joints, ÎàèæÖßØæ L>ì5áàèæÖßØæ ì6áàèæÖßØæ?Í are joint torques due to passive
muscle forces, Â L >à5 à6?Í are joint angles, Â6 L>ñ5 ñ6?Í are angle velocities, and Â7 L >ñ65 ñ66?Í are
angle accelerations.
/:à;Â7 E %kàáà6oÂ6 E $Â6 L ÎE ÎàèæÖßØæ. (3)
The manipulator inertia matrix /, Coriolis and centrifugal
matrix %, and viscosity matrix $ are given as follows
/:à; L dÙ E tÚ ���à6 Ü E Ú ���à6Ü E Ú ���à6 Ü
h, (4)
%kàá à6o L HFÚà66 ���à6 FÚkà65 E à66o ���à6Úà65 ���à6 r
I, (5)
$ L d>55 >56>65 >66
h, (6)
with the following constants:
Ù L +5 E +6 EI5.Ú56 EI6k.56 E .Ú66 o, (7)
Ú L I6.5.Ú6, (8)
Ü L +6 EI6.Ú66 . (9)
D. Muscle modeling
The muscle lengths � L >H5á H6á H7á H8á H9á H:?Í are expressed
as follows:
ÕÖÖÖÔÖÖÖÓ
H5 L F=5à5 E¥>56 F =56 E =5:è F ������ Ô-Õ-;H6 L =6à5 E¥>66 F =66 E =6:�6 F ������
Ô.
Õ.;
H7 L F=7à6 E ¥>76 F =76 E =7:è F ������Ô/Õ/;
H8 L =8à6 E ¥>86 F =86 E =8:�6F ������ Ô0
Õ0;
H9 L =95 @�6F à5AE ¥.56 E :=95 F =96;6 E =96 @�
6F à6A
H: L =:5à5 E ¥.56 E :=:5 F =:6;6 E =:6à6 ÙÖÖÖØÖÖÖ×
, (10)
where =518á>518á=95á=96,�=:5 and =:6 represent the
moment levers of each muscle. The levers are assumed to be
constant in the arm's workspace, independent of the joint
angles. The moment lever matrix is then given by
9Í L HF=5��=6������r������r���=95��=:5�����r����r����=7���=8��=96��=:6
I. (11)
In our model, we used only passive muscle force of Hill-
type model for force generation [8] to simulate muscle
tightness conditions. Since the muscles can only exert
positive forces, the passive muscle force is given by (12),
considering inequality constraints. The shape of the
exponential was determined by the variable -æÛ and the
scaling factor (4 defined the nominal passive muscle force.
(:H; L P r�á H O H4
¿,
ؼÞÓ?5
FA¼ÞÓ:×7×,;,ä1�×, F sG á H R H4
�, (12)
Fig. 1. Planar human arm model. (a) Kinematic model of a human arm
modeled as a planar two-link manipulator. (b) Six muscle model with four
monoarticular muscles (1-pectoralis major, 2-posterior deltoid, 3-brachialis,
4-lateral head of triceps brachii) and two biarticular muscles (5-biceps
brachii, 6-long head of triceps).
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