International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 13 (2016) pp 7943-7946
© Research India Publications. http://www.ripublication.com
Trajectory Planning For Exoskeleton Robot By Using Cubic And Quintic
Polynomial Equation
Sari Abdo Ali 1, Khalil Azha Mohd Annuar 2, 3, Muhammad Fahmi Miskon1, 3
1
Fakulti Kejuruteraan Elektrik, Universiti Teknikal Malaysia Melaka.
2
Fakulti Teknologi Kejuruteraan, Universiti Teknikal Malaysia Melaka.
3
Center of Excellence in Robotic and Industrial Automation, Universiti Teknikal Malaysia Melaka.
Hang Tuah Jaya, 76100 Durian Tunggal, Melaka, Malaysia.
twice in a week. Such treatment cost a lot of money. But using
robotic system devices reduce the number to therapist for
assisting one patient [6-8].
Robotic systems are programed to walk in smooth motion
similar to human walking. Exoskeleton device is an example
of robotic systems which is a structure designed to support the
weak part of the body [9]. There are many types of
exoskeleton robots. Treadmill gait trainer is one of these
robots. It is designed to improve the mobility of the legs with
no loads on them. The foot plate gait trainer is similar to
treadmill trainer but with different function. It reduces the
weight of the body during training. The overground gait
trainer left the patient pu from the ground. Unlike the other
two types, overground trainer enable the patient can control
the movement of their legs.
Abstract
This paper present the trajectory generation for the knee joint.
The study of human walking cycle uses quintic polynomial
equation and cubic polynomial equation. The walking cycle is
divided into eight sub-phases gaits. In this paper, we are using
the quantic and cubic equation in order to generate the same
profile as the normal human walking for position, velocity and
acceleration. The generated signal will be used to control a
device to duplicate and copy the knee movement for a normal
person during walking. Then a comparison between the real
data of human walking and the data gained from the quantic
and cubic equations during the phases of the gait walking
cycle will be shown in graphs using matlab.
Keywords: Cubic polynomial equation; quantic polynomial
equation; Exoskeleton devices; trajectory generation.
INTRODUCTION
Nowadays, a lot of efforts in robotics are presented to help
patients in their rehabilitation treatment. Different devices are
invented each year. Most of these devices mainly divided in to
two parts, devices that treat the upper limb and devices that
treat the lower limb of the body. These robotic systems that
exist previously to solve problems in damaged parts of the
lower limb of human’s body, such as the knee, the ankle and
the hip. Based on the assumption that all of these systems are
generally created, to do specific function like helping people
who have permanent injuries due to strokes or accidents in
their therapy to regain the walking ability [1-2].
Understanding the system of walking is an easy task but when
it comes to make the machines adapt the walking, the process
become more complex and challengeable. In order to simplify
the operation, the walking process is divided into different
gait phases [3]. Human walking is a repeated pattern of
movement. From the first initial contact phase to last swing,
the weight of the body is shifted from one leg to the other.
While having a problem in one side of the body, it is hard to
maintain balanced normal walking [4].
In the last years, rehabilitation therapy treatment is widely
using robotic systems because of the success these systems.
Increasing the stroke patients made the medical centers search
for more efficient and reliable methods to treat patients rather
than the traditional methods [5]. Traditional rehabilitation
therapy treatment needs tree to four therapists to help one
patient and manually move the patient effected parts such as
the legs. Moreover, this treatment should be done at least
Figure 1: Gait phases for normal human walking
TRAJECTORY PLANNING
Trajectory planning is an important part in robotics. It is used
to design a path for an electrical motor or a manipulator to
move to a desired movement with specific velocity and
acceleration [10]. The planning can be point to point or
predefined path. It also might be in work space or in joint
space. It is essential to determine two features for trajectory
planning; the motion law and the geometrical path. There are
many ways to plan the trajectories beside the methods
discussed in this paper.
Linear polynomial is the simplest method. It has constant
velocity. The motion can be determined from the initial and
final points directly. Parabolic polynomial is another method
which has a constant acceleration. It is known in [11] as
gravitational polynomial trajectory. Acceleration in parabolic
polynomial is categorized by absolute value with positive sign
for deceleration and negative for acceleration [11].
The constraints also must be considered for achieving smooth
continuity for the trajectory [12]. The smooth motion is a
major target while planning the trajectory for the walking
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International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 13 (2016) pp7943-7946
© Research India Publications. http://www.ripublication.com
motion especially when shifting from one phase gait to the
other. The position profile and the velocity profile have to be
continuous with the time variable. Figure2, 3, 4 show that
knee position, velocity and acceleration graphs have
continuous functions with time[13].
For k+ 0, 1, 2, 3 the position equation for the knee is shown in
(2)
(2)
The velocity of knee profile is shown in (3) which produced
by deriving equation (2)
(3)
The acceleration equation is the second derivative of equation
(2) which is shown in (4)
(4)
For smooth motion the cubic polynomial has four constraints.
Two of them come from the initial value and the final value of
the velocity which is equal to zero.
Figure 2: Continuous knee position for walking.
(5)
The parameters,
can be determined by
applying the constraints equation (5) in equations (2), (3) and
(4) that result equation (6).
(6)
Figure 3: Continuous knee velocity for walking.
GENERATING TRAJECTORY FOR WALKING
MOTION
USING
QUINTIC
POLYNOMIAL
EQUATION
A fifth degree quintic polynomial equation is expressed as
presented in equation (7). Quintic polynomial equation
represents the knee joint position profile.
(7)
For k = 0, 1, 2, 3, 4, 5 the position quintic equation for the
knee is shown in (8)
(8)
The derivative of position equation (8) produce forth degree
velocity profile equation as shown in equation (9).
Figure 4: Continuous knee acceleration graph.
(9)
The acceleration profile equation is obtained by differentiating
equation (9) which result equation (10).
GENERATING TRAJECTORY FOR WALKING
MOTION BY
USING CUBIC POLYNOMIAL
EQUATION
Cubic polynomial is third degree equation used to generate
trajectory for the knee movement during walking motion.
Equation (1) represents the cubic equation of the knee position
profile [12].
(10)
The six constraints of quantic polynomial are shown in (11).
(1)
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International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 13 (2016) pp7943-7946
© Research India Publications. http://www.ripublication.com
(11)
The constraints can be used to determine the values of,
. By applying equations (8), (9), (10) into
(11) we can get the unknown coefficients as shown in (12).
Figure 6: Cubic polynomial, quintic polynomial and
reference trajectory angular velocity at knee joint for one gait
cycle.
(12)
RESULT
The data in table 1 is an experimental data from [13] as well
as the reference profile for the human walking gaits. The
cubic polynomial coefficients in table 2 are obtained by
following the equations in section 3. Table 3 contains the
coefficients of the quantic polynomial equation in section 4.
For cubic trajectory, during the initial contact gait cycle time
is between 0s and 0, 1s. Inserting the coefficients, the equation
become as shown in (13)
Equation (13) draws the first part of the position profile in
figure (). Cubic profile is drawn by using four equations while
the quantic profile is drawn be six equations. As shown is
figure5 and figure 6, cubic polynomial trajectory give smooth
continuous position and velocity profiles. Whereas, the
acceleration profile is discontinuous profile which result
impulse jerking.
The jerking in the acceleration will lead to vibration in the real
application. For this reason, the proposed quintic coefficients
in table 3 and the quintic equation provide smooth motion and
continuous profile in position, velocity and acceleration.
Figure 7: Cubic polynomial, quintic polynomial and
reference trajectory angular acceleration at knee joint for one
gait cycle.
Table 1: Human walking gait sub-phases trajectory for the
knee joint flexion position, angular velocity values and
angular acceleration [13].
Gait cycle
Figure 5: Cubic polynomial, quintic polynomial and
reference trajectory flexion position at knee joint for one gait
cycle.
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Time Position(rad)
Ang.
Ang.
(100%)
Velocity. Acceleration.
(rad/s)
(rad/s/s)
Initial Contact
0
-0.0105
1.76
40.55
Opp. Toe Off
0.1
0.2444
2.07
-44.04
Heel Rise
0.3
0.1431
-0.73
0.45
Opp. I.C.
0.5
0.2827
3.83
40.04
Toe Off
0.6
0.8308
6.02
-21.92
Feet Adjacent
0.73
1.1153
-2.69
-68.11
Tibia Vertical
0.87
0.3944
-7.37
4.76
Initial Contact
1
-0.0105
1.76
40.55
International Journal of Applied Engineering Research ISSN 0973-4562 Volume 11, Number 13 (2016) pp7943-7946
© Research India Publications. http://www.ripublication.com
Table 2: Coefficients of the cubic polynomial trajectory for
the knee joint flexion angle.
REFERENCES
[1]
Gait cycle
M.H. Rahman, J.P. Kenné, P.S. Archambault,
"Exoskeleton Robot for Rehabilitation of Elbow and
Forearm
Movements"
18th
Mediterranean
Conference on Control & Automation Congress
Palace Hotel, Marrakech, Morocco June 23-25, 2010
[2]
Robert Riener, Martin Anderschitz, Gery Colombo,
Volker Dietz, " Patient-Cooperative Strategies for
Robot-Aided Treadmill Training: First Experimental
Results ", 380 IEEE VOL. 13, NO. 3, SEPTEMBER
2005.
[3]
J. Perry, “Gait analysis: Normal and pathological
function, ” SLACK, 1992, 2nd edition 2010.
[4]
G. J. Gelderblom, M. De Wilt, G. Cremers, and A.
Rensma, “Rehabilitation robotics in robotics for
healthcare; a roadmap study for the European
Commission, ” IEEE International Conference on
Rehabilitation Robotics, pp. 834, Japan, June 2009.
[5]
R. Jimenez-Fabian and O. Verlinden, “Review of
control algorithms for robotic ankle systems in
lower-limb orthoses, prostheses, and exoskeletons, ”
vol. 34, no. 4, pp. 397-408, 2012.
[6]
P. E. Martin, and A. Hreljac, “The relationship
between smoothness and economy during walking, ”
Biological Cybernetics, vol. 69, pp. 213-218, 1993.
[7] G. J. Gelderblom, M. De Wilt, G. Cremers, and A.
Rensma, “Rehabilitation robotics in robotics for
healthcare; a roadmap study for the European
Commission, ” in Proceedings of the IEEE
International Conference on Rehabilitation Robotics,
(ICORR ’09), pp. 834–838, Kyoto, Japan, June 2009.
[8]
J. Nutt, C. Marsden, and P. Thompson, “Human
walking and higher-level gait disorders, particularly
in the elderly, ” Neurology, vol. 43, no. 2, pp. 268268, 1993..
[9]
O. Franch, L. Calandre, J. Álvarez-Linera, E. D.
Louis, F. Bermejo-Pareja, and J. Benito-León, “Gait
disorders of unknown cause in the elderly: Clinical
and MRI findings, ” J. Neurol. Sci., vol. 280, no. 1,
pp. 84-86, 2009.
[10]
S. Murray and M. Goldfarb, “Towards the use of a
lower limb exoskeleton for locomotion assistance in
individuals with neuromuscular locomotor deficits, ”
in Proc. IEEE Annual International Conference on
Engineering in Medicine and Biology Society, 2012,
pp. 1912-1915.
[11]
Luigi Biagiotti, Claudio Melchiorri, " trajectory
planning for automatic machines and robots" page
15-39 Springer, 2008.
[12]
J. J. Craig, “Introduction to Robotics: Mechanics and
Control 3rd, ” Prentice Hall, vol. 1, no. 3, p. 203,
2004.
[13]
D. A. Winter, Biomechanics Motor Control Human
Movement, 3rd ed., John Wiley and Sons, New
York, 2004
Time
(100%)
Initial Contact 0-0.1
-0.0105
1.7600
20.5454
Opp. Toe Off
0.1
-0.2679
8.7627
-42.2844 58.8073
Heel Rise
0.3
-0.9119
11.5971
-39.7122 42.5935
Opp. I.C.
0.5
19.1538 -107.0812 194.2122 -111.0680
Toe Off
0.6
2.8855
Feet Adjacent
0.73
-11.8969 44.7959
Tibia Vertical
0.87-1 3.8373
-35.1667
27.1999
90.0665
-126.6360
-61.9381
-45.7907 12.1152
-67.7031 36.6554
Table 3: Coefficients of the cubic polynomial trajectory for
the knee joint flexion angle.
Gait
Time
sub(x100%)
phase
Initial 0 – 0.1 -0.01
20.3
-164.3
-0.0794 +2.217
41.803
-444.978 1411.101 -1488.63
0.3
4.923
-65.986
367.147 -1008.11 1336.611 -671.55
Opp.
I.C.
0.5
154.5
1376.8
-4842
Toe
Off
0.6
66.0669 -326.24
Feet
Adj
0.73
-2088.5 13030.346 -32523.1 40619.708 25381.874 6344.372
Tibia
Ver
0.87-1
-2443.2 12055.904 -23482.4 22557.024 -10676.3 1988.917
Opp.
Toe
0.1
Heel
Rise
1.8
8369.3
833.8
7078.1
-4304.2
-2340.4
398.5529 339.1582 -955.088 482.867
CONCLUSION
To conclude, generating trajectory using proposed quintic
polynomial will provide smooth motion with almost identical
position, velocity and acceleration as the normal human gait
cycle. Also, it is used to avoid the jerks which occurred in
cubic polynomial as shown in figure7. The jerking in the
cubic equation is due to the linearity of the acceleration
equation. The result shows that the proposed quintic
polynomial can be used to generate a motion that is accurately
matched to real human knee motion.
ACKNOWLEDGMENT
The authors would like to thank for the support given to this
research by Ministry of Higher Education Malaysia,
Universiti Teknikal Malaysia Melaka (UTeM) and UTeM
Zamalah
Scheme
for
support
this
under
RAGS/1/2015/TK0/FTK/03/B00118 project.
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