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Volume 15 Issue 2 1 ©2015 IJAMS
DYNAMIC ANALYSIS OF SERIAL ROBOT
MANIPULATORS
Hatem Al-Dois1, A.K. Jha
2, R. B. Mishra
2
1Faculty of Engineering and Architecture, University of Ibb, Ibb, YEMEN
[email protected] 2Indian Institute of Technology (Banaras Hindu University), Varanasi, 221005, U.P., INDIA
[email protected]; [email protected] Abstract: This paper describes a method to analyze the dynamic performance of serial robot manipulators. The considered measures
are the quadratic average of joint torques and the positioning and orientation accuracy of the robot’s end-effector. A set of exciting
trajectories is designed and optimized along which the robot performance is assessed. The initial points that construct the exciting
trajectories are taken from the analysis of the robot workspace. A probabilistic method is adopted to simulate the uncertainty in
control factors of robot models. An illustrative example on Unimation Puma 560 robot is provided to demonstrate the proposed
approach. Modeling, simulation, and optimization are done in MATLAB®. The expected outcome is a framework for the dynamic
analysis of serial robot manipulators.
Keywords: Optimization. Dynamic analysis; Control factors; Noise factors; Exciting trajectories; Trajectory optimization;
Positioning accuracy; Orientation accuracy.
1. Introduction
The growing demand on fast and accurate robot
manipulators requires a continuous development
of robot performance measures. A variety of
indices and measures have been proposed in the
literature to characterize the performance of
robotic mechanisms which can be broadly
categorized into two groups kinematic-based and
dynamic-based measures. Kinematic-based
measures are oriented, in general, towards
manipulator structure which can be conveniently
characterized based on workspace capabilities
and, thus, attempt to tackle global properties.
Dynamic-based measures tend to be more
interested in local characteristics at or near a
particular configuration of the manipulator [1].
Few of dynamic measures are cited below.
Khatib [2] formulated the operational space
dynamics using the kinetic energy matrix which
provides a description of the mechanisms inertial
properties as perceived at the end-effector. Asada
[3] introduced the generalized inertia ellipsoid,
(GIE), a geometric presentation of the kinetic
energy matrix. Yoshikawa [4, 5] proposed the
dynamic manipulability measure and dynamic
manipulability ellipsoid for measuring the ease of
changing the end-effector configuration by a set of
joint torques with fixed magnitude. Tourassis and
Neuman [6] studied the internal characteristics of
the dynamic equations of manipulators. Khatib
and Burdick [7] proposed dynamic model in
operational space which is the space spanned by
the vector representing the pose of the end
effector. Graettinger and Krogh [8] introduced the
acceleration radius to represent the minimum
isotropic acceleration over a number of
configurations.
In 1990 Ma and Angeles [9] presented the concept
of dynamic conditioning index (DCI) which
measures the difference between an inertia matrix
and a reference inertia matrix from a configuration
where the inertial properties are isotropic. Khatib
[10] proposed a visualization of the inertial
IJAMS
Hatem Al-Dois, A.K. Jha, R.B. Mishra
Volume 15 Issue 2 2 ©2015 IJAMS
properties, the belted inertia ellipsoid, which
describes the actual magnitude of effective mass
and inertia in different directions. Zhao et. al [11]
proposed a performance criteria which measured
the distance between the current actuator torque
state and the torque boundaries. Bowling [1]
extended the acceleration isotropy to include
isotropic angular velocity and acceleration in
addition to linear velocity and acceleration for
redundant and nonredundant manipulators. Rao
and Bhatti [12] proposed a method to construct
manipulator kinematic and dynamic model using a
probabilistic approach to study the effect of
various factors that produce uncertainties in the
behavior of a manipulator. Voglewede and Ebert–
Uphoff [13] proposed performance indices based
on joint stiffness and link inertia with the aim of
establishing a distance to singularity for any robot
posture. Bowling and Khatib [14] proposed a
general framework for capturing the dynamic
capability of ageneral manipulator that includes
the velocity and acceleration characteristics of the
end-effector, taking into account factors such as
torque and velocity limits of the actuators.
Ma and Yunong et al. [15, 16] characterize the
robot performance by the amount of joint torques
required to perform a task. Similar to other
dynamic measures, the required joint torques are
regarded as a local measure since they evaluate
the robot performance for a given configuration or
along a prescribed path. Therefore, such a
measure should be evaluated on either a standard
trajectory or an exciting trajectory.
In this paper, an optimization approach is
proposed to optimize a set of exciting trajectories
through which the dynamic performance of the
robot is measured. Exciting trajectories are mainly
used for parametric identification of robot
manipulator. However, they can also be used to
assess the robot performance when the force the
robot joints to exert maximum torques or to move
in maximum velocities. Two performance
measures are considered: the quadratic average of
joint torques and the positioning and orientation
accuracy. The initial points of the optimized
trajectories are defined from the analysis of the
robot workspace. To simulate the variability in
robot parameters, a probabilistic method is
implemented. An illustrative example on Puma
560 robot is given to demonstrate the proposed
approach.
The rest of the paper is arranged as follows. The
problem is formulated in section 2. The robot
kinematic and dynamic models are discussed in
section 3. The static and kinetostatic analysis are
highlighted in section 4. Section 5 discusses the
dynamic analysis of serial robot manipulators.
Section 6 outlines the proposed exciting
trajectories. The illustration on Puma 560 robot is
given in section 7, and finally the paper is
summarized in section 8.
2. Problem Formulation
It is required to characterize the dynamic
performance of serial robot manipulators based on
the analysis of quadratic average of joint torques
(QAT) and the positioning and orientation
accuracy. As the main drawbacks of the proposed
measures are their dependency on the motion
profile, a set of optimized exciting trajectories is
designed along which the performance indices are
measured. The exciting trajectories rule is to
withdraw maximum torques or force the robot to
move in its maximum end-effector velocity [17].
To evaluate the robot performance in different
situations, different excitation methods are
implemented. Points that construct the optimal
trajectory are selected based on the analysis of the
robot workspace. Quan et al. [17] selected the
path points according to the static torque analysis.
The method is extended to consider the kinematic
and dynamic manipulability of the robot tip. To
simulate the real manipulator performance, errors
due to uncertainty in robot parameters are
modeled using a probabilistic approach. The
simulation results of each task are the
performance measures in forms of QAT and
positioning and orientation accuracy considering
the manipulator dynamics. The average of these
measures can then be considered as the overall
performance measure of the robot.
3. Kinematic and Dynamic Modeling
The kinematic model describes the geometric
relation between joints variables and the end-
Dynamic Analysis of Serial Robot Manipulators
Volume 15 Issue 2 3 ©2015 IJAMS
effector position and orientation. The kinematic
model can be extracted from the well-known D-H
convention [18]. The dynamic model relates robot
motion to joint torques. The governing equations
of motion can be derived using Newton-Euler
method or Lagrange-Euler method [19] which
appear in the form
𝜏 = 𝑀 𝑞 𝑞 + 𝑞, 𝑞 + 𝐺 𝑞 + 𝐼(𝑎𝑐𝑡 )𝑞 + 𝜏𝑓
…(1)
where 𝜏 is the joint torque vector; 𝑀 is the joint
space symmetric inertia matrix; is the vector of
centrifugal and Coriolis forces; 𝐺 is the gravity
force vector; 𝐼(𝑎𝑐𝑡 ) is the vector of actuators
inertias; 𝜏𝑓 is the vector of friction torques ; and
𝑞, 𝑞 ,𝑞 are the joint position, velocity, acceleration
vectors, respectively.
4. Static and Kinetostatic Analysis
The static analysis evaluates the performance of a
manipulator with zero joint velocities and
accelerations. Thus, the performance measure
depends only on the robot configuration. Two
measures are considered: the input joint torques
and the kinematic manipulability.
4.1. Static Torques
Static torques or torques against gravity are the
torques required to prevent the robot arm from
collapsing under gravity. Static torques are good
indicators for the robot capabilities at zero or slow
motions. By setting to zero at each time step of the
task, static torques can be calculated from (1); this
yields to
𝜏 = 𝐺 𝑞 …(2)
Static torques can be evaluated at each point in the
robot workspace. For spatial type manipulators it
is only necessary to consider cross-section area of
the workspace because of the symmetry. Figures
(1, 2) show in Cartesian space the torques required
against gravity at joints 2 and 3 of Puma 560 robot
with elbow-up configuration, respectively. Figures
(3,4) depict the gravity torques at joints 2 and 3 as
a function of joints 2 and 3, respectively.
Figure 1: Static Torques, Joint 2 of Puma 560
Figure 2: Static Torques, Joint 3 of Puma 560
Figure 3: Torques at Joint 2 as function of Joints 2 &3
Hatem Al-Dois, A.K. Jha, R.B. Mishra
Volume 15 Issue 2 4 ©2015 IJAMS
Figure 4: Torques at Joint 3 as function of Joints 2 &3
4.2. Kinematic Manipulability
Yoshikawa [5] proposed the kinematic
manipulability as a performance measure of the
easiness of changing arbitrarily the position and
orientation of the end-effector at the tip of the
manipulator. When the Jacobian of the
manipulator loses its full rank, the manipulator
loses one of its degrees of freedom. The
manipulability for redundant robot manipulators is
defined as
𝑤𝑘 = 𝑑𝑒𝑡( 𝐽 𝐽𝑇) …(3)
where J is the Jacobian matrix. The kinematic
manipulability for nonredundant manipulators is
reduced to
𝑤𝑘 = | 𝑑𝑒𝑡 𝐽| …(4)
The manipulability measure is very useful for
manipulator designing, task planning and fast
recovery ability from the escapable singular points
for robot manipulators. To obtain a maximally
large well-conditioned workspace, the
manipulability measure can be used as design
criteria. The objective of a well-designed robot
manipulator is to maximize the volume of
reachable workspace characterized by high values
of manipulability measure [5].
The kinematic manipulability of Puma 560 is
depicted in Figure (5) considering elbow-up
configuration. The kinematic manipulability is
also shown in Figure (6) at different values of
joints 2 and 3.
Figure 5: Kinematic manipulability of Puma 560
Figure 6: Kinematic Manipulability for Puma 560 at
Different Values of Joints 2 & 3
5. Dynamic Analysis
A robot dynamic measure characterizes the robot
performance considering its dynamics. Three
measures are considered: dynamic manipulability,
quadratic average of torques, and positioning and
orientation accuracy.
5.1. Dynamic Manipulability
The dynamic manipulability provides a measure
of the ability of performing end-effector
accelerations along each task-space direction for a
given set of joint torques [5]. Equation (1)
represents the dynamic model of an open loop
manipulator in joint space with n-rigid links
whose end-effector acts in an m-dimensional task
Dynamic Analysis of Serial Robot Manipulators
Volume 15 Issue 2 5 ©2015 IJAMS
space. Considering only translational end-effector
accelerations and following the methodology
proposed by Chiacchio et al. [20], the dynamic
manipulability ellipsoid for redundant manipulator
whose tip variables (m) are less than or equal to
the robot’s degrees of freedom (n) can be obtained
as
(𝑥 + 𝐽 𝑀−1𝑔)𝑇 𝐽𝑄+𝑇
𝑄 𝐽𝑄+ 𝑥 + 𝐽 𝑀−1𝑔 ≤ 1
…(5)
where 𝑥 is the (𝑚 × 1) vector of end-effector
accelerations; 𝐽 is the (𝑚 × 𝑛) Jacobian matrix; M
is the (𝑛 × 𝑛) symmetric, positive definite inertia
matrix; g is the (𝑛 × 1) vector of gravitational
torques; and JQ+ is the weighted pseudo-inverse of
the Jacobian given by
𝐽𝑄+ = 𝑄−1 𝐽𝑇 ( 𝐽 𝑄−1 𝐽𝑇 )−1 …(6)
where
𝑄 = 𝑀 𝐿−2𝑀 …(7)
and L is the scaling matrix given by
𝐿 = diag ( 𝜏1𝑚𝑎𝑥 ,… , 𝜏𝑛
𝑚𝑎𝑥 ) …(8)
where τimax is the maximum allowed torque at
the robot joint. The dynamic manipulability which
is proportional to the volume of the ellipsoid and
equal to the product of the singular values of
matrix can be interpreted as
𝑤𝑑 = 𝐽𝑄+𝑇
𝑄 𝐽𝑄+ …(9)
For non-redundant manipulator where m = n this
measure is reduced to
𝑤𝑑 = 𝑑𝑒𝑡( 𝐽𝑇 𝑄 𝐽−1) …(10)
Figure (7) shows the dynamic manipulability of
Puma 560 manipulator workspace with elbow-up
configuration and zero payload, and Figure (8)
shows the dynamic manipulability in joint space at
different values of joint 2 and 3.
5.2. Quadratic Average of Joint Torques
(QAT)
QAT is measured along a prescribed trajectory. It can
be calculated from
QAT = τi(t)
τimax
2
dt
n
i=1
T
0
…(11)
where i is the ith robot joint and τi is the torque
required at joint i. 𝜏𝑖max is introduced to
normalize the torques at each robot joint since
some joints require much higher torques than
others [21]. The squared term casts the problem of
optimizing actuator torques in the optimized
trajectory as a constrained time-varying quadratic
programming (QP). QP function describes the
physical costs better than norms and provides,
with a positive definite Hessian, a global solution
for the optimization problem for which many
efficient solvers exist. QP has been widely used in
robotic trajectory and control optimization
problems, see for example [21, 22]. As a
performance measure it is always preferred to
minimize QAT for a given task. Minimizing joint
torques produces smoothing effect preferable for
joint motors and helps to avoid excitation [23].
Figure 7: Dynamic manipulability of Puma 560
Hatem Al-Dois, A.K. Jha, R.B. Mishra
Volume 15 Issue 2 6 ©2015 IJAMS
Figure 8: Dynamic Manipulability for Puma 560 at
Different Values of Joints 2 & 3
5.3. Position and Orientation Accuracy
Robots are required to deliver motions in high
positioning and orientation accuracy; however,
this accuracy is affected by the control capability
and the uncertainty in robot parameters or the
‘Noise Factors’. Noise in robot parameters are
resulted from the computational error, uncertainty
in manufacturing and assembly, joint clearances
and misalignment, fluctuation in joint torques,
flexibility effect of links, gear backlash, sensor
errors and control errors and host of other static
and dynamic errors.
Positioning and orientation accuracy are treated as
dynamic measures since they are evaluated along
specified trajectories. Robot parameters affected
by noise are incorporated in the robot models
while calculating the positioning and orientation
error. Figure (9) shows the steps followed to
simulate the robot performance in terms of
positioning and orientation accuracy. First the
coordinates of the initial and final poses (positions
and orientations) are determined in Cartesian
space using the (X, Y, Z, roll, pitch, yaw)
representation. The required trajectory is planned
either in the Cartesian space for the robot’s end-
effector or in the joint space for the robot joints. If
the motion is planned in joint space, the initial and
final coordinates are converted to the joint space
using inverse kinematics model and then the
trajectory is planned for them. If the trajectory is
planned in Cartesian space, the pose is converted
into the joint space at each time step using the
inverse kinematics model.
Figure 9: Scheme for the Calculation of End-Effector
Position and Orientation Accuracy
The effect of noise is incorporated in robot
parameters using the probabilistic approach
discussed next. The torque vector is calculated
using the inverse dynamic model. The forward
dynamic Model is, then, used to calculate the joint
accelerations which can be integrated numerically
to calculate the joint velocities and positions
incorporating errors. The joint space positions are
then converted to the end-effector poses using the
forward kinematics model. The function ODE45
from MALAB is used to integrate the joint
acceleration in the forward dynamic model with a
resolution of 0.001 s.
The position error of the robot’s end-effector (ԑ𝑝)
is the vector distance between the actual point
reached by the end-effector (𝑥, 𝑦, 𝑧) and the
desired point (𝑥∗, 𝑦∗, 𝑧∗) as follows [24]
ԑ𝑝 = (𝑥∗ − 𝑥)2 + (𝑦∗ − 𝑦)2+(𝑧∗ − 𝑧)2…(12)
Dynamic Analysis of Serial Robot Manipulators
Volume 15 Issue 2 7 ©2015 IJAMS
The orientation error calculation is quite more
complicated than the calculations of positioning
error. Many researchers studied the rotational
error from the Euler angles representation. Several
papers deal with the accuracy of robots with
several orientational DOFs (Jelenkovic and Budin,
2002; Kim and Choi, 2000; Merlet and Daney,
2007; Zhao et al., 2002). In this study the
orientation error of the end-effector (ԑ𝑜𝑟𝑖𝑒𝑛 ) is
measured from the roll, pitch, yaw representation
of the end-effector orientation in the actual
orientation (𝑅, 𝑃, 𝑌) and desired orientation
𝑅∗,𝑃∗, 𝑌∗ [25] as follows
ԑ𝑜𝑟𝑖𝑒𝑛 = (𝑅∗ − 𝑅)2 + (𝑃∗ − 𝑃)2+(𝑌∗ − 𝑌)2
…(13)
It is always preferred to reduce the positioning
and orientation error to minimum.
5.3.1 Method to Simulate Variance in
Control and Noise Variance
It is difficult to model Noise in robot parameters
because of the non-linear highly coupling nature
of robot dynamics. We assume that the variations
of the parameter are to follow Gaussian
distributions with nonzero mean and non-identical
standard deviations denoted by
𝜎𝑙 ,𝜎𝑚𝜎𝛼 ,𝜎𝜃 , 𝜎𝑑 ,𝜎𝑃 , 𝜎𝜏 [24]. Hence, the variations
in length, mass of the links, joint angles, and
payload are assumed to follow the Gaussian
distribution and standard deviations are dependent
on the specified tolerance. Errors due to
fluctuations in the supplied torques are assumed to
follow Gaussian stochastic process with Markov
property in which the future values of a Markov
process depend only on immediate past or present
but not all past events. The weight parameter φ1
for first order process is taken as 0.8 to satisfy the
stationary condition [12, 24, 26]. Table (1) shows
the noise factors and their standard deviations.
Table 1: Noise Factors
Sl.
No.
Noise
Factor
SD
1 𝜎𝑙 0.0001 m
2 𝜎𝑚 0.005 kg
3 𝜎𝛼 0.1 deg
4 𝜎𝜃 0.1 deg
5 𝜎𝑑 0.0001 m
6 𝜎𝜏 0.1 N.m
6. Exciting Trajectories
The robot dynamic performance is evaluated
along specified trajectories. The generation of an
optimal excitation trajectory involves nonlinear
optimization with motion constraints i.e.,
constraints on joint positions, velocities and
accelerations, and on the robot end effector
position in the Cartesian space in order to avoid
collision with the environment and the robot itself.
Several approaches for trajectory excitation have
been proposed in the literature. In this paper two
types of exciting trajectories are used: cubic spline
in Cartesian space and Fourier-based trajectory
PTP in Cartesian space.
6.1. Cubic Spline Trajectory in
Cartesian Space
The cubic spline trajectory is designed in
Cartesian space so that that required joint torques
should be maximum. The initial, via, and final
points of the spline are selected from the static
torque analysis of the robot workspace [17]. The
spline should pass from high static torque
distribution area to low torque distribution area. If
the robot workspace is symmetric such as the
workspace of Puma 560, the points can be
selected in the X-Z plane of the robot workspace
and then rotated an angle 𝛼 to the required
positions around Z-axis i.e.
𝑃 = 𝑟𝑜𝑡_𝑧 𝛼 ∗ 𝑃 …(14)
The trajectory is made exciting by finding the
optimal positions of the points so that the
trajectory exerts maximum torques. A search
space is defined for the spline knot points in the
Hatem Al-Dois, A.K. Jha, R.B. Mishra
Volume 15 Issue 2 8 ©2015 IJAMS
Cartesian space which considers the feasible
regions and singularity avoidance areas. Then, a
direct search method can be implemented to find
the optimal positions of these points. After finding
the optimal cubic spline trajectory, the robot
performance can be evaluated accordingly.
6.2. Fourier-Based PTP Optimized
Trajectory in Cartesian Space
The Frudenstein 1-3-5 Fourier-series-based PTP
trajectory is implemented [27]. The position,
velocity, acceleration for each axis is calculated as
follows
𝑞 𝑡 = 𝑞0 + 𝑡 − 𝑡0
𝑇−
2𝜋𝛼
𝑠𝑖𝑛2𝜋 𝑡 − 𝑡0
𝑇+
1
54𝑠𝑖𝑛
6𝜋 𝑡 − 𝑡0
𝑇
+1
1250𝑠𝑖𝑛
10𝜋 𝑡 − 𝑡0
𝑇
𝑞 𝑡 =
𝑇 1 − 𝛼
𝑐𝑜𝑠2𝜋 𝑡 − 𝑡0
𝑇+
1
18𝑐𝑜𝑠
6𝜋 𝑡 − 𝑡0
𝑇
+1
250𝑐𝑜𝑠
10𝜋 𝑡 − 𝑡0
𝑇
𝑞 𝑡 =2𝜋
𝑇2𝛼 𝑠𝑖𝑛
2𝜋(𝑡 − 𝑡0)
𝑇+
1
6𝑠𝑖𝑛
6𝜋(𝑡 − 𝑡0)
𝑇+
1
50𝑠𝑖𝑛
10𝜋(𝑡 − 𝑡0)
𝑇
…(15)
where is the traveled joint distance and 𝛼 =
0.9438. The initial and final points of the
trajectory are selected from the analysis of the
kinematic and dynamic manipulability of the
manipulator. Four trajectories can be selected
where each one passes from a high to low
kinematic and dynamic manipulability points in a
quarter of the robot’s work volume.
Similarly, if the robot workspace is symmetric
with respect to the kinematic and dynamic
manipulability such as the workspace of Puma
560, the initial and final points can be selected in
the X-Z plane and transformed to the required
quarter by rotating them the required angle around
Z-axis,
To make the trajectory an exciting trajectory, the
optimal time method is adopted so that the motion
is traveled in a minimum time. For that, the
normalization of time scale is introduced where
the problem is solved for fixed final time [28], i.e.
for 𝑞 𝑥 : 0 ≤ 𝑥 ≤ 1. Therefore, equation (1)
becomes
𝜏𝑖 𝑥 = 1
𝑇2 𝑀𝑖𝑗 𝑞 𝑥 𝑞𝑗′′ 𝑥 +𝑖(𝑞 𝑥 ,𝑞 ′ 𝑥 )
𝑛
𝑗=1
+ 𝐺𝑖(𝑞 𝑥 )
…(16)
Note that the gravity component is position
dependent and not affected by the time scaling.
This process allows translating the geometric,
kinematic, and dynamic constraints into bounds
on admissible values of the optimal transfer time
𝑇𝑞 of the parameter (x), for instance,
𝑇𝑣 = max𝑖=1,..,𝑛 max𝑥∈ 0,1 𝑞 ′ (𝑥)
𝑞 𝑖max …(17)
𝑇𝑎 = max𝑖=1,..,𝑛 max𝑥∈ 0,1 𝑞 ′′ (𝑥)
𝑞 𝑖max
1/2
…(18)
𝑇𝐸𝐸𝑣 = 𝑣
𝑣max …(19)
where 𝑇𝑣, 𝑇𝑎 , and 𝑇𝐸𝐸𝑣 are the minimum times
due to constraints on joint velocities , joint
accelerations, and end-effector velocity,
respectively The minimum task time 𝑇𝑞 of the
selected trajectory 𝑞(𝑥) should satisfy
𝑇𝑞 = min [ 𝑇𝑣 ,𝑇𝑎 ,𝑇𝐸𝐸𝑣 ] …(20)
After finding the minimum time Fourier-based
PTP trajectories, the robot performance is
evaluated along them in terms of QAT and
positioning and orientation accuracy.
7. Numerical Example
The proposed frame work is evaluated for Puma
560 robot shown in Figure (10). The kinematic
and dynamic models are taken from Armstrong et
al. [29]. The performance is evaluated with zero
payload. The friction is neglected and the
following constraints are imposed
Dynamic Analysis of Serial Robot Manipulators
Volume 15 Issue 2 9 ©2015 IJAMS
𝑞 𝑖𝑚𝑎𝑥 = 8 m/s
𝑞 𝑖𝑚𝑎𝑥 = 30 m/s2
𝜏𝑚𝑎𝑥 = [100,100,50,50,50,50] N. m𝐸𝐸𝑣_𝑚𝑎𝑥 = 1 m/s
…(21)
𝐸𝐸𝑣 is Calculated from
𝐸𝐸𝑣 = 𝑥 2 + 𝑦 2 + 𝑧 2 …(22)
where 𝑥 , 𝑦 , 𝑧 are linear velocities of 𝑥, 𝑦, and 𝑧
axis, respectively.
Figure 10 : Puma 560 Robot Manipulator
While evaluating the robot performance, the
control is assumed perfect and the elbow-up
configuration is always selected while solving the
inverse kinematic problem. The orientation
planning is done using rotation around fixed
vector in the space approach, then, the trapezoidal
velocity profile is applied to 𝛼 [30].
7.1. Cubic spline
The problem is formulated as finding the optimal
points in the workspace within specific areas so
that joints exert maximum torques. The initial
positions of the spline points are selected from the
static torque analysis, Figures (1-4). The
Objective function is taken as
𝑎𝑏𝑠 𝜏𝑗𝑘𝑚𝑎𝑥 → max𝑘 for j=1,..,6 …(23)
subjected to
τik max ≤ τimax
|EEvk max| ≤ EEvmax
|q i| ≤ q imax
|q i| ≤ q ima x
xkmin≤ xmin ≤ xkmax
ykmin≤ ymin ≤ ykmax
zkmin≤ zmin ≤ zkmax
…(24)
where k is the spline segment and i is the ith joint.
The search space, ∆𝑘, is taken as ∓0.1 m from the
initial position in all directions. The segment
traveling times for the cubic spline are (1.75, 1.75,
1.75, 1.75, 1.75) s. ‘goalatain’ from MATLAB
optimization tool is used to search the optimal
points within their search ranges.
Two cubic spline trajectories are optimized and
along them the manipulator performance is
evaluated. Table (2) lists the initial and optimized
points of the two trajectories and the measured
QAT, ԑ𝑝 and ԑ𝑜𝑟𝑖𝑒𝑛 .
Table 2: Results of Optimal Cubic Spline Trajectories
Tra
jecto
ry
Initial
Points
in XZ-
Plan
(m,m,m)
Rot
.
An
gle
(rad
)
Optimal
Points
in XYZ-
space
(m,m,m)
Q
AT
Pos.
Erro
r
ԑ𝑝
(m)
Orie
nt.
Erro
r
ԑ𝒐𝒓𝒊𝒆𝒏
(rad)
I 0.71, 0, 0.35 0.40, 0, 0.10
0.71, 0, 0.30
0.40, 0, 0.60 0.60, 0, 0.30
0 π/4
π
π/2 3π/4
0.81, 0, 0.30 0.31, 0.31, 0.05
-0.78, 0, -0.29
0, 0.44, 0.61 -0.42, 0.42, -
0.38
1.452
0.0002 0.4396
II 0.20, 0, 0.20
0.60, 0, 0.40 0.40, 0, 0.70
0.30, 0, 0
0.40, 0, 0.70
3π/2
2π 7π/4
0
5π/4
0, -0.17, 0.13
0.65, 0, -0.40 0.29, -0.29,
0.76
-0.35, 0, 0 -0.22, -0.22, -
0.8
1.31
7
0.0011 0.0312
Figure (11) shows the two optimized trajectories
in Cartesian space. Figures (12a-c) show the joint
velocities, accelerations, and torques for the first
three joints; Figures (13a-b) depict the Cartesian
velocities and accelerations and Figure (14) shows
the end-effector velocity. It can be noticed from
all figures that the motion constraints have not
been violated during the optimization process. The
resultant QAT is 1.452 and 1.317, the positioning
Hatem Al-Dois, A.K. Jha, R.B. Mishra
Volume 15 Issue 2 10 ©2015 IJAMS
error is 0.0002 m and 0.0011 m, and the
orientation error is 0.4396 rad and 0.0312 rad
along the first and second optimized trajectories,
respectively.
(a)
(b) Figure 11: The Cubic Spline Trajectories, (a) XY
Plane, (b) XZ Plane
Table 3: Results of Optimal Fourier Series Trajectories
Trajectory I II III IV
Initial Pos. in
XZplane,α (m, m, rad)
0.7, 0, 0 0.25, -0.25, 𝜋
2 0.5, 0.5,
3𝜋
4 0.6, 0.25,
3𝜋
2
Initial Pos.
in XYZ Space (m, m, m)
0.7, 0, 0 0, 0.25, -0.25 -0.354, 0.354,
0.5
0, -0.6,
0.25
Final Position
XZPlane,α (m, m, rad)
0.25, -
0.25, 𝜋
2
0.5, 0.5, 3𝜋
4 0.2, 0.75,
5𝜋
4 0.25, -
0.75,
0
Final Position XYZ space (m, m, m)
0, 0.25, -
0.25
-0.354, 0.354, 0.5
-0.141, -0.141, 0.75
0.25, 0, -0.75
Final Orient.
R, P, Y (rad, rad, rad)
𝜋
2,
𝜋
2, 𝜋
2
𝜋
2, 𝜋
2, 𝜋
2
𝜋
2, 𝜋
2, 𝜋
2
𝜋
2, 𝜋
2, 𝜋
2
Optimal Time
(s) 1.5684 1.6713 1.1875 2.3854
QAT 0.3578 0.2953 0.2419 0.3317
Pos.Error (m)
0.0005 0.0005 0.0001 0.0006
Orien. Error (rad)
0.093 0.3714 0.1276 0.3166
(a)
(b)
(c) Figure 12: Cubic Spline Trajectories, (a) Joint
Velocities, (b) Joint Accelerations, (c) Joint Torques
(a)
Dynamic Analysis of Serial Robot Manipulators
Volume 15 Issue 2 11 ©2015 IJAMS
(b)
Figure 13: Cubic Spline Trajectories, (a) Cartesian
Velocities, (b) Cartesian Accelerations
Figure 14: The End-Effector Velocity for the Cubic
Spline Trajectories
7.2. PTP Fourier-Based Trajectories,
Time Optimized
Four Fourier series trajectories are optimized. The
initial and final points are taken from the analysis
of kinematic and dynamic manipulability of the
robot’s workspace, Figures (5-8). The initial
(R,P,Y) are set equal to (0,0,0). Each trajectory is
optimized with respect to time. Table (3) indicates
the four trajectories and their optimized times,
QAT, and positioning and orientation accuracy.
Figure (15) shows the four trajectories in the work
space. Figures (16-17) show the kinematic and
dynamic manipulability of the first trajectory.
Figures (18a-d) show the joint position, velocity,
acceleration, and torques and Figure (19a-b) show
the Cartesian velocities and accelerations. The
end-effector velocity is depicted in Figure (20).
All motion constraints are maintained during the
time optimization. It can also be noticed how the
maximum end-effector velocity is saturated due to
the time optimization. The maximum QAT is
0.3578 and maximum positioning and orientation
errors are 0.0006 m and 0.3714 rad, respectively.
Figure 15: The Optimal Fourier-Based Trajectories
Figure 16: The Kinematic Manipulability of the First
PTP Trajectory
Figure 17: The Dynamic Manipulability of the First
PTP Trajectory
Hatem Al-Dois, A.K. Jha, R.B. Mishra
Volume 15 Issue 2 12 ©2015 IJAMS
(a)
(b)
(c)
Figure 18: Fourier-Based Trajectories, (a) Joint
Velocities, (b) Joint Accelerations, (c) Joint Torques
(a)
(b)
Figure 19: Fourier-Based Trajectories, (a) Cartesian
Velocities, (b) Cartesian Accelerations
Figure 20 : The End-Effector Velocity for the Fourier-
Based Trajectories
8. Summary
A systematic approach is proposed to analyze the
dynamic performance of serial robot
manipulators. The approach is summarized in
designing deferent optimized exciting trajectories
through which the performance of the manipulator
is evaluated. The initial and final points of the
constructed trajectories are selected based on the
kinematic and kinetostatic analysis of the robot
workspace. Two performance indices are
considered: the quadratic average of joints torques
and the positioning and orientation accuracy.
Errors due to variability in robot parameters are
modeled using a probabilistic approach to
simulate a realistic robot behavior. The proposed
method was applied to Puma 560 manipulator to
demonstrate the feasibility of the approach.
9. References
1. Bowling, A., Analysis of robotic
manipulator dynamic performance:
acceleration and force capabilities, in
Department of Mechanical
Engineering+1998, Stanford University.
Dynamic Analysis of Serial Robot Manipulators
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2. Khatib, O. Dynamic control of manipulators
in operational space. in Sixth CISM-
IFToMM World Congress on the Theory of
Machines and Mechanisms. 1983. New
Delhi, India.
3. Asada, H., A geometrical representation of
manipulator dynamics and its application to
arm design. , Measurement, and Control.
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4. Yoshikawa, T. Dynamic manipulability of
robot manipulators. in 1985 IEEE
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USA.
5. Yoshikawa, T., Manipulability of Robotic
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6. Tourassis, V.D. and C.P. Neuman, The
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9. Ma, O. and J. Angeles. Optimum design of
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Biography
Hatem Al-Dois: received his B.SC. degree in
production engineering from
University of Technology, Baghdad,
Iraq, M.Tech. in mechanical
engineering from National Institute
of Technology, Warangal, A.P.,
India, and PhD from IIT(BHU),
Varanasi, U.P., India. Mr. Al-Dois is a faculty
member at the Faculty of Engineering and
Architecture, University of Ibb, Yemen. His areas
of interest include computer integrated
manufacturing and robot dynamics.
A. K. Jha: received his undergraduate and
postgraduate degrees in
mechanical engineering from IT-
BHU and PhD from BIT, Mesra,
Ranchi, India. Dr. Jha has served
more than 15 years in a foundry-
forge plant of a heavy industry in
India and presently serving department of
mechanical engineering, IIT-BHU as Professor.
He has authored more than 65 research papers
appeared in national and international journals and
proceedings of seminars/conferences. His areas of
specialization are material processing and
computer integrated systems. He has supervised
several postgraduate and doctoral level
researchers. Dr. Jha is also member of several
professional bodies and committees in India.
R. B. Mishra: received his B.Sc. in electrical
engineering from BIT Sindri and
M.Tech. in control system from
IT-BHU and PhD (AI in
medicine) from IT-BHU. Dr.
Mishra has 33 years teaching
experience and presently serving
Dynamic Analysis of Serial Robot Manipulators
Volume 15 Issue 2 15 ©2015 IJAMS
as a professor and Head in the department of
computer engineering, IIT-BHU. His areas of
research interest include artificial intelligence and
multiagent systems and their applications in
medical computing, machine translation, e-
commerce, robotics and intelligent tutoring
systems. He has authored around 200 papers in
journals and conferences. Dr. Mishra has
supervised more than 25 postgraduate students
and 12 PhD students and currently 7 are in
progress. He has visited USA and UK-University
of Bath under INSA fellowship.