DYNAMIC ANALYSIS OF SERIAL ROBOT …ispe-usa.com/docs/images/1_Dynamic_Analysis.pdfDynamic Analysis of Serial Robot Manipulators Volume 15 Issue 2 3 ©2015 IJAMS effector position

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


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


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



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



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



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)



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



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



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



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)



(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



(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

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manipulator dynamic performance:

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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



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.

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