Journal of Robotics, Networking and Artificial Life, Vol. 4, No. 2 (September 2017) 138–141___________________________________________________________________________________________________________Dynamic Model and Motion Control of a Robotic Manipulator
Jinho Kim, Kevin Chang, Brian Schwarz
Department of Mechanical Engineering, University of Maryland, Baltimore County (UMBC), 1000 Hilltop Circle,Baltimore, Maryland, USA, 21250
Andrew S. Lee, S. Andrew Gadsden
School of Engineering, University of Guelph,Guelph, Ontario, Canada, N1G 2W1
Mohammad Al-Shabi
Department of Mechanical Engineering, University of Sharjah, University City Rd, Sharjah, United Arab Emirates
E-mail: umbcjhkim@umbc.edu, alee32@uoguelph.ca, kevicha1@umbc.edu, schwarz4@umbc.edu,malshabi@sharjah.ac.ae, and gadsden@uoguelph.ca

Abstract
This paper presents the dynamic modeling and motion control of a three-link rotary robotic manipulator, also
known as an RRR robot. The Kinect motion capture system by Microsoft is used in conjunction with themanipulator. A camera is used to capture the motion of a user’s arm and tracks certain angles made by parts of thearm. We consider a pinhole camera model to generate reference angles in our simulations. These desired angles are
fed into the controller and are used by the RRR robot in an effort to copy the movement of the user. A proportionalderivative (PD) controller is developed and applied to the manipulator for improved trajectory tracking. The RRR
robot is dynamically modeled and the results of the proposed control strategy demonstrate good trajectory
following. Researchers can implement this system as a relatively inexpensive starting point for their activities.
Keywords: Three-link robotic manipulator, RRR robot manipulator, Motion capture, PD controller.

vehicles with robotic manipulators are used in bomb
disarming, search and rescue, and reconnaissance. To
conduct surgery or translate human movement with arobotic manipulator on military missions remotely, it isimportant to control the robotic manipulator in sync
with the movement of the operator [5-13].
In this brief paper, we consider controlling a system
which consists of a camera and robotic manipulator.
The pinhole camera model extracts reference angles
from simulations. The desired angles are fed into thecontroller and are utilized by the robotic manipulator inan effort to mimic the movement of the user. Aproportional-derivative (PD) controller is designed and1. Introduction
Robotic manipulators have been developed and utilized
in a variety of sectors, including: industry, medical, andmilitary. In recent years, due to their potential to
improve precision and enhance robustness, robotic
manipulators have been applied to not only military
missions but also medical devices. There are many
research topics for medical devices with robotic
manipulators such as human-robot collaboration, motion
planning, and simultaneous control [1-4]. In the military
field, we can easily perform robotic manipulator
applications as well. For instance, autonomous ground

Copyright © 2017, the Authors. Published by Atlantis Press.
This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
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Journal of Robotics, Networking and Artificial Life, Vol. 4, No. 2 (September 2017) 138–141___________________________________________________________________________________________________________

𝑝1 = (𝑢1,𝑣1 ) and 𝑝2 = (𝑢2, 𝑣2 ) . The angle can becalculated as follows:𝑣 −𝑣
𝜃 = tan−1 � 2 1 �(2)applied to the robotic manipulator for trajectory
tracking. The paper is organized as follows. In Section
2, the pinhole camera model is introduced and the threelink robotic manipulator model is derived using theclassical Euler-Lagrange equation. Based on thesemodels, a PD controller is designed and shown inSection 3. The simulation results are presented anddiscussed in Section 4. Finally, the paper concludes andfuture work is described.

𝑢2−𝑢1

3. Dynamics and PD Controller for the RRR
Robotic Manipulator

In this section, we derive the dynamics of a three-linkrobotic manipulator and design a PD controller for it.
An overview of the system is depicted in Fig. 2.

2. Pinhole Camera Model
The pinhole camera model is used to obtain reference
angles. The camera captures the motion of the user’sarm. The shoulder, elbow, and wrist angles are extracted
through image processing. The pinhole camera model isshown in Fig. 1.
Note that we define 𝕀 as the inertial coordinate

Figure 2. Overview of the system.

3.1. Dynamics of Robotic Manipulator
The configuration of the three-link robotic manipulator
is shown in Fig. 3. Note that 𝜃𝑖 (𝑖 = 1,2,3) denotes theangles between each link, 𝐿 𝑖 is the length of the link, 𝑙 𝑖is the distance from the centroid to corresponding axis,
𝑚 𝑖 is the mass, and 𝐽𝑖 is the moment of inertia. Let 𝜏𝑖denotes the joint torque and its counterclockwise
rotation is assumed to be positive. The Euler-Lagrange
equation is applied as shown in Eqs. (3-4) where ℒ is
the Lagrangian with kinetic energy 𝒦 and potential
energy 𝒫 of the system [10].

Figure 1. Geometry and coordinate frames for the pinhole
camera model.

𝑑 𝜕ℒ

𝑑𝑑 𝜕𝜃̇𝚤

frame, ℂ as the camera frame, and 𝕊 as the image frame.
Let [𝑋𝐼 , 𝑌𝐼 , 𝑍𝐼 ]𝑇 be the axes of the inertial coordinate
frame 𝕀, and [𝑋𝑐𝑐𝑐, 𝑌𝑐𝑐𝑐 ,𝑍𝑐𝑐𝑐 ]𝑇 be the position vector
𝑇in the camera frame ℂ . Let �𝑈𝑖𝑖𝑖𝑖𝑖, 𝑉𝑖𝑖𝑎𝑔𝑔� be thecoordinate axes of the image frame 𝕊 and 𝑂𝑆 represents
the principal point where the 𝑧 -axis of the camera
coordinate system intersects the image plane. The focal
length of the camera, 𝑓, is the distance between 𝑂𝑐𝑐𝑐
and 𝑂𝑆 . Finally, the mapping from a point 𝑃 =[𝑥𝑐 , 𝑦𝑐, 𝑧𝑐 ]𝑇 ∈ ℂ onto the image plane can be written asEq.(1).
𝑢𝑓 𝑥𝑝 = � � = �𝑦𝑐 �(1)𝑣𝑧𝑐𝑐−𝜕ℒ𝜕𝜃𝑖= 𝜏𝑖ℒ = 𝒦1 + 𝒦2 + 𝒦3 − 𝒫1 − 𝒫2 − 𝒫3

The user’s arm is projected onto the image plane 𝕊through Eq. (1). The three angles are calculated byextracting three points from the image plane using
simple geometry. Let us define two points in 𝕊 as

(3)

(4)

Figure 3. Configuration of the three-link robotic manipulator.

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Journal of Robotics, Networking and Artificial Life, Vol. 4, No. 2 (September 2017) 138–141___________________________________________________________________________________________________________In this system, 𝒦 and 𝒫 of each link can be written as
per Eqs. (5-6) where ‖𝑣𝑖 ‖ represents the magnitude of
the velocity, 𝑙 𝑖,𝑦 is the position of the centroid that isdependent on joint coordinates.
2
1
1
𝒦𝑖 = 𝑚 𝑖 ‖𝑣𝑖 ‖2 + 𝐼𝑖 �∑𝑖 𝜃̇𝑖 �(5)2In order to control the system, 𝜏𝑖 can be defined as acontrol input 𝑢𝑖 which controls 𝜃𝑖 with a reference input
𝜃𝑟 . The PD controller of the system are given by:
𝑢𝑖 = 𝑘𝑝,𝑖 𝑒𝑖 + 𝑘𝑑,𝑖 𝑒̇𝚤(8)where 𝑘𝑝 and 𝑘𝑑 are proportional and derivative gains,respectively. The angle error, 𝑒𝑖 , denotes 𝜃𝑟,𝑖 − 𝜃𝑖 andthe derivative of the angle error, 𝑒̇𝚤 , denotes 𝜃̇𝑟,𝑖 − 𝜃̇𝑖 .

2𝑃𝑖 = 𝑚 𝑖 𝑔𝑙 𝑖,𝑦(6)From these equations, we can obtain the final dynamics
of the system in a simple form as follows:𝑢 = 𝑀(𝜃)𝜃̈ + 𝐶�𝜃, 𝜃̇� + 𝑔(𝜃)(7)where 𝑀(𝜃) is the inertia matrix, 𝐶�𝜃, 𝜃̇� represents thematrix of coriolis and centrifugal elements, 𝑔(𝜃) is thematrix of gravity elements, and, 𝑢 is control input. The
matrices and their parameters can be written as follows:
𝑎11 𝑎 12 𝑎 13𝑀 (𝜃) = �𝑎 12 𝑎 22 𝑎 23�,𝑎 13 𝑎 23 𝑎 33
𝑏1𝐶�𝜃,𝜃̇� = �𝑏2 �,
𝑏3
𝑔1
𝑔(𝜃) = �𝑔2 �,
𝑔3where,

4. Simulation Results

This section presents simulation results with initialparameters. The simulation used the initial settings
which were defined as follows:
𝑘𝑝 = [37.5, 24.2, 16.4]
𝑘𝑑 = [1.34, 0.87, 1.04]
𝑚 𝑖 = [1𝑘𝑘, 1𝑘𝑘, 1𝑘𝑘]
𝐿 𝑖 = [0.2𝑚, 0.2𝑚, 0.2𝑚] , 𝑙 𝑖 = [0.1𝑚, 0.1𝑚, 0.1𝑚]
𝐽𝑖 = [0.5𝑘𝑘 ∙ 𝑚2 , 0.5𝑘𝑘 ∙ 𝑚2 , 0.5𝑘𝑘 ∙ 𝑚2 ].
We generated movements of arm and extracted theshoulder, elbow, and wrist angles as reference inputs
𝜃𝑟,1 ,𝜃𝑟,2 , 𝜃𝑟,3 . We set the reference inputs as follows:𝜃𝑟,1 = 3 sin(0.1𝜋𝜋)𝜃𝑟,2 = 4 cos(0.1𝜋𝜋)𝜃𝑟,3 = 9 sin(0.15𝜋𝜋)
where 𝑡 is time from 0 to 25 seconds in this simulation.
The history of each angle is shown in Fig. 4 where theblack dashed lines are reference inputs 𝜃𝑟,𝑖 and coloredlines are current angles. As shown in Fig. 4, we canobserve that the trajectories of the system angles follow
the reference angels with satisfactory results. The initialovershoot in Fig. 4 shows acceptable performance of thePD controller for this system.

𝑎11 = 𝑚1 𝑎 2𝑐1 + 𝑚2 (𝑎12 + 𝑎 2𝑐2 + 2𝑎1 𝑎 𝑐2 𝑐2 ) + 𝑚3 (𝑎12 + 𝑎 22 +𝑎 2𝑐3 + 2𝑎1 𝑎 2 𝑐2 + 2𝑎1 𝑎 𝑐3 𝑐23 + 2𝑎 2 𝑎 𝑐3 𝑐3 ) + 𝐼1 + 𝐼2 + 𝐼3
𝑎12 = 𝑚2 (𝑎 2𝑐2 + 𝑎1 𝑎 𝑐2 𝑐2 ) + 𝑚3 (𝑎 22 + 𝑎 2𝑐3 + 𝑎1 𝑎 2 𝑐2 +𝑎1 𝑎 𝑐3 𝑐23 + 2𝑎 2 𝑎 𝑐3𝑐3 ) + 𝐼2 + 𝐼3
𝑎13 = 𝑚3 (𝑎 2𝑐3 + 𝑎1 𝑎 𝑐3 𝑐23 + 𝑎 2 𝑎 𝑐3 𝑐3 ) + 𝐼3
𝑎 22 = 𝑚2 𝑎 2𝑐2 + 𝑚3 (𝑎 22 + 𝑎 2𝑐3 + 2𝑎 2 𝑎 𝑐3 𝑐3 ) + 𝐼2 + 𝐼3
𝑎 23 = 𝑚3 (𝑎 2𝑐3 + 𝑎 2 𝑎 𝑐3 𝑐3 ) + 𝐼3
𝑎 33 = 𝑚3 𝑎 2𝑐3 + 𝐼3
𝑏1 = −𝑚2 𝑎1 𝑎 𝑐2 �2𝜃1̇ + 𝜃̇2 �𝑠2 𝜃2̇ − 𝑚3 �𝑎1 𝑎 2 �2𝜃1̇ + 𝜃̇2 �𝑠2 +𝑎1 𝑎 𝑐3 �2𝜃̇1 + 𝜃̇2 + 𝜃̇3 �𝑠23 + 𝑎 2 𝑎𝑐3 �2𝜃1̇ + 2𝜃̇2 + 𝜃̇3 �𝑠3 �𝑏2 = −𝑚2 �𝑎1 𝑎 𝑐2 �𝜃1̇ 2 + 𝜃̇1 𝜃̇2 �𝑠2 + 𝑎1 𝑎 𝑐2 𝜃1̇ 𝑠2 𝜃2̇ � −𝑚3 �𝑎1 𝑎 2 �𝜃1̇ 2 + 𝜃1̇ 𝜃̇2 � + 𝑎1 𝑎 𝑐3 �𝜃1̇ 2 + 𝜃1̇ 𝜃̇2 + 𝜃1̇ 𝜃̇3 �𝑠23 +𝑎 2 𝑎 𝑐3 �𝜃1̇ 2 + 2𝜃̇1 𝜃̇2 + 𝜃̇1 𝜃3̇ + 𝜃̇22 + 𝜃̇2 𝜃̇3 �𝑠3 + 𝑎1 𝑎 2 𝜃1̇ 𝑠2 𝜃̇2 +𝑎1 𝑎 𝑐3 𝜃1̇ 𝜃̇2 𝑠23 + 𝑎 2 𝑎 𝑐3�2𝜃1̇ + 2𝜃̇2 + 𝜃̇3 �𝑠3 �𝑏3 = −𝑚3 �𝑎1 𝑎 𝑐3 �𝜃1̇ 2 + 𝜃̇1 𝜃̇2 + 𝜃̇1 𝜃̇3 �𝑠23 + 𝑎 2 𝑎 𝑐3 �𝜃1̇ 2 +2𝜃1̇ 𝜃̇2 + 𝜃1̇ 𝜃̇3 + 𝜃2̇ 2 + 𝜃2̇ 𝜃̇3 �𝑠3 + 𝑎1 𝑎 𝑐3 𝜃1̇ 𝜃̇3 𝑠23 +𝑎 2 𝑎 𝑐3 �𝜃1̇ + 𝜃̇2 �𝑠3 𝜃3̇ �
𝑔1 = 𝑔[𝑐1 (𝑚1 𝑎 𝑐1 + 𝑚2 𝑎1 + 𝑚3 𝑎1 ) + 𝑐12 (𝑚2 𝑎 𝑐2 + 𝑚3 𝑎 2 ) +𝑐123 (𝑚3 𝑎 𝑐3 )]𝑔2 = 𝑔 [(𝑚2 𝑎 𝑐2 + 𝑚3 𝑎 2 )𝑐12 + 𝑚3 𝑎 𝑐3𝑐123 ]𝑔3 = 𝑔 (𝑚3 𝑎 𝑐3 𝑐123 ).

3.2. PD Controller

In this subsection, the PD controller for the three-link
manipulator system is presented.

Figure 4. Results of PD controller for RRR robot.

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Journal of Robotics, Networking and Artificial Life, Vol. 4, No. 2 (September 2017) 138–141___________________________________________________________________________________________________________References
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Figure 5. History of the three control inputs.

Figure 6. Error between angles and reference inputs.

The control signals and corresponding tracking errors
are shown in Figs. 5 and 6, respectively. The magnitude
errors are less than one degree which is acceptable for arobotic manipulator with linkage dimensions similar toa human arm.
5. Conclusions
This paper presents a three-link robotic manipulator
combined with a camera for capturing the motion of theuser. For successful control of the system, the pin holecamera model captures the arm motion of the user andextracts the shoulder, elbow, and wrist angles throughimage processing. The dynamics of the system arederived and the PD controller is applied to the angles ofeach link. The simulation results verify the efficacy of a
PD controller and demonstrate satisfactory results for
tracking arm movements.

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