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Cascaded PID Trajectory Tracking Control for
Quadruped Robotic Leg
Ahmed M. El-Dalatony*, Tamer Attia, Hossam Ragheb, and Alhossein M. Sharaf
Military Technical College, Cairo, Egypt;
Email: t_attia@mtc.edu.eg,
h.ragheb@mtc.edu.eg, a.m.sharaf@mtc.edu.eg
*Correspondence: adalatony@gmail.com
Abstract—This paper presents a cascaded Proportional
Integral Derivative (PID) trajectory tracking controller to
control the foot's tip of a quadruped robotic leg. The
proposed robotic leg is designed and developed using electric
Quasi-Direct Drive (QDD) actuators with high efficiency and
torque density. Both the forward and inverse kinematics of
the robotic leg are introduced to generate the desired path
with the associated velocity of the foot's tip. Furthermore, the
cascaded PID trajectory tracking controller is developed as a
low-level controller to control the position and angular
velocity of each leg's joint. Both the numerical simulation and
experimental results showed that the proposed controller
succeeded in tracking the desired trajectory with high
accuracy and robustness of two different types of trajectories.
—trajectory tracking control, quadruped robots,
cascaded PID controller
I. INTRODUCTION
Quadruped robots have been gaining importance in
recent years by enhancing the robot design in terms of
actuation technique, sensory system, advanced control
algorithms, and design optimization to introduce agile
quadrupedal robots such as the Massachusetts Institute of
Technology (MIT) Cheetah [1], ANYmal [2], and Mini-
HyQ [3]. Recently, legged robots can be used in many
applications as it can offer an alternative and excellent
solution to the mobility problems. Scientists are inspired
by what legged animals can do where it can go almost
everywhere and move dynamically with high mobility and
agility to overcome challenging obstacles.
Quadruped robots have recently provided great progress
in dynamic locomotion capabilities by utilizing different
actuation and control approaches. For instance, using
hydraulic actuators which can provide high joint forces,
was fortunate in some quadruped platforms such as IIT’s
HyQ [4] and Boston Dynamics’ Big Dog [5]. On the other
hand, Series Elastic Actuators (SEA) provide higher
impact mitigation properties and better force control
capacity. SEA is suitable for high speed legged locomotion
as demonstrated in ANYmal quadruped robot [2]. Another
actuating approach is using a custom-designed
Manuscript received June 6, 2022; revised July 1, 2022; accepted August
12, 2022.
proprioceptive actuator to acquire impact mitigation
properties and high force control capabilities.
Figure 1. The proposed robotic leg with three Quasi-Direct Drive
(QDD) actuators.
This approach was used in developing the MIT Cheetah
II [6], in which the robot was able to successfully perform
jumping over obstacles and bounding at high speeds of up
to 6 m/s [7].
A noticeable limitation in the field of quadruped robot
is to implement a robust control algorithm. It is required to
support the robot’s body weight, control its attitude, place
the robot’s feet in a certain trajectory and provide
locomotion while keeping the robot balanced during the
planned trajectory. Position control strategies can be
employed through precise kinematic modeling and
trajectory tracking. It requires detailed knowledge about
the internal states of the robot and the surrounding
environment. Position control can be integrated with
impedance control during the swing phase to ensure
flexible interaction between the leg and the ground to
reduce the impact [8].
A different control approach named Model Predictive
Control (MPC) is considered as a powerful approach for
controlling robotic systems. Therefore, it can be used as
motion control for complex nonlinear dynamic systems
such as quadruped locomotion [9]. Another approach is
Zero Moment Point (ZMP) control is considered as an
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International Journal of Mechanical Engineering and Robotics Research Vol. 12, No. 1, January 2023
© 2023 Int. J. Mech. Eng. Rob. Res
doi: 10.18178/ijmerr.12.1.40-47
Keywords
approach to maintaining the balance of a quadruped robot
by keeping its Center of Mass (COM) directly above a
support region that can be previously determined [10].
While the Proportional Integral Derivative (PID)
controllers are also widely used to control machines in
various industrial applications. In robotic systems, a
typical PID controller independently controls the motion
of each joint, whereas model-based controllers are used to
improve the performance of the PID controllers as they are
based on the system dynamics as feedback.
Various control approaches were used in different
quadruped robots. Yasuhiro Fukuoka used a PID controller
for the leg’s joints to simulate a virtual spring–damper
system [11], while the neural model consists of a Central
Pattern Generator (CPG) and some reflexes. The CPG
receives data from sensors and then modifies the active
phase period, the desired angle, and the gains of each joint.
While the MIT cheetah quadruped platform has a control
approach that uses a gait-pattern generator to calculate
each leg’s foot trajectory, the desired speed, duration of
swing, and stance phases for each leg. The desired leg
trajectory serves as an equilibrium point for the leg
impedance controller which controls leg’s mechanical
behavior to mimic it as a spring-damper system positioned
between the hip and the foot.
Boston Dynamics’ quadruped robot BigDog has a
control system that coordinates between the robot’s
kinematics and ground reaction forces at the feet. For
adapting to the terrain changes, it uses sensor information
from joints to determine the feet status by calculating the
load on each leg [5]. While Victor Barasuol proposed a
novel reactive controller framework to cope with terrain
irregularities, trajectory tracking problems, and state
estimation problems [12]. It uses a module to generate an
elliptic feet trajectory and another module for stability
control of the whole robot. Barasuol also proposed a CPG-
based trajectory generation method for adapting smoothly
to terrain irregularities. CPG is becoming a popular model
for the locomotion control of quadruped robots [13].
Sangbae Kim presented an implementation of MPC to
calculate the ground reaction forces for MIT Cheetah III.
The robot was able to perform different gaits including
walking, trotting, flying-trotting, bounding, pacing, and
galloping with a maximum speed of up to 3 m/s. The
controller introduced excellent results because the model
was able to capture all the vital details of locomotion,
especially the ground reaction forces [14]. Hutter proposed
a novel technique for quadruped robot training neural-
network policies and terrain-aware locomotion. This
technique combines model-based motion planning method
and reinforcement learning to estimate the terrain-aware
locomotion and improve locomotion accordingly [15].
In this paper, a fully designed and developed QDD
actuation approach is used for developing a single
quadruped robotic leg with the required control and
sensing systems for investigating the robotic leg
performance with different trajectories and different loads.
Furthermore, a cascaded PID trajectory tracking controller
is presented to control the foot’s tip of the quadruped
robotic leg. First the forward and inverse kinematics
models of the robotic leg is introduced for generating the
desired foot’s tip positions and the associated joints’
angles. Then, trajectory tracking control is proposed as a
low-level controller to control the position, velocity, and
torque for each joint. Finally, experimental results show
the effectiveness of the proposed controller framework in
tracking the desired path of the foot’s tip.
II. ROBOTIC LEG DESIGN
In this section, an overview of the robotic leg system
will be introduced, followed by the mechanical and
electrical design with the actuation and sensory system.
A. Mechanical Design and Actuation
Selecting a suitable actuation technique is a very
challenging process as quadruped locomotion requires a
lot of conflicting characteristics such as high force, high
speed, and high torque density properties. Direct Drive
(DD) actuators have high effective torque with no backlash
and high efficiency but at the cost of low torque and high
speed properties [16]. Another approach is using SEA
which provides precise force control, better impact
absorption, and high torque density [17]. SEA have high
mechanical complexity, control difficulty and cost. SEA
are used in quadruped platforms such as StarlETH [18].
Using proprioceptive QDD robotic actuators with a
relatively low reduction ratio gearbox integrated with a
Brushless Direct Current (BLDC) motor allows for high
efficiency and high torque density actuator [19]. QDD
actuators are used without a dedicated force sensor since
the torque values of any joint can be measured using motor
phase current [20].
Most innovative electric-derived quadruped actuators
use custom-designed electric motors and gears, which
makes the total cost very high. In this paper, an off-the-
shelf electric BLDC motor is used as a base for building
an inexpensive QDD actuator which offers features such
as impact resistance, high torque density, and robust force
control through internal torque control without sensory
feedback.
Quadruped legs have great importance in the legged
locomotion performance, the most important parameters
are having a low mass, low inertia, and maximum range of
motion. In this design approach, changing the knee
actuator position from its natural place at the knee and
moving it up to the hip allows for a low inertia leg. The
design approach is based on most of the modern
quadrupeds that exist in the robotics field, such as Boston
Dynamics Spot [21], Unitree Robotics quadruped robots
Laikago [22], vision 60 of Ghost Robotics [23], and the
MIT Cheetah III [24].
The leg parts are CNC machined from Aluminum Alloy
7075, which is chosen for its lightweight and high-strength
properties. The upper link of the leg consists of two
identical parts and the lower link is machined as one piece.
The leg’s total mass is approximately 1:5 Kg without
actuation joints. For transmitting the torque to the knee
joint, a system of two pulleys and a high shock absorbing
characteristics timing belt is used. The range of motion is
±135° for the Knee Flexion/ Extension (KFE) joint from
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the fully extended lower part of the leg, the Hip
Abduction/Adduction (HAA) joint allows for a motion
range of 90°, and the Hip Flexion/Extension (HFE) joint
allows for a range up to 345°, these wide ranges provide
agility and different motion capabilities for the quadruped
robot as shown in Fig. 1.
B. Electric Design and Sensing
The main electrical configuration of the robotic leg
consists of a PC, a motor controller (ODrive v3.6), two
position sensors (AS5047P), a 24-volt power supply, and
an emergency switch as shown in Fig. 2. The high-
performance motor controller consists of a microcontroller
with two voltage and current feedback sensors and two
motor drivers. The motor controller can control the two
BLDC motors simultaneously and capable of controlling
the actuators’ position, velocity, and torque. Moreover, the
motor controller can read the motors’ positions using an
external encoder for arbitrarily precise motions and is
powered by a 24-volt dedicated power supply. A personal
computer is used to communicate with the motor controller
through serial communication. The PC is used as a high-
level trajectory tracking controller. The position sensor is
a magnetic rotary position sensor used for high speed angle
measurement. The magnetic position sensor is placed
concentrically with a diametric magnet allowing for
position tracking.
Figure 2. Electric diagram circuit and sensory system.
III. ROBOTIC LEG KINEMATICS
Quadruped’s body is mathematically represented as a
rigid box where its four robotic legs are mostly
symmetrically distributed and structured. The robotic leg
consists of rigid links and rotary joints, where the rigid
links are connected through the rotary joints. The
kinematic modeling and analysis are necessary for
modeling and controlling quadruped robots. Where it
studies all the possible motions of the robot, regardless of
the forces and moments that generate these motions.
Kinematics is classified into two types; forward
kinematics and inverse kinematics [25]. It is sufficient to
study the kinematics of a single robotic leg since the
quadruped robot has four similar legs.
A. Forward Kinematics
The basic requirement is to find all the possible
positions that a quadruped foot can achieve. Forward
kinematic analysis is used as a pre-calculation for inverse
kinematic analysis. In 3D space with X-Y-Z frame of
reference, the HAA joint is perpendicular to the HFE joint,
which is connected to the thigh with length (l1) through the
KFE joint. The angle between the thigh and the X-axis is
called (θ1), while the angle between the shin and the thigh
extension is called (θ2). Fig. 3a and Fig. 3b show the
forward kinematic model for a 3-Degree of Freedom (DOF)
and a 2-DOF robotic Leg respectively. The robotic leg
forward kinematics in space can be defined as follows:
(1)
While the forward kinematics in 2D plane is written as:
(2)
The analytic algebraic approach is used for solving the
forward kinematics using Denavit-Hartenberg (DH)
notation for describing the legged robots [26]. The
relationship between two coordinate frames has four
parameters; two rotations and two displacements
respectively. The algebraic solution is obtained by
defining these four parameters for each leg. The path that
forms the hip joint to the tip was described using the matrix
transformations, by multiplying all the transform matrices.
The final matrix describes the kinematic model, where (i)
is the joint angle, (αi) is the link twist, (ai) is the link length,
and (di) is the link offset. Table I describes the DH
Parameters of the proposed robotic leg.
(a) 3D Kinematics. (b) 2D Kinematics.
Figure 3. Free-body diagram of the 3-DOF and 2-DOF legs.
TABLE I. THE FOUR DH PARAMETERS.
αi-1 ai-1 di
Link 1 0 l1 0
Link 2 0 l2 0
The 4 DH parameters (θi, αi-1, ai-1, and di) are
associated with each link, where θi is rotation about the z-
axis, or called the joint angle and it equals zero if the joint
(i) is a prism. While (di) is the distance on the Z-axis or
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called the joint offset which equals zero, and it can be a
variable if the joint (i) is prismatic. The parameter (ai-1) is
the length of each link, and (αi-1) is the angle between two
successive Z-axes or called the twist angle which equals
zero. The homogeneous transformation A represents the
pose of the foot tip of the robotic leg, and by using the DH
parameters, the foot pose can be written as a sequence of
elementary homogeneous transformations as in (3).
By expanding and multiplying all the transformations,
the 3×3 homogeneous transformation matrix is
representing the pose of the leg’s foot tip as follows:
(3)
(4)
Where c (.) and s (.) denote cos (.) and sin (.) respectively.
Solving for the developed 2DOF quadruped robotic leg in
(X-Y) coordinates results in the same solution from the
geometric approach as derived in (2).
B. Inverse Kinematic
The inverse kinematics approach is to find the angles of
the robotic leg’s joints that map the desired position of the
foot’s tip. By using the geometric inverse kinematic
approach in the following equations with simplifying them
using trigonometric, we can get the values of the required
joint angles for a 3-DOF robotic leg. These angles are the
HAA angle, HFE angle, and KFE which are denoted as θ1,
θ2 and θ3 respectively as follows:
(5)
(6)
(7)
Where:
(8)
For a 2-DOF robotic leg, using (2), the solution of
inverse kinematics can be computed geometrically as
follows, then the angles of joints can be calculated as
follows:
(9)
By using (2) which represents the foot pose in 2D, and
considering the angles of joints are function of time as
follows:
(10)
The linear relationship between the quadruped leg foot
velocity (v) or the spatial velocity, and the joint angles rate
of change () can be calculated as follow:
(11)
Where (J) is the Jacobian matrix which can be calculated
as follows:
(12)
From (11) and (12), the foot tip spatial velocity can be
estimated as:
(13)
Therefore, the angular velocity of the joints θ can be
calculated as follows:
(14)
IV. TRAJECTORY TRACKING CONTROL
Proper foot trajectory tracking is a vital aspect to have
efficient locomotion of the quadruped robot. First, the
desired trajectory has to be sampled with multiple points,
and the foot’s tip should follow the desired points’ path
with a specific velocity. The trajectory might be a smooth
polynomial interpolation of these points, some examples
of these paths are cubic and Bernstein polynomials
trajectories [27]. Multiple foot trajectories are studied
along with their description equations, and distinct
characteristics, some examples of these trajectories are
cycloidal, Fourier-Series trajectory, elliptical, and a
combination between cubic polynomial and straight line.
The trajectory tracking control consists of two levels;
the high-level controller, and the low-level controller.
− In the high-level controller, inverse kinematics is used
to generate the desired angles for each joint with the
associated velocity and torque. Then, the desired joints’
angles are sent to the low-level controller to control
the position, velocity, and torque for each joint.
− The low-level controller is a cascaded PID controller
as shown in Fig. 4 where the desired joints’ angles are
sent directly into the position PID controller. Then, the
output is the velocity command associated with the
desired joints’ angular velocities as a feed-forward to
the velocity PID controller.
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Finally, the output from the velocity PID controller is
the torque command associated with the desired torque as
a feed-forward to the torque PID controller. As a result, the
appropriate position, velocity, and torque will be achieved
by the driver of the motor.
Figure 4. The PID control diagram for robotic leg trajectory tracking.
V. NUMERICAL AND EXPERIMENTAL RESULTS
This section demonstrates the performance of the
cascaded PID controller as described in section IV using
numerical simulations and experiments. Where, two
different trajectories are considered to assess the robotic
leg performance: the full circle and a combination between
cubic polynomial (swing phase) and straight line (stance
phase). The proposed trajectories are verified in both
simulation and experiments. The results are compared with
the reference trajectories.
A.
Simulation Results
For the quadruped robot to perform a certain task such
as different gaits of locomotion, the task is transformed
into a path planning problem aiming to generate a set of
points from the start to the goal. From that path the
trajectories can be generated to schedule or follow the
desired reference path including information about the
pose, the speed, and the acceleration. Then, the low level
control capabilities are used to perform the trajectory
tracking. Numerical simulation is performed using
MATLAB to generate and execute the required motion
trajectories. The robotic leg model is represented in 2D as
two links which are connected together by joints allowing
rotational motion. The two links are considered as a
kinematic chain with the foot tip as an end-effector. After
assigning the robotic leg variables such as the links’ angles,
the desired trajectories are provided along with boundary
conditions, trajectory Cartesian points, and sampling rate.
MATLAB software is also used to guarantee that these
trajectories are within the allowable workspace of the leg’s
foot as shown in Fig. 5.
The first trajectory represents a full circle with a
diameter of 200 mm. The second trajectory consists of an
energy efficient combination of two phases, a cubic
polynomial for the swing phase and a straight line for the
stance phase. The step length is 400 mm, and the step
height is 100 mm. The results demonstrate the simulation
of different trajectories tracking.
(a) (b)
Figure 5. The simulation results of tracking: (a) Full circle trajectory (b)
A combination between cubic polynomial and straight-line trajectory
Fig. 5a shows the full circle trajectory, where the
proposed cascaded PID controller succeeded to track the
desired circular trajectory. Also, Fig. 5b shows that the
controller is able to track the combination trajectory
between cubic polynomial and a straight line which
confirm the robustness of the proposed controller to track
different trajectories.
B. Experimental Results
To evaluate the performance of the proposed robotic leg
and the controller framework, a 2-DOF robotic leg
prototype is developed, and attached to a stand with
vertical linear sliding as shown in Fig. 6a. The ranges of
the robotic leg joints are demonstrated in Fig. 6b, where
the KFE has a ±135° range of motion from fully extended,
and the HAA joint allows for a ±90° from the initial
position, and the HFE joint can provide up to ±345°. Table
II shows the characteristics of the developed 2-DOF
quadruped robotic leg.
(a) The developed test rig (b) The robotic leg joints’ ranges
Figure 6. The robotic leg test rig and the ranges of the joints
The test rig is used for performing various trajectories
tracking experiments. For the full circle trajectory, the
desired circle’s diameter used in this experiment is 200
mm, which is used as an input for the high-level controller.
Fig. 7 shows the experimental results, where Fig. 7a
shows the desired and actual angles of the HFE and KFE
joints, Fig. 7b shows the desired and actual trajectory of
the foot’s tip, Fig. 7c shows the torque of each joint, and
Fig. 7d shows the angular velocity of HFE and KFE joints.
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(a) The desired and actual angle of each joint (b) The desired and actual foot trajectory
(c) The torque of each joint (d) The angular velocity of each joint
Figure 7. Full circle trajectory results
(a) The desired and actual angle of each joint (b) The desired and actual foot trajectory
(c) The torque of each joint (d) The angular velocity of each joint
Figure 8. A combination between cubic polynomial and straight-line trajectory results
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From these results, it can be seen that the proposed
cascaded PID trajectory tracking controller succeeded to
track the desired circular trajectory with high accuracy.
TABLE II. THE ROBOTIC LEG CHARACTERISTICS.
Parameter Value
Actuator Mass
Actuator Outer Diameter
Actuator Axial length
Actuator Maximum Torque
2-DOF Leg Total Mass
Upper Link (Hip) Length
Lower Link (Shin) Length
550 g
94 mm
42 mm
17 Nm
1.6 Kg
200 mm
200 mm
Similarly, the experiment results of the generated
trajectory from the combination between a cubic
polynomial for the swing phase and a straight line for the
stance phase can be demonstrated in Fig. 8. It can be seen
that the cascaded PID trajectory tracking controller
succeeded to track the desired combined trajectory with
high accuracy. Based on these experimental results, we can
conclude that the proposed trajectory tracking controller
has high accuracy and is robust to perform different
trajectories tracking.
VI. CONCLUSIONS
In this paper, a quadruped robotic leg is introduced with
its mechanical and electric design. The robotic leg
kinematics is presented with its two types; forward and
inverse kinematics. A cascaded PID trajectory tracking
controller is developed to control the position, velocity,
and torque of each robotic leg joint to perform tracking of
the desired trajectory. The developed quadruped robotic
leg is used for implying trajectory tracking framework via
numerical simulation and experimental tests using the
proposed controller framework, which succeeded in
tracking different desired trajectories with high accuracy
such as full circle trajectory and a combination between
cubic polynomial and straight line trajectory.
CONFLICT OF INTEREST
The authors declare no conflict of interest.
AUTHOR CONTRIBUTIONS
El-Dalatony conducted the research and wrote the
original draft, Dr. Attia proposed the project idea and
analyzed the results, Prof. Ragheb revised the results and
the final draft, Prof. Sharaf revised the manuscript and
approved the final version
ACKNOWLEDGMENT
The authors wish to express their gratitude to the
Egyptian Ministry of Defense (MOD) for the financial
support extended to this research project. A special word
of thanks goes to the Egyptian Academy of Scientific
Research and Technology for support.
REFERENCES
[1] D. J. Hyun, S. Seok, J. Lee, and S. Kim, “High speed trot-running:
Implementation of a hierarchical controller using proprioceptive
impedance control on the mit cheetah,” The International Journal
of Robotics Research, vol. 33, no. 11, pp. 1417–1445, 2014.
[2] M. Hutter, C. Gehring, D. Jud, A. Lauber, C. D. Bellicoso, V.
Tsounis, J. Hwangbo, K. Bodie, P. Fankhauser, M. Bloesch et al.,
“Anymala highly mobile and dynamic quadrupedal robot,”
IEEE/RSJ International Conference on Intelligent Robots and
Systems (IROS), IEEE, 2016, pp. 38–44.
[3] H. Khan, S. Kitano, M. Frigerio, M. Camurri, V. Barasuol, R.
Featherstone, D. G. Caldwell, and C. Semini, “Development of the
lightweight hydraulic quadruped robot—minihyq,” IEEE
International Conference on Technologies for Practical Robot
Applications (TePRA). IEEE, 2015, pp. 1–6.
[4] C. Semini, N. G. Tsagarakis, E. Guglielmino, M. Focchi, F.
Cannella, and D. G. Caldwell, “Design of hyq–a hydraulically and
electrically actuated quadruped robot,” in Proc. the Institution of
Mechanical Engineers, Part I: Journal of Systems and Control
Engineering, vol. 225, no. 6, pp. 831–849, 2011.
[5] M. Raibert, K. Blankespoor, G. Nelson, and R. Playter, “Bigdog,
the rough-terrain quadruped robot,” IFAC Proceedings Volumes,
vol. 41, no. 2, pp. 10 822–10 825, 2008.
[6] H.-W. Park, S. Park, and S. Kim, “Variable-speed quadrupedal
bounding using impulse planning: Untethered high-speed 3d
running of mit cheetah 2,” IEEE International Conference on
Robotics and Automation (ICRA). IEEE, 2015, pp. 5163–5170.
[7] H. W. Park, P. M. Wensing, and S. Kim, “High-speed bounding
with the mit cheetah 2: Control design and experiments,” The
International Journal of Robotics Research, vol. 36, no. 2, pp. 167–
192, 2017.
[8] G. Xin, W. Wolfslag, H. C. Lin, C. Tiseo, and M. Mistry, “An
optimization-based locomotion controller for quadruped robots
leveraging Cartesian impedance control,” Frontiers in Robotics and
AI, vol. 7, p. 48, 2020.
[9] Y. Tassa, T. Erez, and E. Todorov, “Synthesis and stabilization of
complex behaviors through online trajectory optimization,”
IEEE/RSJ International Conference on Intelligent Robots and
Systems, IEEE, 2012, pp. 4906–4913.
[10] R. Orsolino, M. Focchi, D. G. Caldwell, and C. Semini, “A
combined limit cycle-zero moment point based approach for omni-
directional quadrupedal bounding,” in Human-Centric Robotics:
Proceedings of CLAWAR 2017: 20th International Conference on
Climbing and Walking Robots and the Support Technologies for
Mobile Machines. World Scientific, 2018, pp. 407–414.
[11] Y. Fukuoka, H. Kimura, and A. H. Cohen, “Adaptive dynamic
walking of a quadruped robot on irregular terrain based on
biological concepts,” The International Journal of Robotics
Research, vol. 22, no. 3-4, pp. 187–202, 2003.
[12] V. Barasuol, J. Buchli, C. Semini, M. Frigerio, E. R. De Pieri, and
D. G. Caldwell, “A reactive controller framework for quadrupedal
locomotion on challenging terrain,” IEEE International Conference
on Robotics and Automation. IEEE, 2013, pp. 2554–2561.
[13] Z. Ma, Y. Liang, and H. Tian, “Research on gait planning algorithm
of quadruped robot based on central pattern generator,” in Proc. 39th
Chinese Control Conference (CCC). IEEE, 2020, pp. 3948–3953.
[14] J. Di Carlo, P. M. Wensing, B. Katz, G. Bledt, and S. Kim,
“Dynamic locomotion in the mit cheetah 3 through convex model-
predictive control,” IEEE/RSJ International Conference on
Intelligent Robots and Systems (IROS). IEEE, 2018, pp. 1–9.
[15] V. Tsounis, M. Alge, J. Lee, F. Farshidian, and M. Hutter,
“Deepgait: Planning and control of quadrupedal gaits using deep
reinforcement learning,” IEEE Robotics and Automation Letters,
vol. 5, no. 2, pp. 3699–3706, 2020.
[16] D. J. Blackman, J. V. Nicholson, C. Ordonez, B. D. Miller, and J.
E. Clark, “Gait development on minitaur, a direct drive quadrupedal
robot,” Unmanned Systems Technology XVIII, vol. 9837. SPIE,
2016, pp.141–155.
[17] C. Lee, S. Kwak, J. Kwak, and S. Oh, “Generalization of series
elastic actuator configurations and dynamic behavior comparison,”
Actuators, vol. 6, no. 3. MDPI, 2017, p. 26.
[18] M. Hutter, C. Gehring, M. Bloesch, M. A. Hoepflinger, C. D. Remy,
and R. Siegwart, “Starleth: A compliant quadrupedal robot for fast,
efficient, and versatile locomotion,” Adaptive Mobile Robotics.
World Scientific, 2012, pp. 483–490.
46
International Journal of Mechanical Engineering and Robotics Research Vol. 12, No. 1, January 2023
© 2023 Int. J. Mech. Eng. Rob. Res
[19] A. T. Spr¨owitz, A. Tuleu, M. Ajallooeian, M. Vespignani, R.
M¨ockel, P. Eckert, M. D’Haene, J. Degrave, A. Nordmann, B.
Schrauwen et al.,“Oncilla robot: a versatile open-source quadruped
research robot with compliant pantograph legs,” Frontiers in
Robotics and AI, vol. 5, p. 67, 2018.
[20] P. M. Wensing, A. Wang, S. Seok, D. Otten, J. Lang, and S. Kim,
“Proprioceptive actuator design in the mit cheetah: Impact
mitigation and high-bandwidth physical interaction for dynamic
legged robots,” IEEE Transactions on Robotics, vol. 33, no. 3, pp.
509–522, 2017.
[21] “Bostondynamics website. [Onlnie]. Available:
https://www.bostondynamics.com.
[22] “Unitree website,” [Onlnie]. Available: http://www.unitree.cc/.
[23] “Ghost robotics website. [Onlnie]. Available:
https://www.ghostrobotics.io/.
[24] G. Bledt, M. J. Powell, B. Katz, J. Di Carlo, P. M. Wensing, and S.
Kim, “Mit cheetah 3: Design and control of a robust, dynamic
quadruped robot,” IEEE/RSJ International Conference on
Intelligent Robots and Systems (IROS). IEEE, 2018, pp. 2245–2252.
[25] M. W. Spong, S. A. Hutchinson, and M. Vidyasagar, “Robot
modeling and control,” IEEE Control Systems, vol. 26, no. 6, pp.
113–115, 2006.
[26] S. K. Saha, Introduction to Robotics, Tata McGraw-Hill Education,
2014.
[27] Ü, Dinçer and M. C¸ evik, “Improved trajectory planning of an
industrial parallel mechanism by a composite polynomial consisting
of bezier curves and cubic polynomials,” Mechanism and Machine
Theory, vol. 132, pp. 248–263, 2019.
Copyright © 2023 by the authors. This is an open access article
distributed under the Creative Commons Attribution License (CC BY-
NC-ND 4.0), which permits use, distribution and reproduction in any
medium, provided that the article is properly cited, the use is non-
commercial and no modifications or adaptations are made.
Ahmed M. El-Dalatony received BSc in
Mechanical Engineering from Military
Technical College (MTC), Cairo, Egypt. He
is interested in robotics, navigation,
estimation and control. He is now pursuing
a master's degree in Mechanical
Engineering. Military Technical College,
Cairo, Egypt
Dr. Tamer Attia received the B.Sc. (with
honors) and the M.Sc. degrees in
mechanical engineering, both from The
Military Technical College, Egypt in 2006
and 2013 respectively, and his Ph.D. in 2018
in mechanical engineering from Virginia
Tech, USA. Currently, he is an assistant
professor at the automotive engineering
department in MTC. His research interest
includes the design and development of
robotic systems, Bayesian robotics, vehicle dynamics, estimation and
control, navigation and motion planning.
Associate Professor Hossam Ragheb is an
Associate Professor in Automotive
Engineering Department at Military
Technical College (MTC). He received his
BSc in Mechanical Engineering from
Military Technical College in Cairo in
2000. In 2007, he achieved his MSc in Off-
road vehicle mobility from MTC. In
November 2014, he finished his PhD from
University of Ontario Institute of
Technology (UOIT), Canada. His research interests are off-road
vehicles mobility, Tire mechanics, Tire-soft and hard soils interaction,
Multi wheels military vehicles dynamics, Traction control systems,
Vehicle dynamics, Ground vehicles aerodynamics and unmanned
ground vehicles.
Professor Alhossein Mostafa Sharaf
received BSc in Mechanical Engineering
from Military Technical College (MTC),
Cairo in 1993 with grade excellent with
honors. In 1998, he achieved MSc in heavy
vehicles dynamics from MTC. In 2007, he
finished PhD from Loughborough
University, UK. From January to July,
2016, he awarded a Post-Doctoral
Fellowship from Ontario Tech University, Canada. Currently, he is a
professor of Automotive Engineering, MTC. He contributed to
several research projects including: Advanced drivetrain systems for
off-road vehicle, Vehicle driving simulators, Heavy vehicle
dynamics, Vehicle system dynamics stability and control issues,
Electric and Electric-Hybrid Vehicles, Autonomous Ground
Vehicles.
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International Journal of Mechanical Engineering and Robotics Research Vol. 12, No. 1, January 2023
© 2023 Int. J. Mech. Eng. Rob. Res
