=Paper=
{{Paper
|id=Vol-2843/paper17
|storemode=property
|title=Control system for a group of industrial mobile robots for moving large-sized objects (paper)
|pdfUrl=https://ceur-ws.org/Vol-2843/paper017.pdf
|volume=Vol-2843
|authors=Dmitrii Shabanov,Nikolai Pirogov,Ayaulym Kuanyshova,Valentin Kim
}}
==Control system for a group of industrial mobile robots for moving large-sized objects (paper)==
https://ceur-ws.org/Vol-2843/paper017.pdf
Control system for a group of industrial mobile robots for
moving large-sized objects*
Dmitrii Shabanov1, Nikolai Pirogov2, Ayaulym Kuanyshova1 and Valentin Kim1
1
Peter the Great St.Petersburg Polytechnic University, 29, Polytechnicheskaya st.,
St.Petersburg, 195251, Russian Federation
2
Chemnitz University of Technology, 62, Str. der Nationen, Chemnitz, 09111, Germany
dsb956@yandex.ru
Abstract. The article is devoted to the control system for a group of industrial
wheeled mobile robots (WMR) for moving large-sized objects along a trajec-
tory. The concept of a variable configuration transport cell (VCTC) consisting
of a cargo and a group of WMRs is proposed. The formed transport cell belongs
to the class of multi-wheeled vehicle. Dynamic control system which gives or-
ders to WMRs in the form of force vectors is proposed. The dynamic control
system allows to take into account additional cargo movement efficiency crite-
ria. The VCTC mathematical model is developed. The inverse kinematics prob-
lem was solved and a feedback controller was proposed to be used in trajectory
tracking controller. The research carried out in Simulink proved that the new
concept can form the basis for a system, efficiently using drives. Efficiency can
be achieved by the wheels slippage avoidance and by controlling the robots
charge level during and after motion.
Keywords: wheeled mobile robot (WMR), trajectory tracking, variable con-
figuration system, industrial transport system, multi-wheeled vehicle.
1 Introduction
This article deals with the industrial transport system based on the wheeled mobile
robots (WMRs). The use of WMRs provides high flexibility and system's fault toler-
ance. A tractor robot with a trailer is the most flexible solution nowadays. However,
such systems have the following disadvantages: cargo’s weight limit, low efficiency
at low-weight cargos transporting, and extremely low maneuverability. The imple-
mentation of “tractor-trailer robot” solution requires a difficult analysis of kinematics
[1] and the development of a specific control system [2-4].
The proposed concept of a variable configuration transport cell (VCTC) allows to
increase the transport system's flexibility and does not have the disadvantages men-
tioned above. The VCTC concept consists in moving the cargo by required number of
single-type robots along a given trajectory.
*
Copyright © 2021 for this paper by its authors. Use permitted under Creative Commons License Attribu-
tion 4.0 International (CC BY 4.0).
The VCTC concept is partly based on the retractable football field [5], self-
propelled modular transporter (SPMT) [6] (Fig. 1) and R. Stetter’s industrial ro-
bot [7].
Fig. 1. Traction modules of the "Gazprom Arena" stadium and SCHEUERLE SPMT.
The SPMT and the retractable football field are considered to be a dynamic system
due to providing load distribution between the drives/modules. The additional cargo
movement efficiency criteria as the wheel slip avoidance or the robots charge level
control are taken into account due to the dynamic approach. In such dynamic systems,
the torque generated by motors should not affect the motion trajectory. The trajectory
is set by a guide rail at the football field and by the wheels angle position at the
SPMT.
2 Materials and methods
2.1 Variable Configuration Transport Cell
The Variable Configuration Transport Cell (VCTC) consists of a table, cargo and
WMRs. Fig. 2 shows the VCTC kinematic diagram. The cargo is located on a table 1
with triple swivel casters 2. The differential drive WMRs 3 equipped with the cou-
pling mechanisms 4 and the spring mechanisms 5 to ensure the drive wheels 6 trac-
tion with floor. The coupling mechanism allows the robots to turn to the desired angu-
lar position by creating the gearmotors’ 7 velocity difference. The triple swivel casters
8 are required for robot's independent movement apart from the VCTC.
Fig. 2. Kinematic diagram of a Variable Configuration Transport Cell.
In case of a heavy-weight cargo, it can be advisable to replace the triple swivel
casters with an air cushion [8]. In case of a low-weight cargo (which can be lifted by
spring mechanisms), it is possible to replace triple swivel casters with support legs.
The VCTC configuration is defined as “variable”, because the WMRs number and
docking places are determined by a cargo mass and a trajectory’s feature. The WMRs
universality improves the industrial transport system's flexibility: same robots move
cargos of any size and weight.
The proposed system allows complex motion performing. Robots take such posi-
tions to get each wheels' instant center of rotation (ICR) converged in one point. Fig.
3 shows the examples of possible transport cell motion and ICR positions.
Fig. 3. Possible VCTC motions: linear motion (1, 2); rotation around its own center (3); rota-
tion around adjusted center (4); complex motion with ICR movement (5)
2.2 Control system structure and dynamic approach
Fig. 4 shows the hierarchy of the transport system control based on hierarchy de-
scribed in [9]. The VCTC control system solves only the Low-level control problems.
High-level control is realized by a centralized intelligent industrial system. The cen-
tralized system selects robots for moving to a table, docking with it and forming the
VCTC. Once the robot has finished the cargo's movement, it undocks and perform the
following assigned tasks.
Consequently, the problem of VCTC is trajectory tracking. i.e. the VCTCs motion
must comply with the law q*(t) = [x*(t), y*(t), φ*(t)].
Fig. 4. Hierarchy of the transport system control
There are exist several basic approaches of trajectory tracking commonly based on:
the robot's moving to a reference point or its motion at a reference velocity and direc-
tion, or the synthesis of both approaches [10-11]. If apply these approaches to the
VCTC control, each of them will fully determine the WMRs reference values accord-
ing to the law q*(t), however the traction force generating by robots will not be con-
trolled. This leads to uneven loading and wheels slipping, as well as uneven discharge
of robots. It can lead to any of the robot’s complete discharge before the motion ends.
Fig. 5 shows the VCTC control system's general structure. Proposed control system
based on the dynamic approach implies giving an order to the WMRs in the form of
the force vector Fi. The power required to generate is determined from the law q*(t)
and can be obtained by various combinations of the Fi forces. Such variability allows
to take into account additional criteria of VCTC motion efficiency.
The variability allows to use Torque Vectoring Strategy [12] and other methods of
torque distribution (for example, [13]) for VCTC energy efficiency and slippage pro-
tection. An advanced risk assessment system [14] can be used to control the robot's
charge level.
Moreover, it is important to have an optimal level of robot's charge after finishing
cargo’s movement. Robots may have an equal “middle” charge level sufficient to
return to a charging station, but insufficient to complete the next task. Instead, it is
possible to get half minimum charge level robots and half high-level charge robots to
complete a next task.
Fig. 5. VCTC control system's general structure.
The current positions and velocities of the robots and the VCTC are known via
feedback sensors. The ICR desired position qICR* fully determines the robots desired
angular position. The “Distribution” part determines combination of robots forces Fi
required to obtain TICR*. The rolling resistance forces and inertial forces are not taken
into account in the force Fi* because they could be overcome by other robots’ forces.
Inertial characteristics are taken into account in m and I determination.
Only the “VCTC position control” and “VCTC velocity control” parts are consid-
ered in the article further. The WMR internal control system and the “Distribution”
part were implemented to make experiments in Simulink.
2.3 Mathematical model
Fig. 6 shows a transport cell, which position is defined by the vector q(t) = [x(t), y(t),
φ(t)]. The variables x and y indicate the center mass C position in a global coordinate
system in the movement plane. The current position q(t) is calculated using the robot's
feedback sensors output values.
VCTC-fixed coordinate system m is centered at the C point. The angular position φ
is defined as the angle between Xm and X axes. The WMRs (2) move the cargo using
joints formed by coupling mechanisms (3) with the Fi force at the φi angle. The joints
are located at [xmi, ymi] in the m coordinate system. The equivalent mass m and
equivalent moment of inertia I are determined from the VCTC mass characteristics
and the rotors inertia.
Fig. 6. Mathematical model.
The following values are shown on Fig. 6:
V – the VCTC line velocity;
ω – the VCTC angular velocity relative to the center of mass;
Frr – the robot's equivalent wheel rolling resistance force;
qICR = [xICR, yICR] – the position of the instant center of rotation (ICR);
RICR – the distance between the VCTC center of mass and the ICR;
RICRi – the i robot’s lever arm relative to the ICR;
ωICR – VCTC rotation velocity relative to the ICR (numerically equal to ωVCTC);
IICR – the equivalent moment of inertia of the VCTC relative to the ICR.
The weight is assumed to be evenly distributed over the robots’ wheels. This as-
sumption allows to simplify the rolling resistance force calculation and at the same
time does not affect the control system design and experiments conduction in the
Simulink.
The VCTC linear motion kinetic energy can flow into rotational kinetic energy and
back during robot's rotation i.e., while the ICR change. Therefore, linear and angular
velocities are calculated from kinetic energy. The kinetic energy can be calculated as
the integral of the powers developing by the traction and rolling resistance forces.
Performing calculations relative to the ICR make possible to represent complex mo-
tion as rotational motion:
t n
Ek (t ) ( Fi (t ) RICRi (t ) ICR (t ) Frr RICRi (t ) ωICR (t ))dt (1)
0
i 1
Lever arms are better to calculate in matrix form:
cos( ( t )) sin( (t ))
RICRi (t ) xmi y mi x(t ) y (t ) qICR (t ) (2)
sin( (t )) cos( (t ))
As the motion was represented as rotation around the ICR it is possible to calculate
the VCTC's ωICR and ω angular velocities, linear velocity V and position q.
mV 2 (t ) I 2 (t ) I ICR (t )ICR
2
(t )
Ek ( t ) (3)
2 2 2
I ICR ( t ) I m R IC
2
R (t ) (4)
RICR ( t ) x ( t ) y ( t ) q ICR ( t ) (5)
0.5
2 Ek (t )
(t ) ICR (t ) (6)
I ICR (t )
V (t ) ω (t ) RICR (t ) (7)
t t t
(t ) (t )dt ; x(t ) VX (t )dt ; y(t ) VY (t )dt (8)
0 0 0
2.4 Trajectory tracking control
Inverse kinematics-based method (IK-method). At any instant of time t, there is a
value τ(t) that q*(τ(t)) is the closest point to q(t) located on the trajectory (Fig. 7). In
case the VCTC follows the trajectory precisely then τ(t)=t. The q*(τ(t)+dt) and
q*(τ(t)+2dt) are the desired VCTC positions during the dt and 2dt times accordingly.
The ICR desired position is calculated from linear and angular velocities ratio. The
velocities are calculated as the ratio of distance between q*(τ(t)+dt) and q(t) to the
time.
Fig. 7. Vectors used in the IK-method.
q * ( (t ) dt ) q (t )
*
q IK (t ) VIK* X (t ),VIK* Y (t ),IK
*
(t ) (9)
dt
V * (t ) cos( / 2) sin( / 2)
*
RICR (t ) IK (10)
IK*
(t ) sin( / 2) cos( / 2)
*
qICR (t ) x (t ) y (t ) RICR
*
(t ) (11)
The desired torque TICR*(t) can be calculated in the same way. The current value of
kinetic energy Ek(t) can be calculated from the VCTC's current velocities. It is possi-
ble to calculate the required kinetic energy Ek*(τ(t)+dt) by the difference between
states q*(τ(t)+dt) and q*(τ(t)+2dt):
q* ( (t ) 2dt ) q* ( (t ) dt )
q * ( (t ) dt )
dt (12)
VX* ( (t ) dt ),VY* ( (t ) dt ), * ( (t ) dt )
mV *2 ( (t ) dt ) I *2 ( (t ) dt )
Ek* ( (t ) dt ) (13)
2 2
E ( (t ) dt ) Ek (t )
*
E k TICR ICR t => TICR
*
(t ) k (14)
(t ) dt
Torque controller. The IK-method eliminates deviations from the trajectory but
can't eliminate the t-τ(t) error. The value t-τ(t) and the torque TICR*(t) are related by
second-order differential equation. Therefore, for the error elimination a two-circuit
controller with kinetic energy control is needed. For the system's correct work at the
low RICR(t) and at the infinitely large one, the controller’s output multiplies by
RICR(t). Fig. 8 shows the torque controller and mathematically it is described as fol-
lows:
T * (t ) PID( E * ( (t ) dt ) E (t ) PID(t (t ))) R* (t )
ICR k k
(15)
ICR
Fig. 8. Torque controller.
The way of VCTC's entering the trajectory also impacts on a t-τ(t) error. The de-
scribed controller will move the VCTC to the trajectory's nearest point. Then VCTC
will make a sharp turn and start moving along the trajectory. It is possible to enter the
trajectory smoothly by tangential as an alternative option. Fig. 11 shows a simulation
results of both ways of trajectory entering. For a smooth trajectory entering it is re-
quired to replace τ(t) with t in the IK-method.
Smooth trajectory entering may lead to a collision with another object located near
the trajectory. Moving to the nearest trajectory point can result in high accelerations
occurrence during sharp turn. Smooth trajectory entering provides a significantly less
t-τ(t) error.
Feedback control method. The IK-method can be modified to get some problems
solved. For example, the trajectory tracking control's stiffness can be reduced to in-
crease smooth motion. Moreover, modified controller can form the basis for a colli-
sion avoidance controller. It is proposed to use a feedback controller with a structure
similar to the inverse model.
The initial values qICRst*(t) and Ek st*(t) are calculated by the equations (9,10,12,13)
with replacing q(t) by q*(τ(t)). The current VCTC position error can be calculated as
the difference between the desired and current positions: qe(t) = q*(τ(t))- q(t) = [xe(t),
ye(t), φe(t)]. The linear error consists of the component qeC(t) collinear to RICRst*(t) and
the component qeP(t) perpendicular to RICRst*(t) (Fig. 9). The initial reference values
changes according to the technology presented in the in Table 1 to eliminate the er-
rors.
Table 1. Errors elimination technology.
Error Way to eliminate the error Controller
Increase or decrease of the distance
φe(t) RICR*(t)= RICRst*(t)(1-PID(φe(t)))
to the ICR
Moving the ICR along an arc by the
qeC(t) α(t)=PID(qeC(t))
angle α
qeP(t)
Change of kinetic energy by torque Ek fb*(t)= Ek st*(t)+PID(qeP(t))+PID(t - τ(t))
t - τ(t)
Fig. 9. Components of qe(t) errors and ICR rotation.
Moving the ICR along an arc by the angle α rotates the linear velocity vector at the
same angle (Fig. 9). The desired position of ICR is calculated as follows:
*
qICR (t ) x (t ) y (t ) RICR
*
(t ) (16)
cos( (t )) sin( (t ))
(t ) RICRst (t ) 1 PID(e (t ))
* *
RICR (17)
sin( (t )) cos( (t ))
3 Results
The proposed system was simulated in Simulink environment. The VCTC has the
following parameters:
─ equivalent mass m=1000 kg;
─ equivalent moment of inertia I=450 kg·m2;
─ amount of traction robots – 4 pc.;
─ one robot’s maximum traction force Fmax=130 N;
Robots’ positions are shown in Fig. 10. The trajectory was set by a Bezier curve to
exclude instant velocity changes. The curve is divided into three sections (Fig. 11): in
section I the transport cell rotates with an angular velocity π/12 rad/s; in section II the
cell's rotation smoothly decelerates; in section III the transport cell maintains an angu-
lar position of 1.5 π rad. Transport cell's initial deviation is [2 0.5] (m). Linear speed
of motion: 1m/s.
Fig. 10. Robots positions.
Fig. 11. Transport cell’s motion.
The simulation results are shown in Fig. 11 and Fig. 12. The IK-method and feed-
back controller's graphs are look similar. Therefore, the IK-method’s graphs only are
shown.
The letter A on the graphs shows a cell’s motion with a sharp trajectory entering
without t-τ(t) delay control; the letter B shows a cell’s motion with a smooth trajec-
tory entering with a t-τ(t) delay control. The ICR motion trajectory also shown on Fig.
11. The maximal deviations after entering the trajectory are shown in Table 2.
Fig. 12. qe and t - τ(t) errors.
Table 2. Maximal errors qe.
Control method xe ,m ye ,m φe ,rad
-2 -2
IK-method 2·10 1·10 1.6·10-2
Feedback controller 6·10-2 4·10-2 8·10-2
4 Discussion
The obtained results show the possibility of using the ICR position and the total
torque relative to ICR as the WMR’s reference values. The obtained error values
show the upper bound of the system accuracy. The system's accuracy level е (maxi-
mal deviations from the trajectory) on a straight trajectory with further turn along the
radius R with the velocity V can be evaluated by the ratio:
e k V / R, (18)
Where coefficient k=0.5 m·s is found from the experiment with parameters V=1m/s
and R=7m. The found error value is e=7.2·10-2m.
The robot's angular position control system has a significant impact on the accu-
racy. Therefore, it is necessary to use highly effective robot’s angular position con-
trolling methods such as reinforcement learning [15].
It is mathematically possible to convert the IK-method to the feedback controller
form. It allows to estimate the controller coefficients optimal values. However, some
of the values will be related with variables (e.g. velocities). This demonstrates the
potential efficiency of open-loop adaptive controller [16] use in solving the VCTC
motion problem.
5 Conclusion
The variable configuration transport cell (VCTC) advantages were shown in compari-
son with traditional WMR-based industrial transport systems.
To carry out the experiments in Simulink the VCTC's mathematical model was devel-
oped. The results show the dynamic approach's ability to solve the VCTC motion
problem. The dynamic approach implies giving orders to the mobile robots in the
form of desired angular positions and desired traction forces values. It allows to dis-
tribute tractive force between robots without affecting velocity and direction of mo-
tion. This distribution can provide the drives efficient use by taking into account a
number of additional criteria, such as:
─ no wheel slippage
─ providing robot’s desired charge level after the motion’s end
─ providing robot’s required charge level during the motion.
The simulation results show a high accuracy of a given trajectories following by
the proposed system.
References
1. Michalek, M.: Trailer-Maneuverability in N-Trailer Structures. IEEE Robotics and Auto-
mation Letters, 5(4), 5105-5112 (2020).
2. Ritzen, P. et al: Trailer Steering Control of a Tractor–Trailer Robot. IEEE Transactions on
Control Systems Technology, 24(4), 1240-1252 (2015).
3. Binh, N.T. et al: An Adaptive Backstepping Trajectory Tracking Control of a Tractor
Trailer Wheeled Mobile Robot. Int. J. of Control, Automation, and Systems, 17, 465–473
(2019).
4. Kassaeiyan, P. et al: Control of tractor-trailer wheeled robots considering self-collision ef-
fect and actuator saturation limitations. Mechanical Systems and Signal Processing, 127,
388–411 (2019).
5. Volkov, A.N. et al: Retractable football field for indoor stadium. Patent Russian Federa-
tion no. RU 161988 U1 (2016).
6. Lavrikov, A.A. et al: Control of Movement of a Loosely Coupled Self-Propelled Modular
Transporter. J. of Machinery Manufacture and Reliabilit, 48(1), 20-25 (2019).
7. Stetter, R., Ziemniak, P. et al: Development, Realization and Control of a Mobile Robot.
In: Obdržálek, D., Gottscheber, A. (eds.) Int. Conf. on Research and Education in Robotics
– EUROBOT 2010, 130-140, Springer, Heidelberg (2011).
8. Kochneva, O.V. et al: Mobile robotic air cushion system. In: IOP Conf. Ser.: Earth and
Environmental Science, 539, 012119 (2020).
9. Adinandra, S., Kosti´c, D. et al.: Towards a Flexible Transportation in Distribution Centers
- Low-level Motion Control Approach. In: Proceedings of the 8th Int. Conf. on Informatics
in Control, Automation and Robotics, 2, 155-160. SciTePress, Setúbal (2011).
10. Klanˇcar, G., Zdešar, A., Blažiˇc, S. et al: Wheeled Mobile Robotics. 1st edn. Butterworth-
Heinemann, Kidlington (2017).
11. Corke, P.: Robotics, Vision and Control. Fundamental Algorithms in MATLAB. 1st edn.
Springer, Heidelberg (2011).
12. Ghosh, J., Tonoli, A. et al: A Torque Vectoring Strategy for Improving the Performance of
a Rear Wheel Drive Electric Vehicle. In: 2015 IEEE Vehicle Power and Propulsion Conf.,
IEEE, Montreal (2015).
13. Park, J. et al: Torque Distribution Algorithm for an Independently Driven Electric Vehicle
Using a Fuzzy Control Method. Energies, 8(8), 8537-8561 (2015).
14. Berenz, V. et al.: Autonomous battery management for mobile robots based on risk and
gain assessment. Artificial Intelligence Review, 37(3), 217–237 (2012).
15. Dziomin, U. et al: A Multi-Agent Reinforcement Learning Approach for the Efficient
Control of Mobile Robot. In: The 7th IEEE Int. Conf. on Intelligent Data Acquisition and
Advanced Computing Systems: Technology and Applications, 867-873, IEEE, Berlin
(2013).
16. Landau, I.D. et al: Adaptive Control. Algorithms, Analysis and Applications. 2nd edn.
Springer, London (2011).