Hindawi Publishing CorporationThe Scientific World JournalVolume 2013, Article ID 723645, 17 pageshttp://dx.doi.org/10.1155/2013/723645
Research ArticleConstruction of a WMR for Trajectory Tracking Control:Experimental Results
R. Silva-Ortigoza,1 C. Márquez-Sánchez,1 M. Marcelino-Aranda,2
M. Marciano-Melchor,1 G. Silva-Ortigoza,3 R. Bautista-Quintero,4 E. R. Ramos-Silvestre,5
J. C. Rivera-Díaz,6 and D. Muñoz-Carrillo1
1 Instituto Politecnico Nacional, CIDETEC, Area de Mecatronica, Unidad Profesional Adolfo Lopez Mateos,07700 Mexico, DF, Mexico
2 Instituto Politecnico Nacional, UPIICSA, Seccion de Estudios de Posgrado e Investigacion, 08400 Mexico, DF, Mexico3 Benemerita Universidad Autonoma de Puebla, Facultad de Ciencias Fısico Matematicas, 72001 Puebla, PUE, Mexico4 Instituto Tecnologico de Culiacan, Departamento de Metal-Mecanica, 80220 Culiacan, SIN, Mexico5 Universidad Privada del Valle, Facultad de Informatica y Electronica, Tiquipaya, CBBA, Bolivia6Centro Nacional de Actualizacion Docente, Area de Maquinas, 13420 Mexico, DF, Mexico
Correspondence should be addressed to R. Silva-Ortigoza; [email protected]
This paper reports a solution for trajectory tracking control of a differential drive wheeled mobile robot (WMR) based on a hier-archical approach. The general design and construction of the WMR are described. The hierarchical controller proposed has twocomponents: a high-level control and a low-level control. The high-level control law is based on an input-output linearizationscheme for the robot kinematic model, which provides the desired angular velocity profiles that the WMR has to track in orderto achieve the desired position (𝑥∗, 𝑦∗) and orientation (𝜑∗). Then, a low-level control law, based on a proportional integral (PI)approach, is designed to control the velocity of the WMR wheels to ensure those tracking features. Regarding the trajectories, thispaper provides the solution or the following cases: (1) time-varying parametric trajectories such as straight lines and parabolas and(2) smooth curves fitted by cubic splines which are generated by the desired data points {(𝑥∗
1, 𝑦∗
1), . . . , (𝑥
∗
𝑛, 𝑦∗
𝑛)}. A straightforward
algorithm is developed for constructing the cubic splines. Finally, this paper includes an experimental validation of the proposedtechnique by employing a DS1104 dSPACE electronic board along with MATLAB/Simulink software.
1. Introduction
In the last decades, the control of wheeled mobile robots(WMRs) has been an interesting topic for research [1]. Thedifferential robot configuration studied in this paper hasnonholonomic constraints [2]. In order to improve the auton-omy of the mobile robots, the literature in this field has gen-erally focused on solving the following problems: (1) mobilerobot positioning, (2) stabilization, (3) trajectory trackingcontrol, (4) planning the trajectories, and (5) obstacle avoid-ance.
In the robot stabilization problem, according to [3], it isknown that a nonholonomic system cannot be asymptotically
stabilized at an equilibrium point using a differentiable con-trol law, despite the system’s being completely controllable.Accordingly, the stabilization of nonholonomic systems canonly be achieved by nondifferentiable control laws [4] ortime-dependent ones [5–9].
On the other hand, the trajectory tracking task in non-holonomic systems can be performed through differentiablecontrol laws. In [10], a hierarchical control scheme based ontwo levels (high-level and low-level) is presented for the tra-jectory tracking control of a car-trailer system.The high-levelcontrol is based on a time-varying linear quadratic regulator,which provides the desired angular velocity profiles that thesystem has to track in order to achieve the desired trajectory.
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Stage 1:subsystems
Subsystem aActuators
sensorsand
Subsystem bMechanical
design
Stage 2:power system
Source circuit
Optoisolatorcircuit circuit circuit
H-bridge Encoder
Stage 3:acquisition andcontrol system
circuitInterface
PC DS1104
Figure 1: General block diagram of the WMR prototype.
Then, a low-level control is designed for controlling the trac-tion and the steering motors by using a proportional integralderivative (PID) control. Experimental results from the appli-cation of this tracking control scheme are presented. Simi-larly, in [11], the control of a differential WMR is proposedvia a variable structure control scheme, based on external andinternal control strategies. The external control is associatedto the kinematic model of the mobile robot, which is respon-sible for generating the desired angular velocity profiles forthe motors, via an input-output linearization scheme. Mean-while, the internal control is meant to track the velocitiesimposed by the external control via a PID control. In [12],via the differential flatness property of the WMR kinematicmodel, the dynamic controller design for trajectory trackingtasks was presented. The basis of flatness-based control canbe found in the work by Fliess et al. [13]. Likewise, in [14], aslidingmode control in combinationwith differential flatnesswas presented for different types ofWMRs for tracking prob-lems.These controllers were implemented through computersimulations. In [15], a control algorithm to achieve trajectorytracking of a WMR with nonholonomic constraints wasproposed by using the computed torque method and thetheory of sliding mode control. The proposed algorithm wasimplemented on a vision-based mobile robot system. In [16],an input-output linearization design was presented in whichthe actuators are controlled by a sensorless scheme. Morerecently, in [17], a formal justification was presented for useof velocity and electric current inner loops, driven by pro-portional controllers, when a trajectory tracking task is to besolved. In this work, the dynamics of the brushed DCmotorsused as actuators is taken into account to perform the stabilityanalysis.
Different approaches to the trajectory generation problemhave beenwidely studied (see, e.g., [18]) where the “path plan-ning” problem is related to generating a sequence of pointsthrough which the robot must pass (either an end-effector ofan industrial robot or a mobile trajectory); see [19]. Anotherapproach is known as “trajectory planning and optimization,”which includes a structure that integrates the dynamics withthe path planning. Multiobjective optimization can beaddressed, such as the cost efficient use of energy, the tasktime, and other similar constraints; see [20]. Although thetrajectory generation can be solved regardless of dynamicconsideration, real constraint issues make this approachimpractical for many applications. An ideal trajectory imple-mentation with smooth transitions (human-like) as well asbeing robust against external disturbances must be designedunder multiobjective contradictory goals. A well known
implementation for this problem is based on splines (see[21–23]) which can be designed in such a way that boththe velocity and acceleration transitions follow the dynamicrequirements. For instance, in [24, 25], an algorithm for pathplanning for mobile robots used in football competitions waspresented. In [26], a technique was shown for path planningbased on splines, which was used to simulate differentialmobile robots.
The contribution of this research is to present an inte-gration of both theoretical and practical knowledge in orderto close the gap between the two. The relevant and modernopen-architecture testbed described in this paper deals witha synergetic combination of an effective control approach anda state-of-the-art technology for rapid prototyping that com-bines a hard real-time control implementationwith a software(MATLAB/Simulink) widely used for these applications. Inthis context, the present paper has three aims: (1) to describethe construction of a differential driveWMR, (2) to show theimplementation of a real-time hierarchical control strategyin order to carry out a trajectory tracking control task, and(3) to present a methodology for generating theWMR trajec-tories (based on cubic splines) which are constructed fromthe desired data points {(𝑥∗
1, 𝑦∗1), . . . , (𝑥∗
𝑛, 𝑦∗𝑛)}. Furthermore,
time-varying parametric trajectories such as straight linesand parabolic curves, are also implemented.
To this end, the present paper is organized as follows.Section 2 presents the general description of the WMR con-struction. Section 3 describes the hierarchical controller lawfor the kinematic model for the trajectory tracking task.Section 4 gives an algorithm for generating smooth curvesbased on specific points given in the 𝑋-𝑌 plane via cubicsplines. Section 5 shows the real-time control implementa-tion for the WMR. Finally, some conclusions and prospectsfor future research are presented in Section 6.
2. Construction of the WMR
A mobile robot is, in general, composed of two mechanicalsubsystems: (1) actuators and sensors and (2) mechanicaldesign. The control of these subsystems requires two elec-tronic stages: (1) the power stage and (2) the acquisition andcontrol stage. The interaction of these stages is shown in theblock diagram presented in Figure 1. The electronic powerstage (stage 2) allows interaction between the electronic con-trol interface (stage 3) and the two dynamic subsystems (stage1). This interaction includes the communication system, thepower supply, and the conditioning circuits (in order to inter-connect the electronic board DS1104). The strategy used in
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the PC (using MATLAB/Simulink) keeps the whole systemunder control, taking into consideration the physical restric-tions. Likewise, the control stage comprises the controlstrategies used to integrate the functioning of each subsystem(based on mathematic models of the plant).
2.1. Stage 1: Subsystems. This part describes the WMR dy-namic subsystems a and b, which include actuators, sensors,and the mechanical structure. Subsystem a (motors and sen-sors) allows propulsion of theWMR in a specified workspaceand discrete position sensing. The subsystem b describes themechanical design.
2.1.1. Subsystem a: Actuators and Sensors. The prototypedescribed in this work uses DC motors as actuators. In orderto estimate the mechanical capacity of these actuators, it isproposed to use the maximum velocity, acceleration, andweight of the WMR system as
𝜐 = 1m/s, 𝑎 = 1m/s2, 𝑚 = 46.74 kg. (1)
Based on the mass 𝑚 and maximum acceleration 𝑎, it ispossible to calculate the required force 𝐹 for a given displace-ment as
𝐹 = 𝑚𝑎 = (46.74 kg) (1m/s2) = 46.74N. (2)
Since two motors are used, each of them has an associatedforce, 𝐹rm for the right wheel motor and 𝐹lm for the left wheelmotor. One motor provides half of the total force required,that is,
𝐹rm = 𝐹lm =𝐹
2=46.74N2
= 23.37N. (3)
Since the wheel diameter is 0.15m (𝑟 = 0.075m), the requiredtorque for each motor is given by
𝜏rm = 𝐹rm𝑟 = (23.37N) (0.075m) = 1.75Nm,
𝜏lm = 𝐹lm𝑟 = (23.37N) (0.075m) = 1.75Nm.(4)
Since 𝜐 = 𝜔w𝑟, the angular velocity is related to each wheeland it is determined by
𝜔w =𝜐
𝑟=1m/s0.075m
≈ 13.33 rad/s ≈ 127.35 rpm. (5)
Hence, the power required, 𝑃w, is calculated as follows:
𝑃w = 𝜏rm𝜔w = (1.75Nm) (13.33 rad/s) ≈ 23.37W. (6)
The energy must be transmitted to each wheel of the WMR;however, the power required is higher than the calculatedpower since there is a loss due to the gearbox coupling. Thegearbox efficiency is around 70%, so that the estimation of thepower required (𝑃
𝑑) is given by
𝑃𝑑=𝑃w0.7=23.37W0.7
= 33.39W. (7)
Figure 2: Materials used for the prototype construction.
Based on the data sheet of the motor [27] and the golden rulefor selecting an appropriatemotor, it is recommended to havebetween 1.5 and 2 times the required power 𝑃
𝑑, and thus
50.09W < 𝑃𝑑< 66.78W, (8)
so that themotors selected areGNM3150 (24V, 55W) and thegearbox chosen for each motor is G 2.6. The second goldenrule given by the manufacturer shows the importance ofmaintaining the desired velocity 𝜔
𝑑between 65% and 90% of
the no-load motor velocity (i.e., 3526 rpm), and thus
2291.9 rpm < 𝜔𝑑< 3173.4 rpm. (9)
In order to satisfy this condition, the reduction ratio shouldbe at least 20 : 1, since
𝜔𝑚= 20𝜔w = (20) (127.35 rpm) = 2547 rpm. (10)
Finally, these selected motors, including the gearboxes, arethe series GNM 3150 (24V, 55W) Engel brand and a gearboxG 2.6. The power for the motors is supplied with two YUASAbatteries 12 V @ 12Ah.
Regarding the prototype sensors, it uses two incrementaloptical encoders. These sensors allow estimating the angularvelocity of the wheels, and consequently the robot is able toperform the trajectory tracking task.Themanufacturer of thesensors is the KoreanAutonics, and themodel series is E50S8.These sensor are powered by the YUASA battery previouslypresented.The configuration of these sensors in the prototypecan be seen in Figure 9.
2.1.2. Subsystem b: Mechanical Design. Once the componentswere selected (motors, encoders, and batteries), the softwareSolidWorks is used to integrate all themechanical parts of theWMR into a virtual model. The 3D visual feature of this soft-ware allows the correct component distribution in the WMRin accordance with the design requirements. Also, the soft-ware can be used for specifying the material properties to themodel of each part of theWMR.Theparts built for this proto-type, from thematerials previously selected (see Figure 2, andVideos S1, S2 in Supplementary Material available online athttp://dx.doi.org/10.1155/2013/723645), are as follows.
(i) The WMR base is made of aluminum sheet, which isused to support all components (see Figure 3).
(ii) The wheels used to provide traction to the WMR areconstructed with an aluminum rod (see Figure 4).
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(a) Base in SolidWorks (b) Real picture of the base
Figure 3: Base design.
(a) Wheel in SolidWorks (b) Real wheels of the WMR
Figure 4: Traction wheels design.
(a) SolidWorks design (b) Coupling system
Figure 5: Join coupling system.
(iii) The axle housings are made of yellow brass: they areused for coupling between the shaft of the motor andthe wheels (see Figure 5).
(iv) Supports for the ball casters: they are two rectangularpieces made of aluminum used for assembly theWMR structure (see Figure 6).
(v) Six aluminum bars join the ball caster supports withthe base of the WMR (see Figure 7).
(vi) Ball-bearings support: the ball bearings allow rotationwithout translation of the traction wheels due to theshaft motor rotation. Supports have two functions:keeping the ball-bearing embedded into it and joiningthe base with the motors. It was made of aluminumcut to the appropriate dimensions. Amechanical lathewas used for making the box in the supports in orderto place the ball bearing (see Figure 8).
To conclude this subsection, Figure 9 shows a bottomview of the final mechanical design made in SolidWorks, aswell as pictures of three different views of the prototype.
2.2. Stage 2: Power System. Electronic design is an essentialstage for theWMRsince its correct functioning depends on it.This section describes the most important features related tothe interconnection between the data acquisition board (fromthe computer) and the conditioning signals (to the WMR).
A block diagram of stage 2 is shown in Figure 10. Thisblock makes reference to four substages, numbered from 1to 4: (source circuit, optoisolator circuit, power circuit, andencoder circuit). In substage 1, the power supply (source cir-cuit) distributes the different voltages to the general electronicsystem. Substage 2 allows electrical signal isolation betweenthe electronic board DS1104 and substage 3. Also, this part of
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(a) SolidWorks design (b) Supports and ball casters
Figure 6: Ball-caster support.
(a) SolidWorks design (b) Join bars, support, and ball casters
Figure 7: Holding bars design.
(a) SolidWorks design (b) Real holders
Figure 8: Ball-bearing supports.
the system allows a voltage to be applied across the motorin either direction. This circuit is based on the H-bridgeLMD18200. Substage 4 is used to acquire the encoders’ sig-nals, which can then be used to estimate the position in theworkspace.
2.3. Stage 3: DataAcquisition andControl. This stage containsthe last subsystem presented in the general description of theWMR (see Figure 1). As was mentioned before, the DS1104board performs the acquisition and control of theWMR.Thisboard was selected due to the integration software betweenthe MATLAB and the board firmware. The high program-ming level of Simulink (as part of MATLAB) is a practicalchoice for programming complex control strategies in agraphic environment. This stage also includes the interfacecircuit (see Figure 1).This circuit establishes the communica-tion between the acquisition board and theWMR. It containsan integrated circuit (74HC541) as a buffer that not only
allows the signal bit interchange between the WMR and theDS1104 but also activates two signals DIR and PWM motor(LEDs indicators). Lastly, Figure 11(a) shows the final WMRmechanical design and Figure 11(b) is the real WMR withthe instrumentation included (see Video S3 placed online asSupplementary Material).
3. Control for the WMR
In this section, two separate controllers are proposed, the firstfor the kinematic model of the WMR and the second for theDCmotors (WMRactuators).The integration of both controlstrategies is based on the hierarchical control as in [10, 16].
3.1. Control for theWMRvia Input-Output Linearization. TheWMR studied in this work has two back wheels (left andright) which are identical, completely parallel, inflexible, and
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(a) SolidWorks design (b) Bottom view of the prototype
(c) Isometric view of the prototype (d) Another view of the prototype
Figure 9: Views of both mechanical design and the real prototype.
Substage 1 Source circuit
motorLeft
motorRight
Substage 2 Substage 3
circuitH-bridge
Substage 4
circuitEncoderOptoisolator
circuitdSPACE
circuitH-bridge
PC
circuit circuitEncoder
DS1104
Optoisolator
Figure 10: Block diagram of the power stage.
(a) Prototype 3D view elaborated in SolidWorks (b) Real view of the WMR
Figure 11: The complete prototype.
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x
y
X
Y
P
2l
𝜔l
𝜔𝜑
𝜔r
𝜐
Figure 12: WMR diagram.
joined by a shaft. Moreover, it has front ball casters that makesure that theWMRplatformmoves in a plane. Assuming thatthe movements of theWMR are over the𝑋-𝑌 plane, and thatthere are no slippery condition, the kinematic model is givenby [12]
�� =(𝜔𝑟+ 𝜔𝑙) 𝑟
2cos𝜑,
𝑦 =(𝜔𝑟+ 𝜔𝑙) 𝑟
2sin𝜑,
�� =(𝜔𝑟− 𝜔𝑙) 𝑟
2𝑙,
(11)
where (𝑥, 𝑦) is the position measured at the midpointbetween the two back wheels, 𝜑 is the angle between the axisof symmetry of the WMR with respect to the positive𝑋 axis,and 𝜔
𝑙(𝜔𝑟) is the angular velocity of the left (right) wheel.
Similarly, 𝑟 is the wheel ratio and 2𝑙 is the gap between them(see Figure 12). In (11), the first derivative with respect to time𝑡 is denoted by a dot.
The input control is given by the angular velocity (𝜔𝑟, 𝜔𝑙)
and a strategy based on input-output linearization static feed-back.The control can be implemented for any pair of outputs:(𝑥, 𝑦), (𝑥, 𝜑), or (𝑦, 𝜑). Consequently, in each case there willbe a remaining dynamic (dynamic zero) so that, in order toensure a closed loop stable system, it requires an analysis ofits stability.
The control design is performed over the output variables(𝑥, 𝜑) and the analysis of the remaining dynamic in closedloop is studied, particularly for the state variable 𝑦. First, thekinematic equations that describe the WMR are rewritten intwo subsystems. The first is given by
(��
��) = 𝐴
1(𝜔𝑟
𝜔𝑙
) , 𝐴1= (
𝑟 cos𝜑2
𝑟 cos𝜑2
𝑟
2𝑙−𝑟
2𝑙
) . (12)
This model only includes the state variables to be controlled(𝑥, 𝜑). And, the second is given by
𝑦 =(𝜔𝑟+ 𝜔𝑙) 𝑟
2sin𝜑. (13)
It has a remaining dynamic associated to the state variable 𝑦,once the states (𝑥, 𝜑) have been controlled, so that 𝑥 → 𝑥∗and 𝜑 → 𝜑∗.
Since det(𝐴1) = −𝑟2 cos𝜑/2𝑙, a controller can be pro-
posedwhere the output is (𝑥, 𝜑) exceptwhen𝜑 = 𝑘𝜋/2, where𝑘 = ±1, ±3, ±5, . . ., (i.e., 𝐴
1must be invertible). In order to
obtain the relation between the controllers (𝜔𝑟, 𝜔𝑙) and the
output variables (𝑥, 𝜑) from (12), we have
(𝜔𝑟
𝜔𝑙
) = (
1
𝑟 cos𝜑𝑙
𝑟1
𝑟 cos𝜑−𝑙
𝑟
)(��
��) . (14)
From these input-output relations, we obtain that the controls(𝜔𝑟, 𝜔𝑙) that allow the states (𝑥, 𝜑) to tend asymptotically to
the desired trajectory (𝑥∗, 𝜑∗) can be defined as follows:
(𝜔𝑟
𝜔𝑙
) = (
1
𝑟 cos𝜑𝑙
𝑟1
𝑟 cos𝜑−𝑙
𝑟
)(𝑢𝑥
𝑢𝜑
) , (15)
where 𝑢𝑥and 𝑢
𝜑are the two auxiliary control variables, writ-
ten as
𝑢𝑥= ��∗
− 𝛼𝑥(𝑥 − 𝑥
∗
) ,
𝑢𝜑= ��∗
− 𝛼𝜑(𝜑 − 𝜑
∗
) .(16)
Hence, the tracking error dynamics in closed loop is deter-mined by the following linear differential equation system:
𝑒𝑥+ 𝛼𝑥𝑒𝑥= 0,
𝑒𝜑+ 𝛼𝜑𝑒𝜑= 0,
(17)
where 𝑒𝑥= 𝑥−𝑥∗ and 𝑒
𝜑= 𝜑−𝜑∗ denote the tracking errors of
the variables 𝑥 and 𝜑, respectively, and 𝛼𝑥and 𝛼𝜑are two pos-
itive constants. From (17) it is observed that (𝑒𝑥, 𝑒𝜑) → (0, 0)
as 𝑡 → ∞, thus (𝑥, 𝜑) → (𝑥∗, 𝜑∗), which is the final controlrequirement.
Now, regarding the remaining dynamic associated withthe state variable 𝑦, we shall analyze the behavior of this vari-able when (𝑥, 𝜑) → (𝑥∗, 𝜑∗). For this purpose, in (13), 𝜑 isreplaced by 𝜑∗ and the controls 𝜔
𝑟and 𝜔
𝑙, which are given by
(15) and (16) respectively, are substituted, obtaining
𝑦 = [��∗
− 𝛼𝑥(𝑥 − 𝑥
∗
)] tan𝜑∗. (18)
Thus, when 𝑡 → ∞,
𝑦 = ��∗ tan𝜑∗. (19)
If 𝑥∗ and 𝑦∗ satisfy
𝑦∗
= 𝑓 (𝑥∗
) , (20)
where 𝑓(𝑥∗) is a smooth function such that 𝑓(0) = 0, then(11) and (20) show that
𝜑∗
= arctan(𝑦∗
��∗) = arctan(
𝑑𝑓 (𝑥∗)
𝑑𝑥∗) , (21)
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0 1 2 3 4 5 6
8
7
6
5
4
3
2
1
0
t (s)
ur(V
),𝜛r
(rad
/s)
ur step𝜛r measured𝜛r estimated
(a) Right motor
0 1 2 3 4 5 6
8
7
6
5
4
3
2
1
0
t (s)
ul(
V),𝜛l
(rad
/s)
ul step𝜛l measured𝜛l estimated
(b) Left motor
Figure 13: Step input response 𝑢𝑟(𝑡) = 𝑢
𝑙(𝑡) = 6V.
and (19) can then be simplified as
𝑦 = ��∗ tan𝜑∗ = ��∗ (
𝑑𝑓 (𝑥∗)
𝑑𝑥∗) =𝑑𝑓 (𝑥∗)
𝑑𝑡. (22)
Now, integrating gives
𝑦 (𝑡) = 𝑓 (𝑥∗
(𝑡)) = 𝑦∗
, (23)
where the constant of integration has been chosen in such away that 𝑦(0) = 𝑓(0) = 0.
In conclusion, it has been shown that under the initialcondition (𝑥, 𝑦, 𝜑) = (0, 0, 0), the controllers (15) arrangethat the state-space variables (𝑥, 𝑦, 𝜑) tend to (𝑥∗, 𝑦∗, 𝜑∗),respectively.
3.2. PI Control for the DC Motor. In general, due to the factthat motor manufacturers do not always providing the dy-namic parameters of their products, there are different strate-gies for experimentally obtaining such parameters. This isbeyond the scope of this paper; however, there is a reductionfrom a second-order linear system (motor model: voltage toangular position) to a first-order linear system (motormodel:voltage to angular velocity).This can be obtainedwith the fol-lowing transfer function: 𝐺(𝑠) = 𝜛(𝑠)/𝑢(𝑠). For that, Laplacetransformation of the motor model is obtained, that is,
𝐺 (𝑠) =𝜛 (𝑠)
𝑢 (𝑠)=
𝑛𝑘𝑚
(𝐽𝑠 + 𝑏) (𝐿𝑠 + 𝑅) + 𝑛2𝑘𝑒𝑘𝑚
. (24)
Assuming that inductance may be neglected, 𝐿 ≈ 0, (24) sim-plifies
𝐺 (𝑠) =𝜛 (𝑠)
𝑢 (𝑠)=𝐾
𝜏𝑠 + 1, (25)
where
𝐾 =𝑛𝑘𝑚
(𝑏𝑅 + 𝑛2𝑘𝑒𝑘𝑚), 𝜏 =
𝐽𝑅
(𝑏𝑅 + 𝑛2𝑘𝑒𝑘𝑚), (26)
with 𝑢 being the armature motor voltage, 𝑘𝑒the counter-
electromotive force constant, 𝑘𝑚the torque constant, 𝑅 the
armature constant, 𝐽 the rotor and inertial load, 𝑏 the Cou-lomb friction coefficient (due to motor and load), and 𝑛represents the gearbox ratio.
In order to characterize the dynamic system (i.e., in orderto obtain the values of the parameters for the DC motor andload), it is proposed to use a step input function with anamplitude of 𝐴 in order to get coefficients 𝐾 and 𝜏 such that
𝑢 (𝑠) =𝐴
𝑠. (27)
Substituting (27) into (25) and assuming that the system is aDCmotor of which the output has the angular velocity 𝜛, weobtain the following expression:
𝜛 (𝑠) =𝐾
𝜏𝑠 + 1
𝐴
𝑠. (28)
Since the coefficients 𝐾 and 𝜏 are obtained experimentallyusing the time representation given by the inverse Laplacetransformation of (28) (see (29)), we have
𝜛 (𝑡) = 𝐾𝐴 (1 − 𝑒−𝑡/𝜏
) . (29)
Once the characterization of themotor parameters𝐾 and 𝜏 isfinished, from (25), with 𝑢(𝑡) = 6V, the experimental resultsare shown in Figure 13. The characterization of the rightmotor is
𝐾𝑟= 0.54, 𝜏
𝑟= 0.10. (30)
The Scientific World Journal 9
Meanwhile for the left motor, we have𝐾𝑙= 0.59, 𝜏
𝑙= 0.10. (31)
Based on this, the temporal representation of the transferfunction for the right and left motors is as follows:
𝑑𝜛𝑟
𝑑𝑡= −10.20𝜛
𝑟+ 5.51𝑢
𝑟,
𝑑𝜛𝑙
𝑑𝑡= −10.20𝜛
𝑙+ 5.99𝑢
𝑙.
(32)
Additionally, Figures 13(a) and 13(b) show the step inputresponse of (29) (𝜛 is labelled as the theoretical value) andthe experimentally obtained values of𝐾 and 𝜏.The validationof this characterization is carried out by comparing thetheoretical and experimental responses.
Since themotors are mechanically attached to the wheels,the angular velocities 𝜛
𝑟and 𝜛
𝑙must be tracked in order to
follow a desired velocity path 𝜛∗𝑟and 𝜛∗
𝑙defined by (15). In
order to attain this goal, a PI control is implemented for eachmotor, so that
𝑢 (𝑡) = 𝐾𝑝𝑒 (𝑡) + 𝐾
𝑖∫𝑡
0
𝑒 (𝑡) 𝑑𝑡, (33)
𝑒 (𝑡) = 𝐸 (𝑡) − 𝑅 (𝑡) = 𝜛∗
− 𝜛, (34)
where 𝑒(𝑡) is the tracking error, 𝐸(𝑡) is the desired value, 𝑅(𝑡)is the signal meant to be controlled, 𝐾
𝑝is the proportional
gain, and 𝐾𝑖is the integral gain.
DCmotors have a nonlinear response due to friction anda dead zone in which the motor requires a certain amount ofvoltage before it can rotate. However, the dead-zone varies,depending on the mechanical load. In practice, the motorselected for this application requires 1.5 V. For this reason,a friction compensator has to be implemented. A frictionmodel can use a linear gain 𝑓
𝜐(which represents the viscous
friction coefficient) and a discontinues shifting phase 𝑓𝑐
where 𝜛 is the angular velocity that has to be compensatedand 𝑉comp(𝜛) is the friction compensator term. In practice,𝑓𝜐represents the system viscous friction, and thus
𝑓𝜐= 𝑏 = 𝑛
2
𝑏𝑚+ 𝑏𝐿, (36)
where 𝑏𝑚is the viscous friction coefficient due to the rotor
and 𝑏𝐿is the viscous friction coefficient due to the load. In
general, these two coefficients values are too small to be esti-mated accurately; hence, 𝑓
𝜐can be neglected, and thus 𝑓
𝑐is
taken into account, so that,𝑉comp (𝜛) = 𝑓𝑐 sign (𝜛) . (37)
Using (33), the PI controller can be redefined in order toconsider the friction compensator 𝑉comp, so that
𝑢 = 𝐾𝑝𝑒 (𝑡) + 𝐾
𝑖∫𝑡
0
𝑒 (𝑡) 𝑑𝑡 + 𝑉comp (𝜛) . (38)
The control diagram for each motor can be seen in Figure 14.
PIcontroller Motor++
𝜛∗e(t)
𝜛
𝜛+−
u(t)
Vcomp (𝜛)
Figure 14: PI control strategy for a DC motor.
3.3. Hierarchical Control Integration. The control law de-scribed before has considered only the kinematic structure oftheWMR, Section 3.1, and the actuator system in Section 3.2,and both systems are basically the WMR. In order to presentclearly the integration of the hierarchical control, Figure 15shows a block diagram for this purpose.
Regarding the DC motor control, the nominal output 𝜛∗used in (38) is determined for the angular path required 𝜛
𝑟
or𝜛𝑙. Since theWMRhas two equal and independent motors
(left and right), the dynamic models are given by
𝑑𝜛𝑟
𝑑𝑡= −10.20𝜛
𝑟+ 5.51𝑢
𝑟,
𝑦𝑟= 𝜛𝑟,
𝑑𝜛𝑙
𝑑𝑡= −10.20𝜛
𝑙+ 5.99𝑢
𝑙,
𝑦𝑙= 𝜛𝑙,
(39)
where 𝜛𝑟and 𝜛
𝑙are the right and left angular velocity of the
motors.Thus, two controls, 𝑢
𝑟and 𝑢
𝑙, are required; according to
Section 3.2, they are given by (38), so that
𝑢𝑟= 𝐾𝑝𝑟𝑒𝑟(𝑡) + 𝐾
𝑖𝑟∫𝑡
0
𝑒𝑟(𝑡) 𝑑𝑡 + 𝑉comp (𝜛𝑟) ,
𝑢𝑙= 𝐾𝑝𝑙𝑒𝑙(𝑡) + 𝐾
𝑖𝑙∫𝑡
0
𝑒𝑙(𝑡) 𝑑𝑡 + 𝑉comp (𝜛𝑙) ,
(40)
with 𝑉comp(𝜛𝑟) and 𝑉comp(𝜛𝑙), defined by
𝑉comp (𝜛𝑟) = 𝑓𝑐 sign (𝜛𝑟) ,
𝑉comp (𝜛𝑙) = 𝑓𝑐 sign (𝜛𝑙) ,(41)
where 𝑢𝑟and 𝑢
𝑙are the control voltages for the right and left
motors, respectively,𝐾𝑝𝑟and𝐾
𝑖𝑟,𝐾𝑝𝑙and𝐾
𝑖𝑙, are the constant
gains (proportional and integral) associated to each motor.Finally, 𝑒
𝑟(𝑡) and 𝑒
𝑙(𝑡) represent the motor tracking errors
defined by
𝑒𝑟(𝑡) = 𝜛
∗
𝑟− 𝜛𝑟,
𝑒𝑙(𝑡) = 𝜛
∗
𝑙− 𝜛𝑙,
(𝜛∗
𝑟, 𝜛∗
𝑙) = (𝜔
𝑟, 𝜔𝑙) .
(42)
It means that the desired angular velocity profiles for themotors, (𝜛∗
𝑟, 𝜛∗𝑙), are determined by the (𝜔
𝑟, 𝜔𝑙).These angu-
lar velocity trajectories are obtained from (15) and (16).
10 The Scientific World Journal
Input-outputlinearization control
Right DCmotor
Left DCmotor
WMR
ur = Kprer(t) + Kir ∫t
0er(t)dt + Vcomp (𝜛r)
ur
𝜔r=𝜛∗ r Vcomp (𝜛r) = fc sign (𝜛r)
𝜛r
𝜛r
x,y,𝜑
x,y,𝜑
𝜛l
𝜛l
𝜔l=𝜛∗ l
x∗, y∗, 𝜑∗
ulul = Kplel(t) + Kil ∫
t
0el(t)dt + Vcomp (𝜛l)
Vcomp (𝜛l) = fc sign (𝜛l)
Figure 15: WMR hierarchical control block diagram.
4. Trajectory Generation via the Method ofInterpolation of Cubic Splines
As can be seen from (15) and (16), the synthesis of theseformulations requires the knowledge of the desired trajectory,i.e., (𝑥∗, 𝑦∗, 𝜑∗). This section presents the path generationproblem solved via the cubic splines interpolation approach,in order to extend the possible trajectories to be followed(compared to the parametric approach widely reported in theliterature).
4.1. Cubic Splines. Polynomial interpolation is based on asubstitution of either a mathematical function or a lookuptable for a polynomial. The bigger the number of points, thehigher the degree of the polynomial, and consequently thegreater the error between the desired trajectory and the inter-polated curve due to oscillations. An alternative method is toemploy several polynomials of lower degree, in subintervalsalong the whole trajectory. This method is the basis of thespline interpolation approach [21–23]. The method dividesthe path to be tracked into subintervals (given by the lookuptable points) and a cubic interpolation joins such subin-tervals. This method is known as polynomial segmentaryapproximation as
𝑆𝑗(𝑥) = 𝑎
𝑗+ 𝑏𝑗(𝑥 − 𝑥
𝑗) + 𝑐𝑗(𝑥 − 𝑥
𝑗)2
+ 𝑑𝑗(𝑥 − 𝑥
𝑗)3
.
(43)
For each 𝑗 = 0, 1, . . . , 𝑛−1 and according to [21], the equationused for cubic spline interpolation is given by 𝐴𝑥 = 𝑏, where
𝐴 =
[[[[
[
1 0 0 . . . 0
ℎ02 (ℎ0+ ℎ1) ℎ
1. . . 0
0 ℎ1
2 (ℎ1+ ℎ2) ℎ
20
. . . . . . . . . . . . . . .
0 . . . ℎ𝑛−2
2 (ℎ𝑛−2+ℎ𝑛−1) ℎ𝑛−1
0 . . . 0 0 1
]]]]
]
,
𝑏 =
[[[[[[[[[[
[
03
ℎ1
(𝑎2− 𝑎1) −3
ℎ0
(𝑎1− 𝑎0)
...3
ℎ𝑛−1
(𝑎𝑛− 𝑎𝑛−1) −3
ℎ𝑛−2
(𝑎𝑛−1− 𝑎𝑛−2)
0
]]]]]]]]]]
]
,
𝑥 = [𝑐0𝑐1𝑐2⋅ ⋅ ⋅ 𝑐𝑛]𝑇
,
(44)
where𝐴 is an (𝑛−1) by (𝑛−1) diagonal matrix, 𝑏 is a constantvector, and 𝑥 is an unknown vector. A numerical method isused to find 𝑐
0, 𝑐1, . . . , 𝑐
𝑛, and based on thismethod, the coeffi-
cients in (43),𝑎𝑗, 𝑏𝑗, 𝑐𝑗, and𝑑
𝑗, are calculated in order to obtain
a cubic polynomials for each segment.
4.2. Interpolation Algorithm. Based on the previous ap-proach, a path planning algorithm is presented in this section.Thismethod generates theCartesian coordinates of the points
The Scientific World Journal 11
along the path to be followed: these points are given by(𝑥∗, 𝑦∗, 𝜑∗), using (15) and (16) in order to allow (𝑥, 𝑦, 𝜑) →(𝑥∗, 𝑦∗, 𝜑∗).
Algorithm 1. In order to build the cubic interpolation func-tion (43), according to [21], theremust be defined the numberof points 𝑛 which satisfy the following condition: 𝑥
0< 𝑥1<
⋅ ⋅ ⋅ < 𝑥𝑛with 𝑆(𝑥
0) = 𝑆(𝑥
𝑛) = 0. The following steps are
suggested to generate (𝑥∗, 𝑦∗, 𝜑∗).
(1) Vector coordinate points are introduced, so that,𝑥0, 𝑥1, . . . , 𝑥
𝑛and 𝑦
0, 𝑦1, . . . , 𝑦
𝑛.
(2) Assign 𝑎0= 𝑓(𝑥
0), 𝑎1= 𝑓(𝑥
1), . . . , 𝑎
𝑛= 𝑓(𝑥
𝑛).
(3) For 𝑖 = 0 until 𝑛 − 1, set ℎ𝑖= 𝑥𝑖+1− 𝑥𝑖.
(4) For 𝑖 = 1 until 𝑛 − 1, set 𝛼𝑖= (3/ℎ
𝑖)(𝑎𝑖+1− 𝑎𝑖) −
(3/ℎ𝑖−1)(𝑎𝑖− 𝑎𝑖−1).
(5) Put 𝑙0= 1, 𝜇
0= 0, 𝑧
0= 0.
(6) For 𝑖 = 1 until 𝑛 − 1, set
𝑙𝑖= 2 (𝑥
𝑖+1− 𝑥𝑖−1) − ℎ𝑖−1𝜇𝑖−1,
𝜇𝑖=ℎ𝑖
𝑙𝑖
,
𝑧𝑖=(𝛼𝑖− ℎ𝑖−1𝑧𝑖−1)
𝑙𝑖
.
(45)
(7) Set 𝑙𝑛= 1, 𝑧
𝑛= 0, 𝑐𝑛= 0.
(8) For 𝑗 = 𝑛 − 1 until 0, set
𝑐𝑗= 𝑧𝑗− 𝜇𝑗𝑐𝑗+1,
𝑏𝑗=(𝑎𝑗+1− 𝑎𝑗)
ℎ𝑗
−ℎ𝑗(𝑐𝑗+1+ 2𝑐𝑗)
3,
𝑑𝑗=(𝑐𝑗+1− 𝑐𝑗)
3ℎ𝑗
.
(46)
(9) If 𝑥 ≤ 𝑥0, then 𝑗 = 1; otherwise, 𝑥 ≥ 𝑥
0and 𝑥 ≤ 𝑥
1
and then 𝑗 = 2; otherwise, 𝑥 ≥ 𝑥𝑛−1
and 𝑥 ≤ 𝑥𝑛and
then 𝑗 = 𝑛 − 1.(10) If 𝑥 ≤ 𝑥
𝑛, then
𝑦𝑗= 𝑎𝑗+ 𝑏𝑗(𝑥 − 𝑥
𝑗) + 𝑐𝑗(𝑥 − 𝑥
𝑗)2
+ 𝑑𝑗(𝑥 − 𝑥
𝑗)3
. (47)
(11) Based on the equation obtained in step 10, the follow-ing expression is obtained:
𝑦𝑗= 𝑏𝑗+ 2𝑐𝑗(𝑥 − 𝑥
𝑗) + 3𝑑
𝑗(𝑥 − 𝑥
𝑗)2
. (48)
(12) 𝜑∗ is obtained as 𝜑∗ = arctan( 𝑦).(13) Based on this, considering the points (1), (10),
and (12), the coordinates obtained are denoted by(𝑥∗, 𝑦∗, 𝜑∗).
This algorithm is encoded inMATLAB in order to imple-ment experiments using the WMR.
5. Experimental Results
This section presents the closed-loop simulation as well asexperimental results in order to validate the behavior of thehierarchical approach presented in Section 3, for both para-metric and generated (based on given points) curves. Thesimulations are implemented in MATLAB/Simulink, and thereal-time experiments are based on the board DS1104 usingthe same software.
According to the WMR dimensions, 𝑟 and 𝑙 have the fol-lowing values:
𝑟 = 0.075m, 𝑙 = 0.22m. (49)
Regarding the motors, whose characterization was presentedin Section 3.2, the right motor has the following parameters:
𝐾𝑟= 0.54, 𝜏
𝑟= 0.10. (50)
The left motor has the parameters
𝐾𝑙= 0.59, 𝜏
𝑙= 0.10. (51)
The controller gains (15) and (16) and the gains of the con-trollers (40), were selected like:
𝛼𝑥= 𝛼𝜑= 2, 𝐾
𝑝𝑟= 𝐾𝑝𝑙= 2, 𝐾
𝑖𝑟= 𝐾𝑖𝑙= 50.
(52)
5.1. Experimental Results Using the Parametric Curve Ap-proach. General results are used when the desired path is aparabola, so that
𝑦∗
(𝑡) = 𝑥∗2
(𝑡) . (53)
The parametrization is chosen as follows:
𝑥∗
(𝑡) = 𝐴 sin(2𝜋𝑃𝑡) . (54)
It is found that 𝜑∗(𝑡) is determined by the following equation:
𝜑∗
(𝑡) = arctan (2𝑥∗ (𝑡)) . (55)
This parametrization allows the displacement of the WMRfrom the origin to the point (𝑥, 𝑦) = (𝐴, 𝐴2), later to (𝑥, 𝑦) =(−𝐴,𝐴2) passing through the origin and finally returning tothe origin in 𝑃 seconds. The parameter values 𝐴 and 𝑃 areassociated to (54), and they are chosen to be
𝐴 = 1m, 𝑃 = 15 s. (56)
In order to simulate the system, an initial condition chosenfor (𝑥, 𝑦, 𝜑) is (0, 0, 0), assuming 𝑡 = 30 s, where 𝑡 is thetotal simulation time. The simulation results are presentedin Figure 16. Similarly, the real-time experiments requirethe same conditions used in the simulation. The data wasobtained using ControlDesk and MATLAB/Simulink alongwith the DS1104 board. These results are presented inFigure 17, and Video S4 in Supplementary Material.
12 The Scientific World Journal
1.4
1.2
1
0.8
0.6
0.4
0.2
0
−0.2
−5
−1 −0.5 0 0.5 1x (m)
(m)
1.5
1
0.5
0
−0.5
−1
0 5 10 15 20 25 30
0 5 10 15 20 25 30
t (s)
t (s)
0 5 10 15 20 25 300 5 10 15 20 25 30t (s)
0
0
5
5
10
10
15 20 25 30t (s)
t (s)
y∗ = f(x∗)
y = f(x)𝜑∗
𝜑
𝜛∗r
𝜛r
𝜛∗l
𝜛l
ulur
8
6
4
2
0
−2
−4
−6
25
20
15
10
5
0
−5
−10
−15
0.4
0.2
0
−0.2
−0.4
−0.6
(rad
)
(V)
𝜐(m
/s)
(rad
/s)
(rad
/s)
Figure 16: Simulation results.
The Scientific World Journal 13
1.4
1.2
1
0.8
0.6
0.4
0.2
0
−0.2−1 −0.5 0 0.5 1
x (m)
(m)
y∗ = f(x∗)
y = f(x)
1.5
1
0.5
0
−0.5
−1
0 5 10 15 20 25 30t (s)
𝜑∗
𝜑
(rad
)
0 5 10 15 20 25 30t (s)
𝜛∗r
𝜛r
12
10
8
6
4
2
0−2
−4
−6
−8
0 5 10 15 20 25 30t (s)
𝜛∗l
𝜛l
8
6
4
2
0
−2
−4
−6
0 5 10 15 20 25 30t (s)
ulur
15
10
5
0
−5
−10
−15
(V)
0 5 10 15 20 25 30t (s)
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
𝜐(m
/s)
(rad
/s)
(rad
/s)
Figure 17: Real-time experimental results.
14 The Scientific World Journal
0 0.5 1 1.50
0.5
1
1.5
2
2.5
3
3.5
4
0 10 20 30 40 50−1.5
−1
−0.5
0
0.5
1
1.5
2
0 10 20 30 40 50
−4
−2
0
2
4
6
8
10
12
0 10 20 30 40 50
−4
−2
0
2
4
6
8
10
12
0 10 20 30 40 50
−5
0
5
10
15
0 10 20 30 40 500
0.05
0.1
0.15
0.2
0.25
0.3
x (m)
(m)
y∗ = f(x∗)
y = f(x)
𝜑∗
𝜑
(rad
)
t (s)
𝜛∗r
𝜛r
t (s)t (s)
𝜛∗l
𝜛l
ulur
(V)
t (s)t (s)
𝜐(m
/s)
(rad
/s)(rad
/s)
Figure 18: Simulation results for a curve obtained from given points.
The Scientific World Journal 15
0 0.5 1 1.50
0.5
1
1.5
2
2.5
3
3.5
4
x (m)
(m)
y∗ = f(x∗)
y = f(x)
0 10 20 30 40 50−1.5
−1
−0.5
0
0.5
1
1.5
2
t (s)
(rad
)
𝜑∗
𝜑
0 10 20 30 40 50
−4
−2
0
2
4
6
8
10
12
𝜛∗r
𝜛r
t (s)0 10 20 30 40 50
−4
−2
0
2
4
6
8
10
12
t (s)
𝜛∗l
𝜛l
0 10 20 30 40 50−5
0
5
10
15
ulur
(V)
t (s)0 10 20 30 40 50
0
0.05
0.1
0.15
0.2
0.25
0.3
t (s)
𝜐(m
/s)
(rad
/s)
(rad
/s)
Figure 19: Experimental results for curve obtained by points.
16 The Scientific World Journal
Table 1: Points used in order to generate the desired trajectory.
𝑥∗ 𝑦∗
𝑥0= 0.0m 𝑦
0= 0.0m
𝑥1= 0.2m 𝑦
1= 0.2m
𝑥2= 0.4m 𝑦
2= 3.5m
𝑥3= 0.8m 𝑦
3= 2.0m
𝑥4= 1.2m 𝑦
4= 0.3m
𝑥5= 1.4m 𝑦
5= 1.0m
5.2. Experimental Results Using Generated Curves Given byPoints. This section presents both the experimental and thesimulation results obtained in Section 3 where the curvegenerated is obtained from coordinate points. The chosenpoints are given by Table 1. Both the control system and theparameters used in the previous subsection are employed,(49)–(52), providing the simulation results shown inFigure 18. The experimental results are shown in Figure 19.
5.3. Discussion of the Experimental Results. Similar results areobtained in both the simulation and the experimental plots.The velocity is estimated based on the position measureddirectly from the encoders attached to the motor shafts. Both𝜛𝑟and 𝜛
𝑙, follow the desired path angular velocity according
to the kinematic structure 𝜛∗𝑟and 𝜛∗
𝑙respectively. Conse-
quently, trajectory tracking task is performed successfully,(i.e., (𝑥, 𝑦, 𝜑) → (𝑥∗, 𝑦∗, 𝜑∗)). In addition, it is shown thatthe control voltages 𝑢
𝑟and 𝑢
𝑙are within the interval (−24V,
+24V), which is convenient since the nominalmotor voltagesare in the range of ±24V. Also, from the results, we find thatthe maximum linear value V is equal to 0.52m/s. Based on allthe previous results, the efficiency of the hierarchical controldesign has excellent performance.
6. Conclusions
This work shows the design and construction of a wheeledmobile robot with differential configuration architecture.In addition, a planning trajectory has been proposed andimplemented with a control law that experimentally showedsuccessful results, for two approaches: smooth parametrictrajectories and interpolated trajectories determined by givenpoints.
More specifically, this control is hierarchically distributedin two controllers: the first controller is meant to usethe kinematic model of the WMR, which is based on alinearization of the input-output plant (𝜔
𝑟, 𝜔𝑙)-(𝑥, 𝜑), that
generates a remaining dynamics for 𝑦 that is stable. Thesecond controller is a PI strategy that determines the outputcontrol signals in order to track the desired velocities 𝜔
𝑟and
𝜔𝑙such that 𝑥 → 𝑥∗, 𝑦 → 𝑦∗, and 𝜑 → 𝜑∗, with the
initial conditions 𝑥 = 0, 𝑦 = 0, and 𝜑 = 0. The maindifference between this hierarchical control and the nonlineartechniques is that these last ones have to commutate betweentwo control laws in order to keep the system stable. In thisapproach, there is a drawback to the use of parametric trajec-tories, which happens when 𝜑 = 𝑘𝜋/2 for 𝑘 = ±1, ±3, ±5, . . ..
This is a significant constraint when the trajectory to be fol-lowed is a parametric curve, since the controller, in some con-ditions, is undetermined. For this reason, the validation of theresults was carried out not only with the use of the parametriccurves approach, but alsowith an interpolationmethod basedon cubic splines, which enables following a wider range oftrajectories than is possible with parametric curves.
The second controller is designed for the model of themotors given by (39), and this control law guarantees thetrajectory tracking given for the desired trajectory establishedfor the first controller. The so-called hierarchical approachwas introduced in [10] and used in [16] and was graphicallypresented in Figure 15.
Likewise, this research found that a control based on aninput-output plant linearization approach for (𝜔
𝑟, 𝜔𝑙)-(𝑦, 𝜑)
is always possible in such a way that 𝜑 = 𝑘𝜋 for 𝑘 = 0, ±1,±2, . . ., since the matrix at these points becomes singular.
The synergetic combination of the theoretical results,implementation techniques, and construction description isthemain contribution of this research.This approach has alsoan important impact for mathematicians willing to explorethe application field aswell as practical researcherswhomightlike to knowmore about the theoretical results described in areal and effective prototype.
Finally, as possible direction for future research, thisapproach can be extended to the problem of real-time obsta-cle avoidance. Similarly, another possible direction for futureresearch is the modification of the second controller (relatedto the actuators), where the armature voltage and current canbe the feedback variables for velocity control instead of theencoders; this then will become a speed sensorless approach.
Conflict of Interests
The authors declare that the research was conducted in theabsence of any commercial, financial, or personal relation-ships that could be construed as a potential conflict of inter-ests.
Acknowledgments
The authors wish to thank the anonymous reviewer for hishelpful comments and critical review. R. Silva-Ortigoza, M.Marcelino-Aranda, and M. Marciano-Melchor acknowledgefinancial support from Secretarıa de Investigacion y Posgradodel Instituto Politecnico Nacional (SIP-IPN), SNI-Mexico,and the programs EDI and COFAA of IPN. The work of C.Marquez-Sanchez was supported by a CONACYT scholar-ship. G. Silva-Ortigoza and R. Bautista Quintero thank SNI-Mexico for financial support. AlsoM.Marcelino-Aranda andG. Silva-Ortigoza acknowledge financial support from IPNthrough the Proyectos de Investigacion en Apoyo a la Consoli-dacion de Profesores SNI del IPN.
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