A twowheeled machine with a handling mechanism in two different directions
 Khaled M. Goher^{1}Email author
Received: 29 May 2016
Accepted: 12 September 2016
Published: 13 October 2016
Abstract
Despite the fact that there are various configurations of selfbalanced twowheeled machines (TWMs), the workspace of such systems is restricted by their current configurations and designs. In this work, the dynamic analysis of a novel configuration of TWMs is introduced that enables handling a payload attached to the intermediate body (IB) in two mutually perpendicular directions. This configuration will enlarge the workspace of the vehicle and increase its flexibility in material handling, objects assembly and similar industrial and service robot applications. The proposed configuration gains advantages of the design of serial arms while occupying a minimum space which is unique feature of TWMs. The proposed machine has five degrees of freedoms (DOFs) that can be useful for industrial applications such as pick and place, material handling and packaging. This machine will provide an advantage over other TWMs in terms of the wider workspace and the increased flexibility in service and industrial applications. Furthermore, the proposed design will add additional challenge of controlling the system to compensate for the change of the location of the COM due to performing tasks of handling in multiple directions.
Keywords
Lagrangian formulation IP new configuration Inverted pendulum Twowheeled vehicle Payload handling PID controlBackground
Twowheeled robots are based on the idea of the inverted pendulum (IP) system. It is a wellidentified benchmark problem that provides many challenges to control design. The IP system is nonlinear, unstable, nonminimum phase and underactuated. The inverted pendulum problem is one of the most wellknown conventional problems in control theory and has been investigated extensively in the literature.
Motion control and stability analysis of a twowheeled vehicle (TWV) are presented by Ren et al. [29] where a selftuning PID control strategy, based on a reduced model, is proposed for implementing a motion control system that stabilizes the TWV and follows the desired motion commands. Chan et al. [5] explored the common methods which have been investigated and the controllers which have been used of twowheeled robots on different types of terrains. Shojaei et al. [30] proposed an adaptive robust tracking controller to cope with both parametric and nonparametric uncertainties of the system occurred due to the integrated kinematic and dynamic trajectory tracking control problem wheeled mobile robots. Deng et al. [11] designed controller based on Lyapunov function candidate and considered virtual forces information including detouring force. Guo et al. [15] design a sliding mode controller for wheeled IP. Li and Kang [23] used the technique of dynamic coupling switching control for a wheeled manipulator. Actuator faults and abnormalities in operation in a twowheeled IP system has been investigated by Tsa et al. [33].
Investigating the parametric and functional uncertainties has been also considered in the literature; Li et al. [20–22] considered the dynamic balance and motion control based on least squares support vector machine for wheeled inverted pendulums (WIP) subjected to dynamics uncertainties. Control algorithms, using Lyapunov synthesis, with the advantage of LSSVM combined with online parameters estimation strategy have been proposed. Based on this approach, the outputs of the system proved to be able to track the given bounded reference signals within a small neighbourhood of zero as well as guarantee semiglobal uniform boundedness of all the closedloop signals. An intelligent backstepping tracking control system is proposed by Chiu et al. [6, 7] for WIPs with unknown system dynamics and external disturbance. An adaptive output recurrent cerebellar model articulation (AORCMAC) is used to copy an ideal backstepping control (IBC), and a compensated controller is designed to compensate for difference between the IBC law and AORCMAC. In a further work by Chiu et al. [6, 7], a novel modelfree intelligent controller to control WIPs has been developed. An adaptive output recurrent cerebellar model articulation controller (AORCMAC) for angle and position control of the WIP without model information has been developed. Lee et al. [19] carried out a historical evolution of IP systems for several designs. Ghaffari et al. [12] used Kane’s and Lagrangian dynamic formulation methods to drive the dynamic model of a selfbalancing twowheeled robot. Ping et al. [26, 27] reviewed various methods of driving the dynamic model and control techniques used for twowheeled robots.
Cui et al. [9] designed a state feedback control for a wheeled IP, and then backsteppingbased adaptive control is designed for output tracking of the system. Brisilla and Sankaranarayanan [4] proposed a nonlinear control strategy for a mobile IP without internal switching between controllers. Chinnadurai et al. [8] used internet on a chip controller to design a twowheel robot using the principle of curvature technique. Dai et al. [10] designed a method based on friction compensation for twowheeled IP. Raffo et al. [28] designed H∞ nonlinear controller to stabilize and control twowheeled machine under the presence of exogenous disturbances. Sun and Li [32] used adaptive neural control and extreme learning machine (ELMs) to develop and implement on twowheeled human transportation system. A novel control scheme is developed based on a singlehidden layer feedforward network approximation capability of combing ELMs to capture vehicle dynamics. Yue et al. [34] investigated error databased trajectory planner and indirect adaptive fuzzy control with the application on twowheeled IP using indirect adaptive fuzzy and sliding mode control approaches, Lyapunov theory and LaSalle’s invariance theorem. Yue et al. [35] designed a composite control approach for balancing and trajectory tracking of twowheeled IP vehicle using adaptive sliding mode, fuzzybased control and adaptive mechanism.
Principle of twowheeled IP with an extended rod
Despite the abovementioned contributions in terms of developing new configurations of TWMs, the dynamic analysis of TWMs with mass balancer in two different directions has not been given too much interest in the literature. A dynamic model of this new configuration will have the potential to form the basis for new applications and exploration of many features of the system as well as the possibility to investigate the impact of various characteristics. In this current work, a novel configuration of TWMs is introduced that enables handling payload attached to the IB in two mutually perpendicular directions. This will allow extension of the workspace of the vehicle and to increase its flexibility in various applications including: material handling, objects assembly and similar industrial and service robots application. The proposed configuration, with similar concept as KOBOKER [3], gains both advantages of serial robots and TWMs that occupy a minimum space due to working on two wheels only.
A model of this new configuration will have the potential to form the basis for new applications and exploration of many features of the system as well as the possibility to investigate the impact of various characteristics. The novel configuration of the vehicle with five DOFs provides the vehicle with an ability to handle objects in two mutually perpendicular directions. This is achieved by either a dualaxes linear actuator or two different actuators that will be able to extend the intermediate body (IB) of the vehicle in two different directions. In this work, five decoupled feedback control loops have been used throughout this work. The developed control strategy, based on loops decoupling, ensures separation of the dynamics due to the high frequency range (tilt angle) from the dynamics of low frequency range (motion of the intermediate body). Various simulation exercises have been considered to test the robustness of the developed control scheme. Even with complicated scenarios of changing the COMs simultaneously in two different directions, the control strategy was able to cope well with such variations. Internal system dynamics have been considered to test the robustness on the control approach. Huang et al. [16] used on the other hand LQR and sliding mode controllers to control the velocity and braking of a twowheeled vehicle. Though the system was developed by Bae and Jung [3], no control has been considered in their work.
The rest of the paper is organized as follows: “Introduction” summarizes relevant contributions in TWMs and the associated control strategies. “System description” section describes the system with the proposed configuration, explanation of the system DOFs and detailed description of a picking and placing scenario while handling an object in a confined space. The mathematical model of the system state space is derived in “Mathematical modelling” section, and a linearized state space model is derived in “State space modelling” section. PID control scheme is designed in “Numerical simulation” section and implemented on the system model based on a set of numerical parameters. Various simulation exercises are used for the numerical validation including either sequentially or simultaneously change of COM of the vehicle in two different directions. Finally, the paper is concluded in “Conclusion” where the work contributions are highlighted and a set of recommendations are formulated for potential future work.
System description
The battery of the twowheeled platform appears in Fig. 6 on the right side of the vehicle. However, in the realized physical system, the attachment of components (battery, electronics, etc.) will be located out in a way to assure uniform distribution of masses around the centre point of interaction of x and y axes.
Advantages of using the proposed design with standard wheels over omnidirectional wheels
There are various types of wheels used in wheeled mobile robots including: standard, castor and omnidirectional wheels. The proposed design in this paper uses two standard wheels powered by two motors. The advantages of the used standard wheels include the simplicity in design and manufacturing and the relatively good reliability. The small size of used wheels (10 cm diameter) helps in providing better stability and stronger grip with the floor. This adds to the stability and rigidity of the entire system while carrying out material handling tasks. The simple manufacturing process of standard wheels assures minimum positioning errors while movement.
Omnidirectional wheels are used in mobile robots doing material handling tasks and other industrial applications, though mobile robots with omnidirectional wheels are controllable with reduced number of actuators and are highly manoeuvrable in narrow or crowded spaces. Accuracy of motion is influenced by systematic errors caused by unavoidable imperfections in the control and mechanical subsystems and nonsystematic caused by unpredictable phenomena such as wheel slippage and surface irregularities. Calibration will be needed to compensate for those errors due to the use of omnidirectional wheels. Other odometry errors, while the robot movement, may also exist due to unequal wheels diameters, joints misalignment, backlash and slippage in encoder pulses [24]. Omnidirectional vehicles are widely used in mobile robots for materials handling vehicles for logistics and wheelchairs. However, they are generally designed for the case of motion on flat, smooth terrain and are not feasible for outdoor usage [17]. Slippage is there when omnidirectional wheels are in motion and manufacturing of those wheels is an expensive and needs high accuracy. Furthermore, there is a poor efficiency because not all the wheels are rotating in the direction of movement, which causes loss from friction, and are more computationally complex because of the angle calculations of movement [25].
Description of the system DOFs
 (a)
The vehicle to start moving on two wheels while keeping a balance condition till reaching a desired location for picking the object. The dominant control efforts during this stage are the two control torque signals from the motors attached to the wheels.
 (b)
Once reaching a suitable position to pick the object, the linear actuators start to work by extending the IB up to the object position by a linear displacement, \( h_{1} \). In this case, the centre of mass (COM) of the vehicle is moving up and the wheels motors must apply the torque necessary to keep a balance condition.
 (c)
Following the extension of the IB in a vertical position, the control system orders the linear actuator to extend the endeffector to extend in a lateral direction to the location of the object. As a consequence, the COM of the entire vehicle is changing its position and it is the responsibility of wheels motors to develop the appropriate motor torque that compensate for this change in the COM position. It is assumed in this stage of the research that the joint at \( O_{1} \) is rigid and the two axes of motion for \( h_{1} \) and \( h_{2} \) are always perpendicular to each other. However, at further stage an active revolute joint should be used to ensure that the motion of \( h_{2} \), to pick/place the object, is always in a horizontal direction. This is to reduce the change in the COM and hence to reduce the control effort required. While picking the object, the vehicle is expected to be subjected to sudden disturbance due to impact with the object. This should be overcome by the control signals from the wheels motors.
 (d)
Following picking up of the object, the endeffector should undergo a reverse motion back to its original position. This motion will be accompanied by readjustment of the COM again to its original position. The linear actuator should apply the appropriate force signal during this stage with the appropriate speed that makes the entire vehicle safe against tipping over. The vehicle needs to keep balancing depending on the torque signals developed by the wheels motors.
 (e)
As the rod of the linear actuator becomes in its original position, the IB begins travelling down to the desired height to place the object in the allocated place. The closer the COM to the chassis, the higher the control effort needs to be exerted by the motor wheels [13, 14].
 (f)
Finally, the endeffector extends till a desired location to place the object. This may include manoeuvring the entire vehicle to adjust the endeffector to do the task appropriately.
 (g)
Switching mechanisms need to be designed as a main part of the control algorithms to determine the sequence of engagement of each individual actuator associated with specific tasks in the abovementioned stages.
Engagement of individual actuators against subtasks
Ser.  Subtask  DOFs associated  Right wheel motor \( \tau_{R} \)  Left wheel motor \( \tau_{L} \)  Linear actuator I \( F_{1} \)  Linear actuator II \( F_{2} \) 

a  Moving till the picking place  \( _{{\delta_{R} }} \), \( _{{\delta_{L} }} \), \( _{\theta } \)  ✓  ✓  ×  × 
b  Extension of the IB  \( _{{\delta_{R} }} \), \( _{{\delta_{L} }} \), \( _{\theta } \), \( _{{h_{1} }} \)  ✓  ✓  ✓  × 
c  Extension of the endeffector  \( _{{\delta_{R} }} \), \( _{{\delta_{L} }} \), \( _{\theta } \), \( _{{h_{2} }} \)  ✓  ✓  ×  ✓ 
d  Reverse motion of the endeffector  \( _{{\delta_{R} }} \), \( _{{\delta_{L} }} \), \( _{\theta } \), \( _{{h_{2} }} \)  ✓  ✓  ×  ✓ 
e  Contraction of the IB  \( _{{\delta_{R} }} \), \( _{{\delta_{L} }} \), \( _{\theta } \), \( _{{h_{1} }} \)  ✓  ✓  ✓  × 
f  Placing of the object  \( _{{\delta_{R} }} \), \( _{{\delta_{L} }} \), \( _{\theta } \), \( _{{h_{2} }} \)  ✓  ✓  ×  × 
As indicated in the table, the wheels motors are always engaged during the entire process as there are always a change in the location of the COM and possibility of external disturbance during the picking and/or placing of the object. For all subtasks, (a–f), the wheels motors need to develop the appropriate torque signal that is sufficient to keep the vehicle balance in an upright vertical position. The engagement of linear actuators will be as when needed during the picking, placing stages to complete both tasks. Switching mechanisms are designed to determine the period of engagement of each individual actuator.
Mathematical modelling
Parameters and description
Parameter  Description  Value  Unit 

\( m_{1} \)  Mass of the chassis  3.1  kg 
\( m_{2} \)  Mass of the linear actuators  0.6  kg 
\( m_{\rm w} \)  Mass of wheel  0.14  kg 
\( g \)  Gravitational acceleration  9.81  m/s^{2} 
\( l \)  Distance of chassis’s centre of mass for wheel axle  0.14  m 
\( R \)  Wheel radius  0.05  m 
\( J_{1} \)  Rotation inertia of  0.068  kg m^{2} 
\( J_{2} \)  Rotation inertia of  0.0093  kg m^{2} 
\( J_{\rm w} \)  Rotation inertia of a wheel  0.000175  kg m^{2} 
\( \mu_{c} \)  Coefficient of friction between chassis and wheel  0.1  Ns/m 
\( \mu_{\rm w} \)  Coefficient of friction between wheel and ground  0  Ns/m 
\( \mu_{1} \)  Coefficient of friction of vertical linear actuator  0.3  Ns/m 
\( \mu_{2} \)  Coefficient of friction of horizontal linear actuator  0.3  Ns/m 
The friction at the mating surfaces has been simplified for the chassis–wheel, wheel–ground interaction and in the linear actuator to follow coulomb frictional model. The values of the coefficients have been selected depending on the type of surfaces. The selected constant values are assumed to be valid under all working conditions of the vehicle and the actuators. This did not take into account variations in speed, path configuration, terrain profile, etc. The constant values have been used to validate the system model. However, modelling interactions between surfaces need to be investigated for various surfaces, various terrain profiles and various operation conditions of the vehicle. The work done by Silva et al. [31] will be considered in future studies as suggested modelling technique for the wheel–ground interaction through modelling of foot–ground interaction of artificial locomotion systems.
Deriving equations of motion
Modelling using Lagrange formulation

\( q_{i} \quad (i = 1,2, \ldots ,n) \) are generalized coordinates such as: \( q_{i} = \left[ {\begin{array}{*{20}c} {h_{1} } & {h_{2} } & \theta & {\delta_{\text{L}} } & {\delta_{\text{R}} } \\ \end{array} } \right] \)

\( f_{i} \) is generalized forces that contain all the given forces in the system acting along the coordinates such as: \( f_{i} = \left[ {\begin{array}{*{20}c} {F_{1} } & {F_{2} } & 0 & {\tau_{\text{L}} } & {\tau_{\text{R}} } \\ \end{array} } \right] \)

\( D \) is the dissipation function and illustrated as \( D = \tfrac{1}{2}bq_{i}^{2} \)
Equations (17–21) represent the nonlinear secondorder differential equations representing the dynamics of the system under consideration.
State space modelling
State space modelling

Right wheel displacement, \( \delta_{\text{R}} \)

Left wheel displacement, \( \delta_{\text{L}} \)

Chassis pitch angle, \( \theta \)

Vertical linear link displacement, \( h_{1} \)

Horizontal linear link displacement, \( h_{2} \)

Right wheel velocity, \( \dot{\delta }_{\text{R}} \)

Left wheel velocity, \( \dot{\delta }_{\text{L}} \)

Chassis angular velocity, \( \dot{\theta } \)

Vertical linear link velocity, \( \dot{h}_{1} \)

Horizontal linear link velocity, \( \dot{h}_{2} \)

\( \tau_{\text{R}} \) and \( \tau_{\text{R}} \) are the required torques for the right and left wheels,

\( F_{1} \) and \( F_{1} \) are the generated linear force by the linear actuator for moving the payload in a vertical and horizontal direction, respectively.
Constants A and B in Eqs. 38 and 39 are described in the "Appendix" at the end of the paper.
Numerical simulation
Openloop system response
Control scheme design
PID control without switching mechanisms
In the following simulation exercises, the developed control schemes are implemented on the system mathematical model identified in “Mathematical modelling” section. First, no switching mechanisms will be considered while running the simulation. The control algorithm and the system behaviour are tested in two various conditions, payload free motion and while considering the activation of the two linear actuators for both the horizontal and vertical motion of the payload. The same exercise is repeated after engaging switching mechanisms that are designed to determine when the linear actuators should start working.
Payload free movement (h _{1} = h _{2} = 0)
Simultaneous horizontal and vertical motion (h _{1} and h _{2} ≠ 0)
Design of switching mechanisms
Three case studies are considered where only one linear actuator is allowed to work at a time in the first two cases and then simultaneously working of the two actuators in the thirds case. Two signals are developed for both \( h_{1} \) and \( h_{2} \) using a signal builder block in MATLAB Simulink^{®}.
Payload vertical movement only
Payload horizontal movement only
Payload simultaneous horizontal and vertical movements

Checking the robustness of the developed control approach. In Fig. 19, the IB leans in the opposite direction to compensate for the change in the position of the COM due to extension of \( h_{2} \). Activating each individual actuator at a certain time tends to act as a sudden disturbance, in particular changing \( h_{2} \) to the system which already achieved a stability.

Adding switching helps the author to conclude that changing does not have significant impact on the output of the system as noticed in Figs. 17 and 19.

Adding switching mechanisms mimics real scenarios in practical applications where not all actuators work at the same time.
The decoupled feedback control is believed that it is not related to the nonsmooth trajectories in Figs. 17 and 19. However, the fluctuations are due to the actuations of the linear actuators either simultaneously or consecutively. Further smoothness of the trajectory tracking can be achieved by minimizing the flexible dynamics items of the change in the tilt angle.
Conclusions

Testing the vehicle in confined space for path tracking and picking and placing an object, this will include consideration of additional weight of an object, tracking a prespecified trajectory, picking and placing the object from a certain location, carrying it and placing in desired location.

Further investigation will include also workspace and kinematics analysis of the vehicle.

Implementation of various optimization tools including bacterial forging (BF), spiral dynamics (SD) and hybrid spiral dynamics bacterial chemotaxis (HSDBC) for better performance of the system and improved energy consumption.

Further investigation of the linear model of the system will be carried out while implementing various control approaches including fuzzy logic control (FLC).
Declarations
Acknowledgements
The author of this paper would like to thank Lincoln University in New Zealand towards supporting this research and offering the funding support for publication.
Competing interests
The author declares that he has no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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