modeling and simulation of three-wheeled mobile robot …asitava/tharak-simulation.pdf · modeling...

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Modeling and Simulation of a Three‐wheeled Mobile Robot on Uneven Terrains with Two‐degree‐of‐freedom Suspension

Mechanisms

Appala Tharakeshwar1 and Ashitava Ghosal2 1Department of Mechanical Engineering, SSJ Engineering College, Hyderabad, India 2Department of Mechanical Engineering, Indian Institute of Science, Bangalore, India

2 Corresponding Author, email: [email protected]

Abstract

A wheeled mobile robot (WMR) will move on an uneven terrain without slip if its torus-shaped wheels tilt in a lateral direction. An independent two degree-of-freedom (DOF) suspension is required to maintain contact with uneven terrain and for lateral tilting. This paper deals with the modeling and simulation of a three-wheeled mobile robot with torus-shaped wheels and four novel two-DOF suspension mechanism concepts. Simulations are performed on an uneven terrain for three representative paths – a straight line, a circular and an `S’ shaped path. Simulations show that a novel concept using double four-bar mechanism performs better than the other three concepts.

Keywords: slip free motion, torus-shaped wheel, two-degree-of-freedom suspension, uneven terrain, wheeled mobile robot

1 Introduction

There is an increasing interest in designing and building robots capable of planetary exploration and moving on rough and uneven terrains. The main candidates for explorations on uneven terrains are legged and wheeled mobile robots. Although legged locomotion is known to give more flexibility, energy efficiency issues dictate the use of wheeled mobile robots (WMRs) in many cases (see, for example, Raibert (1986), Quinn et al. (2002), Siegwart and Nourbakhsh (2004), Iagnemma and Dubowsky (2004)). The most well known instances of WMRs capable of moving on uneven terrain are the series of rovers used for planetary explorations developed by NASA. These are the six-wheeled Rocky, Sojourner and Opportunity rovers with rocker-bogie suspension mechanism (Lindamannon et al., 2006). Some of the others are the five-wheeled Micro5 (Kuroda et al., 1999) with PEGASUS mechanism developed by Institute of Space and Astronautical Science (ISAS), and the four-wheeled NOMAD (Rollins et al., 1998) with transforming chassis developed by NASA. Lee and Velinsky (2009) proposed a three-wheeled mobile robot with omni-directional wheels and ball wheel drive mechanisms to traverse an uneven terrain. In this paper, we present modeling and simulation of a three-wheeled mobile robot with torus-shaped wheels capable of traversing hard uneven terrain without slipping.

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Wheel slips results in wastage of energy and localization errors and it was shown by Waldron (1995) that two wheels connected by a fixed length axle undergo lateral scrub and slipping on uneven terrains. Choi et al. (1999) proposed the concept of variable length axle (VLA) to avoid kinematic slipping. An alternative to VLA was proposed by Chakraborty and Ghosal (2004, 2005). Their concept uses torus-shaped wheels in a three-wheeled mobile robot that allows passive lateral tilting of the rear toroidal wheels. In their work, kinematic and dynamic simulation results demonstrate the capability of a three-wheeled mobile robot to traverse uneven terrain without slipping.

To implement the concept of a WMR with torus-shaped wheels capable of lateral tilting, an independent two degree-of-freedom (DOF) suspension mechanism allowing the toroidal wheel to tilt laterally and move vertically (to adjust to the uneven terrain) is required. Existing one-DOF suspension systems accommodate only wheel vertical travel and ensure that the wheel terrain contact is maintained. These suspensions using leaf springs (for heavy duty vehicles) and double wishbone, MacPherson suspension, etc. with springs, damping and mechanism combinations (for passenger cars) (Dixon, 1996) does not allow variable lateral tilt. The camber angle, equivalent to lateral tilt, provided in existing one-DOF suspension is fixed. In a recent work Tharakeshwar and Ghosal (2013), proposed a modification of a commonly used trailing arm suspension, called the split and fit trailing arm (SFTA) suspension, which enabled the torus-shaped wheel to have the required two-DOF on uneven terrains. Simulation and experimental results of a three-wheeled mobile robot, equipped with the SFTA suspension, traversing an uneven terrain and demonstrated that wheel slip and path deviation is much less with the two-DOF SFTA suspension. In the thesis by Tharakeshwar (2012) six novel concepts for two-DOF suspension mechanisms have been proposed. In this work, we present detailed modeling and simulation results of a three-wheeled mobile robot with torus-shaped wheels attached to the WMR platform with four of the most promising two-DOF suspension mechanisms. The first is a modified shock absorber with revolute joint (SARJ) added in plane perpendicular to the absorber axis. The second concept is the SFTA suspension also discussed in Tharakeshwar and Ghosal (2013). The third suspension concept uses a double four-bar mechanism and is called the D4Bar suspension. The last concept uses 3 parts and 3 links and is called the 3-3 suspension. It is shown that the D4Bar suspension is better than all the other three in terms of reducing wheel slip and path deviation on uneven terrains.

A three-wheeled mobile robot with the four two-DOF suspensions is modeled and simulated in ADAMS/View (2010). For each of the four suspensions, the WMR is made to move on an uneven surface with same motion inputs and the slip velocity of rear wheels, path followed by the centre of mass and the lateral tilt angle are obtained as functions of time. The slip velocity of the wheels with and without the suspension mechanisms is compared and it is shown that the use of two-DOF suspensions results in low wheel slip and less deviation of the WMR centre of mass

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(CM) from the desired path. Simulations show that the D4Bar suspension performs the best in terms of wheel slip.

The paper is organized as follows: In Section 2, the salient aspects of modeling of the uneven terrain, the torus-shaped wheel, the contact between the torus-shaped wheel and the uneven terrain and the three-wheeled mobile robot on uneven terrain is discussed. The modeling of the three-wheeled mobile robot on uneven terrain in ADAMS/View (2010) is also presented in Section 2. In Section 3, four novel suspension mechanisms are proposed for achieving required tilting capability in the rear wheels. In Section 4, simulation details and results obtained are presented and discussed. Finally, the conclusions are presented in Section 5.

2 Modeling of torus‐shaped wheel, uneven terrains and three‐wheeled mobile robot

The kinematics and dynamic equations for a three-wheeled mobile robot with torus-shaped wheel moving on uneven surface is presented in detail in Chakraborty and Ghosal (2004, 2005) and these are discussed in brief here for the sake of completeness. Figure 1 shows the schematic of a torus-shaped wheel in contact with an uneven surface. The parametric equation of a torus-shaped wheel is given by

x = r1 cos (U1), y = cos (V1) (r2 + r1 sin (U1)), z = sin (V1) (r2 + r1 sin (U1)) (1)

where (x,y,z) are the coordinates of an arbitrary point on the torus-shaped wheel with respect to a coordinate system {W} at the centre of the torus-shaped wheel, r1 and r2 are the two radii associated with the torus and (U1, V1) are the two independent parameters.

Figure 1 – Schematic of a torus-shaped wheel on an uneven terrain

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Likewise the uneven terrain can be represented in terms of two independent parameters (U2, V2) with respect to a fixed coordinate system {0}. Denoting the angle between the X-axis of the coordinate systems attached to the wheel and the terrain at the point of contact 0p by ψ, we get five independent parameters which describe the contact between the torus-shaped wheel and the uneven surface. From Montana (1988), the time evolution of the five independent variables can be written as

( 1, 1)T = [M1]-1([K1]+[K*])-1[(-ωy, ωx)T – [K*](Vx,Vy)T]

( 2, 2)T = [M2]-1[R ψ]([K1]+[K*])-1[(-ωy, ωx)T +[K1](Vx,Vy)T] (2)

Ψ& = ωz+ [T1][M1]( 1, 1)T + [T2][M2] ( 2, 2)T

0 = Vz

where i, i , (i = 1, 2) are the rate of change of independent parameters, ωx , ωy and ωz are the angular velocity components, Vx , Vy and Vz are the linear velocity components of the torus-shaped wheel with respect to the ground, the matrices [M1], [M2], [K1], [K2], [K*], [T1], [T2] are determined from the geometrical properties of the wheel and surface at the point of contact and [R ψ ] is a rotation matrix. There are two special cases – pure rolling and pure sliding. If ωx = ωy

= 0, then the surfaces are in pure sliding. Pure rolling and no slip occurs when the linear velocity components Vx = Vy = 0 and, for the motion of WMR over uneven terrain. We are interested in pure rolling without slip. It may be noted that Vz = 0 since the wheel cannot lose contact with terrain surface and the wheel-ground contact has instantaneously three degrees of freedom. The degrees of freedom are instantaneous due to the non-holonomic constraints associated with the no slip conditions (Vx = Vy = 0) which restrict only the velocities and not the position variables. The three degrees of freedom at the wheel-ground contact point are very different from the three degrees of freedom present in the well known spherical joint present in many parallel manipulators and mechanisms.

The surface of the torus-shaped wheels of the three-wheeled mobile robot was generated using equation (1). The uneven surface is assumed to be smooth and hard1 and in ADAMS/View (2010) a smooth 3D surface can be created by extruding a closed spline called ‘profile’ along another open spline called the ‘path’ where the spline is created using chosen discrete data points. We modeled several uneven terrains and in this work, we present representative results for an uneven terrain which has small slopes and smooth peaks (simulation results for other surfaces are available in Tharakeshwar (2012)). In the simulations presented in this work, the uneven surface has a maximum slope/grade of 1 in 4 and the maximum height of the peak is 30

1Terrains with loose soil, dirt, water etc. are not considered in this work.

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mm – this is about 1.5 times the major radius and 2.5 times the minor radius of the torus-shaped wheel used in the three-wheeled mobile robot.

In Chakraborty and Ghosal (2004, 2005), a three-wheeled mobile robot with torus-shaped wheels and moving on uneven terrain has been modeled as instantaneous hybrid-parallel manipulator schematically shown in Figure 2. The rear wheels are driven and each of the two rear wheel has a passive degree of freedom to accommodate lateral tilting. The front wheel can be steered and the roll is a passive degree of freedom. Hence, using the well-known Grubler's criterion

dof = 6 (n - j - 1) + Σfi

with total number of links n as 8, number of joints j as 9 and total number of degree of freedom, , as 15 (3 for each wheel ground contact point and 1 for each of six rotary joint), the degree

of freedom, dof, of equivalent hybrid-parallel manipulator is obtained as 3. It may be noted that the dof is instantaneous since the constraints and degrees of freedom at the wheel-ground contact point are instantaneous.

Figure 2 – A 3-DOF WMR modeled as hybrid-parallel manipulator

In our ADAMS/View (2010) model, the rear torus-shaped wheels are attached to one end of suspension mechanism whereas the other end of suspension mechanism is attached to a rigid platform. The rotations at the rear wheel, 1 and 2, about the axle and perpendicular to the plane of the wheel is provided by an electric motor. Based on terrain geometry and the motion, the rear torus-shaped wheels can also rotate about an axis perpendicular to the axle and lying along the platform. The two lateral tilt at rear wheels, 1 and 2, are passive. Figure 3 shows the line diagram and ADAMS/View model of the three-wheeled mobile robot. The main dimensions of the WMR are chosen as follows: distance between the rear wheel centers is 30 cm, the location of the centre of gravity is 5 cm from the base and 20 cm from the front wheel centre, and the two radii associated with the torus shaped wheel are 20 mm and 12.5 mm, respectively. The two rear

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wheels can have lateral tilts of 30 degrees as angles larger than 30 degrees would be difficult to test in an experimental setup. The three inputs in the ADAMS/View model correspond to the rotation at two rear wheels, 1 and 2, and the steering at front wheel, 3 (see Figure 2).

Figure 3 – Line diagram and ADAMS/View model of the wheeled mobile robot

The mass of platform, wheel and suspension mechanism used in our simulations are 4, 0.25, and 0.25 kg respectively and the aggregate mass of WMR is 5.09 kg. The mass inertia tensor components, in kg-mm2, are IXX = 5964.8, IYY = 2.1E+4, IZZ = 1.7E+4, IXY = 2186.4, IZX = -226.5 and IYZ = 28.1. It may be noted that once the geometry and material is chosen, all the mass and inertia properties are calculated by ADAMS/View directly.

As shown in Chakraborty and Ghosal (2004, 2005), the direct and inverse kinematics equations for the hybrid-parallel mechanism shown in Figure 2 can be written in terms of 21 variables -- fifteen contact variables from each of the three wheel-ground contact point (see equation (2)), three wheel rotations, two lateral tilt of the rear-wheels and the front wheel steering. In addition to the fifteen first-order ordinary differential equations (ODEs) (5 for each wheel as shown in equation (2)), there are three holonomic constraints which represent the constant distances between the three wheel centers and thus the hybrid-parallel robot has three instantaneous degrees of freedom. It is shown that the kinematic equations can be represented by a set of differential-algebraic equations (DAEs) and the direct and inverse kinematics can be solved by integration. Using the kinematic equations Chakraborty and Ghosal (2005) derive the dynamic equations of motion of the three-wheeled mobile robot on uneven terrain. The kinetic and potential energy for each of the WMR components, namely the three torus-shaped wheels, the WMR platform, the links associated with wheel rotation, lateral tilt and steering, are computed

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symbolically. Then the Lagrangian formulation is used to derive a set of 27 second-order ODEs which together with 21 non-holonomic and 3 holonomic constraints (from the kinematics equations) describe the dynamics of the three-wheeled mobile robot on uneven terrain. It may be mentioned that the dynamic model assumes a simple spring as the suspension element and a more realistic suspension with additional links, springs and dampers would lead to even more equations than developed in Chakraborty and Ghosal (2005).

In this work, we have used ADAMS/View (2010) for modeling and simulation instead of the DAEs and the large number of equations of motion and constraints discussed above. The main reasons are a) ease of modeling of all the components, including the two-DOF suspension, of the three-wheeled mobile robot in a CAD software, b) ease of importing the complete model of the three-wheeled mobile robot in ADAMS/View and ease of assigning mass and inertial properties to each of the components of the model, and finally c) ease of simulation using the various readily available integration routines. In ADAMS/View (2010) there is no need to formulate the DAEs and the software allows the user to provide arbitrary inputs, initial conditions etc. once a model is created. In fact, the sophisticated simulation features of ADAMS/View (2010) made us choose this approach instead of deriving DAEs, solving the DAEs and then analyzing simulation output results. A snapshot from a simulation in ADAMS/View of the three-wheeled mobile robot on an uneven surface is shown in Figure 4.

The ADAMS/View (2010) software provides imposition of 3D solid to solid contact constraint with an option for applying a frictional force. It uses iterative refinement to ensure that penetration between geometries is minimal at the contact point. We model the wheel-terrain contact using `contact’ tool of ADAMS/View (2010). The contact parameters like friction, rolling resistance and penetration are set as follows: the penetration is set at 0.001 mm and following the well-known automotive handbook (R Bosch Gmbh, 2007), the static and dynamic resistance at wheel-terrain contact is chosen as 0.9 and 0.8, respectively.

Figure 4 – Three-wheeled mobile robot on uneven terrain surface

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During the simulation, the first step is to ensure that there is contact between the geometries defined in contact statement. If there is no contact, there is no traction force. If contact exists, the geometry modeling system calculates the location of the individual contact points and the outward normal to the two geometries at the contact point. Adams/Solver (C++) (2010) calculates the normal and slip velocities of the contact point. From the slip velocity graphs we can ensure that wheel is in contact with the uneven surface throughout the simulation.

3 Two‐degree‐of‐freedom suspension mechanisms

As discussed earlier the rear wheels must have two degrees of freedom – one to ensure that wheel-ground contact and wheel traction is maintained and a second one for allowing wheel lateral tilt (see Figure 5). The primary role of the suspension mechanism is to permit lateral tilting of a wheel by about 30 degrees on either side in addition to the usual requirement of keeping contact with the uneven terrain at all times. In general, a suspension should also provide superior vehicle stability and meet steering and other requirements – these aspects are not in the scope of this paper.

A key step in any suspension design is to obtain parameters of the springs and dampers present in a suspension. Previous investigators treated the suspension design as an optimal control problem (Hrovat, 1993). Most of these are, however, not applicable to wheeled mobile robots on uneven terrains. We have created models of the three-wheeled mobile robot with suspension mechanisms in ADAMS/View (2010). After extensive simulation trials on flat and uneven terrains, with vertical bumps and ditches, and by examining the deflection of the centre of mass of the WMR and the wheel, we arrived at spring and damping parameters used in the simulations. The spring and damping values used for each of the four suspension mechanisms in the simulations are presented during the description of the suspension mechanisms. It may be noted that springs and dampers used in suspension also play a significant part in the stability and handling of a vehicle and these aspects are not taken into account in this work.

In this section, we describe the proposed two degree-of-freedom suspension mechanisms and the spring parameters used in these suspension mechanisms.

Figure 5 – Suspension requirement on uneven terrain

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3.1 SARJ suspension mechanism In shock absorber with revolute joint (SARJ) suspension mechanism, the WMR Platform is connected with Part1 of the mechanism through a translational joint. The joint axis is denoted as the TJ axis. A compression spring Spring1 with spring stiffness 5 N-mm-1, damping coefficient 0.65 N-s-mm-1 and pre-load of 10 N is used between these two parts. Relative motion between Platform and Part1 is translational for wheel vertical travel. Another part, Part2 of SARJ suspension mechanism, is connected to the Part1 through a revolute joint. The revolute joint axis is denoted as the RJ axis and this is perpendicular to the TJ axis. The revolute joint provides lateral tilt of the wheel. Two stoppers are used to limit the lateral tilt to 30 degrees on either side. A torsion spring, Spring2, with spring stiffness of 20 N-mm-deg-1, damping coefficient 0.65 N-mm-s-deg-1 and pre-load of 20 N-mm is used between Part1 and Part2. The Part2 will act as wheel hub for mounting the Toroidal wheel. As mentioned earlier the parameters for the springs are obtained after extensive simulations. Figure 6 shows the SARJ suspension mechanism.

Figure 6 – The SARJ suspension mechanism

3.2 SFTA suspension mechanism

The existing trailing arm suspension used in automobiles with one degree of freedom is split into two parts A and B. Part A with a depression and Part B with protrusion are connected with a fastener. A torsion spring S1 with spring stiffness 10 N-mm-deg-1, damping coefficient 0.65 N-mm-s-deg-1 and pre load of 845 N-mm is used between platform and Part A. Another torsion spring S2 with spring stiffness 25 N-mm-deg-1, damping coefficient of 0.65 N-mm-s-deg-1 and with a pre-load of 335 N-mm is used between Part A and Part B. The one end of the fastened trailing arm is connected to the wheeled mobile robot body and other end to the wheel. The total assembly will exhibit two degree of freedom, one for vertical bump at revolute joint RJ1 and other for lateral tilt of the wheel at revolute joint RJ2. The wheel is attached to the wheel hub through a revolute joint. The wheel and wheel hub exhibits one degree of freedom signifying the rolling of the wheel. Figure 7 shows the schematic of the SFTA suspension.

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Figure 7 – The SFTA suspension mechanism

3.3 D4Bar suspension mechanism

The double 4-bar suspension mechanism consists of Mount, 4-bar mechanism made by four links, L1, L2, L3 and L4 and Wheel hub. The Mount, in the shape of a bracket is fixed to robot Platform. It holds L1 through a revolute joint at the centre of the link. The link L1 is free to rotate 30 degrees on either side about its centre. The links L2 and L3 are connected to L1 and L4 and forms a 4-bar mechanism. The link L4 is fixed with Wheel hub and the Torus Wheel is free to rotate on Wheel hub because of a revolute joint between Wheel hub and Torus Wheel. In this mechanism three springs, S1, S2 and S3 are used. The spring S1 with a spring stiffness of 5 N-mm-1, damping coefficient of 0.65 N-s-mm-1 and a pre-load of 25 N is connected between L4 and Platform and it accommodates wheel vertical travel. Two similar springs, S2 and S3 with spring stiffness of 9 N-mm-1, damping coefficient of 0.65 N-s-mm-1 and a pre-load of 10 N are connected between two ends of L1 and two slots provided in the Mount. All revolute joint axes except one between wheel hub and wheel is parallel. Figure 8 shows the D4Bar suspension mechanism.

Figure 8 – The D4Bar suspension mechanism

3.4 3‐3 suspension mechanism

The 3-3 suspension mechanism is named because of the three main parts which constitutes the suspension, namely the Base, link and wheel Hub. The Base and wheel Hub is connected by the

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three links L1, L2 and L3. The Base is connected with the robot Platform through a translational joint along TJ axis at spring S2 with spring stiffness of 7.5 N-mm-1, damping coefficient of 0.65 N-s-mm-1 and a pre-load of 20N. The torus shaped wheel is connected with the wheel Hub through a revolute joint. The links L1, L2 and L3 are connected to wheel hub through revolute joints. The other end of link L1 is connected to base through a translational joint and spring S1 with spring stiffness of 5 N-mm-1, damping coefficient of 0.65 N-s-mm-1 and a pre-load of 10 N. The other ends of L2 and L3 are connected to base through a revolute joint. The total assembly will exhibit two degree of freedom one for vertical travel other for lateral tilt of the wheel. Figure 9 shows the 3-3 suspension.

Figure 9 – The 3-3 suspension mechanism

4 Simulation results and discussions

Due to the convention used in ADAMS/View, the robot is in XZ plane and gravity is in negative Y direction – this is different from the convention used in deriving equations (1) and (2). Two kinds of simulation, direct and inverse analysis, are performed. The differential direct analysis of the three-wheeled WMR robot can be stated as follows: Given the actuation rates , and , i.e. angular velocities to the rear wheels and steering to the front wheel, the motion of the WMR is obtained. In the inverse analysis, the path of the centre of mass of the WMR is specified and the wheel motions are determined. In the direct analysis, the velocity of the two rear wheels are chosen to be 3.6 km/h (1 m/sec) and the front steering is done to result in a straight line or a circular trajectory for the WMR – in the case of straight line, the steering input to the front wheel is 0 degree and for the circular trajectory a 30 degree steering input to front wheel is given. For the inverse analysis, the `motion’ generator command in ADAMS/View is used for moving the robot along a chosen `S’ shaped path with specified velocity. It may be noted that in all the three trajectories only the (X, Z) coordinates are specified and the Y coordinate is computed by ADAMS/View (2010) due to the imposed `contacts’ option. In this sense the chosen straight line, circular and `S’ shaped paths are in the X-Z plane and actual path of the centre of mass of the WMR is in 3D space. These direct analysis trajectories are labeled as SIM1 (for straight line) and SIM2 (for circular trajectory) and inverse analysis trajectory is labeled as SIM3. For all the

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suspension mechanisms, the same three reference trajectories, SIM1, SIM2 and SIM3, and the same uneven terrain is used.

The first set of simulation is performed on three wheeled mobile robot with the suspension. The second set of simulation is performed on WMR without suspension or without lateral tilting of the rear wheels. The numerical simulation results for left and right wheel slip velocity, path followed by centre of mass, lateral tilting of the rear wheels are obtained as function of time and finally curve fitting and post-processing were done in MATLAB (2010) for plotting. Keeping in mind space restrictions, we present only the key simulation results for the four suspensions. Complete details and all other simulation results are available in Tharakeshwar (2012).

4.1 SARJ suspension

As shown in Figure 6, the SARJ suspension consists of shock absorber connecting one end rigidly to platform and other end to an additional link with revolute joint. As mentioned earlier, we perform direct and inverse simulations and compare the results for models with and without suspensions and obtain the slip velocity at the wheels and the path followed by the centre of mass (CM) of the WMR platform for three chosen reference paths.

The magnitude of slip velocity at contact/track point of wheel with the uneven terrain for a straight line trajectory, SIM1, for the left and right rear wheel are plotted in Figure 10. From this figure, it is clear that slip is reduced by around 10 to 35% with a maximum reduction in slip of 61 mm/s. For the inverse problem, namely SIM3, there is considerable slip even with the suspension.

Figure 10 – Magnitude of slip velocity for SIM1

In addition to the slip free motion, the second important parameter compared is path followed by the centre of mass of the robot platform. The path followed by robot in SIM3 with SARJ suspension mechanism is shown in Figure 11. Figure 12 shows the error between the desired path and the path traced with and without suspension as a function of the X coordinate. The maximum path deviation in SIM3 is 25 mm and 102 mm with and without suspension, respectively. Other simulation results related to slip velocity, the path deviations for the two other simulations and wheel lateral tilt angle for the SARJ suspension are available in Tharakeshwar (2012).

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Figure 11 – Path followed by centre of mass in SIM3

Figure 12 – Error between desired and path followed by centre of mass in SIM3

4.2 SFTA suspension

The SFTA suspension, shown in Figure 7, is also discussed in Tharakeshwar and Ghosal (2013). The simulation results shown here are for slightly different motion inputs and are presented here for the purpose of comparison with the simulation results from other suspensions. The geometry and other parameters used are the same as in the SARJ suspension.

4.2.1 Slip velocity

The results of the simulation are presented in Figures 13, 14, and 15. Figure 13 show the magnitude of slip velocity for the straight line motion (SIM1) for left and right wheel. It can be seen that there is a reduction in slip of around 35% with a maximum reduction in slip of 130 mm/s. Figures 14 and 15 contain plots of the magnitude of the slip velocity for the circular path (SIM2) and `S’ shaped path (SIM3) simulations and again significant reduction in slip can be seen for the circular and `S’ shaped path when the suspension is used and the rear wheels are allowed to tilt laterally. In several simulations for SIM1 and SIM2, it was observed that there is

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an almost 60 to 70% reduction in slip with suspension where as in inverse analysis simulation there is a lesser 20% reduction in slip.



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4.2.2 Path of centre of mass

The path followed by the centre of mass, with and without suspension, is shown in the Figures 16 and 17. The figures also show the desired (input) straight line, circular trajectory and the `S’ shaped path. From Figures 16 and 17, it is clear that the deviation from a desired path is less when the suspension is present. The maximum path deviation in SIM3 is 50 mm and 56 mm with and without suspension, respectively.

Figure 16 – Path followed by centre of mass in SIM1 (left) and SIM2 (right)

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4.2.3 Lateral tilt angle and trailing arm angle

To study the suspension behavior, wheel lateral tilt angle and trailing arm angle variations are plotted with respect to time. The left plot in Figure 18 shows the lateral tilt angle and the right plot shows the trailing arm angle for the left and right wheel of the WMR executing SIM1. Figure 19 shows the variation of lateral tilt angle for left wheel (LW) and right wheel (RW) when the WMR is executing SIM2 and SIM3. It can be seen that the maximum tilt angle is less than 20 degrees and this meets our requirement of lateral tilt less than 30 degrees.

Figure 18 – Wheel lateral tilt angle and trailing arm angle on SIM1 (SFTA Suspension)

Figure 19 – Wheel lateral tilt angle for SIM2 (left) and SIM3 (right) for SFTA Suspension

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4.3 D4Bar suspension mechanism

The D4Bar suspension was shown in Figure 8. The motion parameters and the reference trajectories used in these simulations are same as those in the earlier suspensions.

4.3.1 Slip velocity

The simulation results for the D4Bar suspension mechanism are presented in Figures 20, 21, and 22. Figure 20 shows the magnitude of slip velocity for the left and right wheel for SIM1 and Figures 21 and 22 shows the magnitude of slip velocity for SIM2 and SIM3, respectively. The plots clearly show that the D4Bar suspension provides a reduction of around 75 to 90% in slip velocity when the direct simulation is performed and around 70 to 80% reduction in slip velocity for inverse simulation. The maximum reduction in slip is about 75 mm/s.



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4.3.2 Path of centre of mass

The path followed by mobile robot in all simulations with D4Bar suspension mechanism is shown in Figures 23 and 24. The maximum path deviation in SIM3 from the desired path (input) is 40 mm and 65 mm with and without suspension, respectively. It is observed from extensive simulations that path following accuracy increases significantly with the D4Bar suspension.

Figure 23 – Path of centre of mass in SIM1 (left) and SIM2 (right) for D4Bar suspension


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4.3.3 Lateral tilt angle

The Figures 25 and 26 show the variation of lateral tilt angles for SIM1, SIM2 and SIM3 with the D4Bar suspension. It can be seen that the maximum lateral tilt is less than ± 20 degrees.

Figure 25 — Wheel lateral tilt angle on SIM1 (left) and SIM2 (right) for D4Bar suspension

Figure 26 — Wheel lateral tilt angle on SIM3 for D4Bar suspension

4.4 The 3‐3 suspension mechanism

The 3-3 suspension mechanism was shown in Figure 9. The motion parameters used in these simulations are same as in the earlier suspensions and again direct and inverse simulations were done for the WMR with this suspension.

Figure 27 shows the magnitude of the slip velocity for SIM1 simulation. It can be seen that the reduction in slip velocity is small in the range of 5% to 10% for this suspension and the maximum reduction in slip velocity is around 40 mm/s. The other simulation results are not shown as this suspension mechanism was not found to reduce wheel slip significantly.

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The path taken by the centre of mass for SIM3 is shown in Figure 28 and it can be seen that there is significant path deviation of about 39 mm and 66 mm with and without suspension for this mechanism.


4.5 Evaluation of suspension mechanisms

There are many ways for evaluating alternatives. The main goal of this work is to develop WMRs and accompanying suspensions with least slip and path deviation on uneven terrain. Hence, the suspension with less slip and less deviation from desired path is preferred. A large requirement of lateral tilt angle makes the design harder and hence suspensions with large lateral deviation were not favored. In addition, smaller number of components and a subjective manufacturability condition was also used to select the most promising suspension mechanisms. Table 1 below gives the evaluation parameters and their values obtained from simulations discussed above. From the table, the SFTA and the D4Bar are the most promising suspension mechanisms for better reduction in slip (60-70% with SFTA, 70-90% with D4Bar). The deviation from the expected path is also less with the SFTA and the D4Bar suspension

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mechanisms. Though the deviation in path is also less with the SARJ suspension, it is more difficult to manufacture than the SFTA or the D4Bar suspension. The D4Bar is superior in terms of reducing wheel slip and appears to be the most promising two degree-of-freedom suspension.

Table 1 – Comparison of suspension mechanisms

Name of the Suspension Mechanism

% Reducti

on of slip

Maximum reduction

in slip (mm/s)

Maximum deviation from desired path in

SIM3 (mm)

Maximum lateral tilt

(deg)

No of components

Manufacturability

SARJ 10-35 61 25 30 11 Difficult

SFTA 60-70 130 50 20 10 Simple

D4Bar 70-90 75 40 30 16 Simple

3-3 10-15 40 39 30 18 Simple

5 Conclusions

This paper deals with the modeling and simulation of a three-wheeled mobile robot capable of traversing uneven terrains without slip. The rear wheels of the proposed three-wheeled mobile robot are capable of lateral tilting which gives it the ability to travel on uneven terrain without slipping. For such a mobile robot, the suspension system must have two degrees of freedom and, in this paper, four two-DOF suspension systems are proposed, modeled, and integrated with the model of the mobile robot. Simulations are performed for each of the proposed suspension systems using ADAMS/View software. It is shown that the mobile robot slip much less when the two-DOF suspension is used and wheel lateral tilt is allowed while it slip significantly more when the two-DOF suspensions is not used and wheel lateral tilt is not allowed. The simulations clearly demonstrate that the wheel lateral tilting indeed leads to reduced slip on uneven terrains. Based on the amount of slip, the deviation of the mobile robot from the desired path and ease of manufacturability, a modified trailing arm suspension mechanism (SFTA) and a suspension mechanism based on four-bar mechanisms (D4Bar) are found to be most promising. With respect to reduction in slip, the D4Bar performs better than all the other three suspension mechanisms.

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