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Research Papers

# Prediction of Wheel Slipping Limits for Mobile RobotsPUBLIC ACCESS

[+] Author and Article Information
Alan Whitman, Garrett Clayton

Center for Nonlinear Dynamics and Control,
Villanova University,
Villanova, PA 19085

Hashem Ashrafiuon

Center for Nonlinear Dynamics and Control,
Villanova University,
Villanova, PA 19085
e-mail: hashem.ashrafiuon@villanova.edu

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT,AND CONTROL. Manuscript received June 17, 2018; final manuscript received October 4, 2018; published online November 22, 2018. Assoc. Editor: Richard Bearee.

J. Dyn. Sys., Meas., Control 141(4), 041002 (Nov 22, 2018) (9 pages) Paper No: DS-18-1289; doi: 10.1115/1.4041664 History: Received June 17, 2018; Revised October 04, 2018

## Abstract

In this paper, we present a method to accurately predict the wheel speed limits at which mobile robots can operate without significant slipping. The method is based on an asymptotic solution of the nonlinear equations of motion. Using this approach, we can predict wheel slipping limits of both the inside and outside wheel when the robot is in a constant circular motion of any radius. The analytical results are supported by experiments, which show that the inside wheel slipping limits for circular motions of various radii occur very close to the predicted values. The method is then applied to predict wheel speed profiles for general motion without slipping and experimentally verified for a sinusoidal path.

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## Introduction

Vehicle models are generally considered underactuated nonholonomic systems since there are fewer actuators than the degrees-of-freedom (DOFs). As a result, these nonlinear systems require special care when developing a closed-loop controller. For instance, in the case of wheeled mobile robots, the closed-loop control is normally developed based on the assumption of rolling without slipping [19]. However, all these closed-loop control methods rely on the availability of absolute position and orientation feedback. Such feedback may be provided using cameras in small indoor experiments [6] or global positioning system (GPS) outdoors.

Absolute position data may not be directly available due to for example loss of GPS signal. Hence, onboard sensors such as inertial measurement unit, vision-based methods such as simultaneous localization and mapping [10,11], and cooperative localization methods [12] have been extensively researched. However, vision-based methods and cooperative localization require additional information that may not be available and in the case of inertial measurement unit, drift can accumulate to the point where localization becomes unacceptable. Thus, it may be of interest to develop open-loop methods that will enable accurate and fast motion control in the short term. In addition, slipping may also degrade the performance of closed-loop control algorithms. As a result, researchers have addressed detecting and modeling approaches [1316].

In a previous work [17], we presented an asymptotic solution for trajectory planning and open-loop control of mobile robots. The asymptotic solution was used to predict accurate stable solutions for nonlinear dynamic models of various forms of vehicles [17,18]. In this work, we use an asymptotic solution of the nonlinear equations of motion to derive an analytical estimate of the wheel speed limits to avoid sliding, as a function of the radius of any curved path. The significance of the solution is that it allows the mobile robot to follow a given path at the maximum possible speed without sliding. For practical purposes, the method depends on an estimation of the friction coefficient. However, it is possible to employ slip analysis and detection methods [19,20] to estimate friction even in the absence of GPS [13]. Note that unlike the proposed method, these slip detection methods may not be useful for short-term accurate trajectory tracking.

Experiments were performed with our in-house differential drive mobile robots on the floor of an indoor laboratory. In the experiments, the robot wheels were used with and without tires. Removal of the tires allowed for slipping at lower speeds and verified the accuracy of the method under different friction coefficients. The experiments showed that the theoretically predicted wheel speeds predicted accurately achieve the desired circular paths. More significantly, the experimental results verified that the method can accurately predict when slipping occurs for robot circular motion of various radii. Finally, the method was applied to a sinusoidal path which has a continuously changing curvature with concave and convex regions. Experimental results showed the mobile robot relatively accurately follow such a path.

## Robot Planar Model

Consider a differential drive robot on a horizontal plane with two rear driving wheels and a passive front spherical ball bearing, as shown in Fig. 1. We assign three reference frames to describe the kinematics of the robot: An inertial reference frame, XY, a body-fixed reference frame $eu−ev$ attached to the center of mass (CM) located at G, and another body-fixed reference frame $eud−evd$ attached to the spherical ball. A comprehensive rigid body model of the robot has 6DOFs. There are the two coordinates of the CM with respect to the XY frame, $x¯,y¯$, the robot yaw angle measured from the X-axis to $eu$, θ, and three angles of the spherical ball. These three angles consist of a rotation about the vertical axis, θd, which measures a deviation of the ball from the robot axis $eu$, a rotation about a rotated horizontal axis, ψr, which measures the rolling of the ball, and a rotation about the rotated vertical axis, ψs, which measures the spin of the ball. These ψr and ψs angles are not shown in the figure. However, if the mass of the ball is negligible in comparison with the masses of the wheels and the frame, $m¯ball≪m¯wheel, the ball's degrees-of-freedom are inconsequential for the robot dynamics. Note that the bar symbol “ ¯ ” is used to denote dimensional quantities.

There are also the two wheels, which can independently rotate about the wheel axis shown parallel to $ev$ at a distance $d¯$ behind the CM. The counterclockwise angular position of the left and right driving wheels is denoted by $ψ±$ where + stands for the left and – for the right wheel. For differential drive mobile robots, these are the two inputs, which are normally maintained using motor controllers running at a significantly higher rate than the robot position tracking controller.

The dynamic equations governing the time evolution of the DOFs are obtained by regarding the robot as consisting of three rigid bodies that are constrained to move relative to each other by frictionless connections; these are the frame and left and right rotating assemblies of the wheels. In this way, we obtain 16 equations that govern 16 unknowns: the 3DOFs and 13 constraint force and torque components. These do not comprise the complete set of 18 equations that could be written; the moment equation for each wheelset in the axle direction involves an external torque that is required to drive the wheel at the specified angular speed and is not of interest in this investigation since we consider the two wheel speeds as the inputs to the system. In writing these equations as a deterministic set, we assume that the contact force between a wheel and the plane is a function of the slip velocity. The driving wheel slip velocities, $v¯±slip$, can be derived from kinematics as Display Formula

(1)$v¯±slip=(v¯u∓h¯θ′+R¯wψ′±) eu+(v¯v−d¯θ′) ev$

where $R¯w$ is the radius of a driving wheel and prime denotes a derivative with respect to dimensional time. The components of the velocity of the frame mass center, $v¯u$ and $v¯v$, are given in terms of velocities in the inertial frame $x¯′$ and $y¯′$ as Display Formula

(2)$v¯u=x¯′ cos θ+y¯′ sin θv¯v=−x¯′ sin θ+y¯′ cos θ$

A simple way to specify the horizontal force components acting at the contact points is by linear functions of the lateral and longitudinal slip velocities Display Formula

(3)$F¯u±=−f¯v¯±slip·eu=−f¯(v¯u∓h¯ω¯−v¯±)F¯v±=−f¯v¯±slip·ev=−f¯(v¯v−d¯ω¯)$

The magnitude of these tractive forces is bounded by $μN¯±$ where $N¯±$ are the normal forces acting on the left and right wheels and μ is a friction coefficient. This limitation in mathematical form is

$|F¯|±≤μN¯±$

or on substituting Eqs. (1) and (3), we find a condition for the slipping of each wheel Display Formula

(4)$(v¯u∓h¯θ′−v¯±)2+(v¯v−d¯θ′)2≤μN¯±/f¯$

The quantity $f¯$ is a measure of the initial slope of the force versus slip velocity curves and we are approximating these curves by two straight lines.

###### Robot Frame Equations of Motion.

The complete set of six robot frame equations of motion are Display Formula

(5)$m¯fa¯G/O·eu=−B¯u+−B¯u−m¯fa¯G/O·ev=−B¯v+−B¯v−0=B¯k++B¯k−+B¯F−m¯fg¯0=(B¯k++B¯k−)d¯−B¯Fl¯0=(B¯k+−B¯k−)h¯B−T¯u+−T¯u−I¯kfω¯′=(B¯u+−B¯u−)h¯B+(B¯v++B¯v−)d¯+T¯k++T¯k−$

where $g¯$ is acceleration due to gravity. The new quantities appearing in Eq. (5) are four horizontal constraint force components, $B¯u±,B¯v±$, three vertical constraint force components $B¯k±, B¯F$, two vertical torque constraint components, $T¯k±$, two horizontal torque components for the $eu$ direction, $T¯u±$, the frame mass and vertical moment of inertia about its centroid, $m¯f,I¯kf=m¯fk¯kf2$, the distance of the outer bearing from the robot axis, $h¯B$ (the location of the rotating mass constraint forces), the frame angular speed Display Formula

(6)$ω¯=θ′$

and the inertial acceleration of the frame mass center Display Formula

(7)$a¯G/O=(v¯u′−v¯vω¯)eu+(v¯v′+v¯uω¯)ev$

###### Rotating Wheel-Axle Equations of Motion.

The ten equations of motion for the two wheel-axle assemblies (five for each wheel-axle) are Display Formula

(8)$m¯w(a¯G/O·eu+d¯ω¯2∓h¯Gω¯′)=B¯u±+F¯u±m¯w(a¯G/O·ev−d¯ω¯′∓h¯Gω¯2)=B¯v±+F¯v±0=−B¯k±+N¯±−m¯wg¯I¯vwψ′±ω¯=∓B¯k±(h¯B−h¯G)±N¯±(h¯−h¯G) +F¯v±R¯w+T¯u±I¯kwω¯′=∓B¯u±(h¯B−h¯G)∓F¯u±(h¯−h¯G)−T¯k±$

Here, $m¯w,I¯kw=m¯wk¯kw2,I¯vw=m¯wk¯vw2$ are the mass, and transverse and axial principal, centroidal, moments of inertia of each driving wheel plus attached axle and motor armature (rotating mass, $k¯kw2=R¯w2/4$ and $k¯vw2=2k¯kw2$ for a thin disk and massless axle armature), $h¯G$ is the distance of the centroid of each rotating mass from the robot axis, and $F¯u±,F¯v±$ are the $eu,ev$ components of the slip force at the left and right wheels.

###### Robot Equations of Motion.

As described at the outset, Eqs. (5) and (8) comprise 16 equations for 16 unknowns. In order to obtain the equations of motion of the system, we must eliminate 11 constraint forces/torques to obtain three equations that specify the 3DOFs of the problem and, in addition, determine the two normal forces $N¯±$ so that we can evaluate the slip force.

We do this by adding the first of Eq. (5) and the sum (left wheel plus right wheel) of the first of Eq. (8). This gives Display Formula

(9)$m¯a¯G/O·eu+2m¯wd¯ω¯2=F¯u++F¯u−$

where $m¯=m¯f+2m¯w$. Next, we do the same thing for the second of Eq. (5) and the sum (left wheel plus right wheel) of the second of Eq. (8) resulting in Display Formula

(10)$m¯a¯G/O·ev−2m¯wd¯ω¯′=F¯v++F¯v−$

Finally, we take the second of Eq. (5) multiplied by $d¯$, the difference (left wheel minus right wheel) of the first of Eq. (8) multiplied by $h¯B$, and the sum (left wheel plus right wheel) of the last of Eq. (8), and add them all to the last of Eq. (5). This gives Display Formula

(11)$m¯fk¯2ω¯′+m¯fa¯G/O·d¯ev=(F¯u−−F¯u+)h¯$

where $k¯=(I¯kf+2I¯kw+2m¯wh¯G2)/m¯f$.

Since the normal forces acting on the rear wheels, $N¯±$, appear in Eq. (4), we need to determine them also. We can do this by making use of the third fourth and fifth of Eq. (5) and sum (left wheel plus right wheel) of the fourth of Eq. (8). Eliminating the sum (left wheel plus right wheel) of the lateral slip force components leaves seven constraint forces and moment components that can be easily determined using these seven equations. In particular, the normal force components are Display Formula

(12)$N¯±=(m¯fR2+m¯w)g¯∓12h¯[(m¯a¯G/O·ev−2m¯wd¯ω¯′)R¯w−I¯vwω¯(ω¯++ω¯−)]$

where $m¯fR=m¯fl¯/(l¯+d¯)$ is the robot frame mass supported by the rear wheels when it is stationary and $ω¯±=ψ′±$.

###### Dimensionless Equations of Motion.

The angular velocity, $ω¯$, varies on, or can be characterized by, a timescale $t¯c$ (for example if $ω¯$ is constant then $t¯c=ω¯−1$). Then we use a characteristic length scale $R¯c$ to make lengths dimensionless and denote the dimensionless quantities without the bar symbol “ ¯ ”, for example $x=x¯/R¯c$. Introducing a dimensionless time $t=t¯/t¯c$, dimensionless speed $v=v¯t¯c/R¯c$, dimensionless angular speed $ω=ω¯t¯c$, and dimensionless force $F=F¯t¯c2/(m¯fR¯c)$, the dimensional equations of motion, Eqs. (9)(11), can be written in dimensionless form as Display Formula

(13)$au=Fu−+Fu+am=Fu−−Fu+av=Fv++Fv−$

where Display Formula

(14)$au=Ia(v˙u−vvω)+2ρwdω2am=[k2ω˙+d(v˙v+vuω)]/hav=Ia(v˙v+vuω)−2ρwdω˙$

$Ia=1+2ρw, ρw=m¯w/m¯f$, and dot represents a derivative with respect to dimensionless time t. The dimensional forces, Eqs. (12) and (3), become in dimensionless form Display Formula

(15)$2N±=ρRG∓P/h2Fu±=−f(vu∓hω−v±)2Fv±=−f(vv−dω)$

where $v±=−Rwψ˙±, f=2f¯t¯c/m¯f, G=g¯t¯c2/R¯c, ρR=(m¯fR+2m¯w)/m¯f$, and $P=avRw−2ρwkvw2ωψ˙S$, with $ψ˙S=(ψ˙++ψ˙−)/2$. On combining the dimensionless equations of motion and forces, the three first-order differential equations for $vu, vv$, and ω can be written as Display Formula

(16)$au∓am=−f(vu∓hω−v±)$
Display Formula
(17)$av=−f(vv−dω)$

## Asymptotic Solution

In the case when $f≫1$, the situation when the longitudinal and lateral slip speeds are small and Eq. (3) is a reasonable model for the slip force, we write Eqs. (16) and (17) as Display Formula

(18)$vu∓hω−v±=−ϵ(au∓am)$
Display Formula
(19)$vv−dω=−ϵav$

where $ϵ=f−1≪1$. In this form, we treat them as algebraic equations for $vu, vv$, and ω, because the derivative terms appear multiplied by small quantities. As such they cannot represent the solution in a short initial interval in which the derivatives are very large and the functions vary from their initial values to those given by Eqs. (18) and (19).

These solutions can be expressed as asymptotic expansions of the small parameter, which we write as Display Formula

(20)$vu=vu0+ϵvu1+O(ϵ2)vv=vv0+ϵvv1+O(ϵ2)ω=ω0+ϵω1+O(ϵ2)$

Substituting into the three equations produces the solution for the dominant order Display Formula

(21)$vu0=vS ω0=vD/h vv0=vDd/h$

where Display Formula

(22)$vS=(v−+v+)/2 vD=(v−−v+)/2$

and $v±=Rwω±$.

As we noted previously, this solution cannot satisfy any initial condition. Indeed, there is an initial layer of order ϵ in time over which the solution varies from the initial conditions to the asymptotic series written in Eq. (20). The dominant order of this solution written in Eq. (21) describes the robot motion when its driving wheels are rolling without slipping. The first-order correction terms describe slipping to dominant order. The solution for these, assuming that both $v−$ and $v+$ are positive are Display Formula

(23)$vu1=−au0 ω1=−am0/h vv1=dω1−av0$

where Display Formula

(24)$av0=(v˙Dd+IavSvD)/hau0=(Iav˙Sh2−vD2d)/h2am0=[(k2+d2)v˙D+vSvDd]/h2$
are the acceleration terms with the zero-order values of the variables, given by Eq. (21) inserted in them.

The dimensionless frame mass center speed is given by $vG=vu2+vv2$ (taking the positive root means that we consider only forward motion) and the angle, αG, between this velocity direction and the robot u axis, $αG=ϕ−θ$, is given by $tan αG=vv/vu$.

As an example, consider a robot whose frame mass center is to move on a circle with radius of curvature $RG$ (this can be positive or negative) and with constant (positive) forward speed $vG$. In this event, $vG=ωRG, vS$ and $vD$ are constants, and the asymptotic solution simplifies to Display Formula

(25)$vu=vS+ϵ{vD2dh2}+O(ϵ2)ω=vDh−ϵ{dvSvDh3}+O(ϵ2)vv=vDdh−ϵ{[d2+Iah2]vSvDh3}+O(ϵ2)$
Note that the signs of ω, $vD$, and $RG$ are all the same and that every point on the rigid robot moves on a circle whose radius depends on the specific point.

These results are useful when we know the wheel speeds, in which case we can determine $vG$ and $RG$. To dominant order, these are Display Formula

(26)$vG∼vS2+(vDd/h)2RG∼sgn(vD)(vS/vD)2h2+d2$

However, we often know the desired trajectory, namely the robot frame mass center speed (or angular speed) and radius of the circle, and wish to know the wheel speeds that will produce it. We can obtain these expressions by inverting the definition for $vG$ along with the third equation of Eq. (25) to give $vS$ and $vD$ in terms of $RG$ and ω. Doing this produces Display Formula

(27)$vS=ωRGS−ϵ{d(vv10−dω10)+RGSvu10RGS}+O(ϵ2)$
Display Formula
(28)$vD=ωh−ϵhω10+O(ϵ2)$

where we have defined $S=1−d2/RG2$; recall that the sign of $RG$ is the same as that of ω, positive for counterclockwise rotation and negative for clockwise rotation. The quantities $vu10,ω10,vv10$ are those that appear in curly brackets in the three equations of (25), respectively, with $vS$ and $vD$ evaluated using their dominant order values Display Formula

(29)$vu10=−ω2H2d/h2ω10=−ω2Hd/h2vv10=−ω2H[d2+Iah2]/h2$

where Display Formula

(30)$H=RGS=RG2−d2$

###### Wheel Sliding Limits.

Here, we rewrite the slip bound Eq. (4) in terms of our dimensionless quantities; after squaring and clearing fractions, it becomes

$f2(vu∓hω−v±)2+f2(vv−dω)2≤(μ2N±)2$

On substituting the first of Eqs. (15), (18), and (19), these inequalities become Display Formula

(31)$(au∓am)2+av2≤μ2(ρRG∓P/h)2$

Making use of the asymptotic expansion, Eq. (20), and substituting its dominant order, Eq. (21) into Eq. (31) gives, to dominant order Display Formula

(32)$(au0∓am0)2+av02≤μ2(ρRG∓P0/h)2+O(ϵ)$

where $P0=av0Rw+2ρwkvw2vSvD/(Rwh)$.

For constant speed circular motion, using the relations in Eq. (24), this simplifies further to Display Formula

(33)$(dh)2[(vD±vS)vD]2+Ia2(vSvD)2≤μ2(A∓vSvDB)2+O(ϵ)$

where

$A=ρRGh B=IaRwh+2ρwkvw2Rwh$

On using the equality here, we find the largest $vS$ for a given $vD$ for which there is no sliding. We can use Eqs. (27) and (28) to write these equations in terms of ω and $RG$Display Formula

(34)$C±ω4±(2AHhBμ2)ω2−μ2A2≤O(ϵ)$

where we have written $C±=(h±H)2d2+(Hh)2(Ia2−μ2B2)$. In this form, the equality produces the largest ω for a given $RG$ for which there is no sliding.

Note that Eqs. (33) and (34) are clearly satisfied for ω = 0. The value we seek is the positive value of ω for which equality is satisfied. Note also that the upper sign corresponds to the left wheel, which for counterclockwise motion around the circle (meaning $sgn(ω)=sgn(RG)>0$) is the inner wheel. Since, for clockwise motion, $sgn(ω)=sgn(RG)<0$, the right wheel is the inner wheel, and the right wheel inequality in that case is identical with the left wheel inequality in the clockwise case we see that the two inequalities are most simply interpreted as written for the inner (upper sign) and outer (lower sign) wheels on the circle.

The desired solutions of the biquadratic Eq. (34) are Display Formula

(35)$|ω|={(ABHhμ2)2+A2μ2C±∓ABHhμ2C±}1/2$

where, taking H > 0, the upper sign refers to the inner wheel sliding and the lower one, the outer wheel.

Let us consider a desired circle of radius RG. The wheel speed ratio to achieve this is given, to dominant order, by the second Eq. (26). Next, we assume counterclockwise circular motion without loss of generality. Then, Eq. (26) can be inverted to determine the required wheel speed ratio, $ψ˙−/ψ˙+>1$, to produce a circular motion of radius RG. Using Eq. (30), the result is given by Display Formula

(36)$ratio=ψ˙−ψ˙+∼|H|+hsgn(RG)|H|−hsgn(RG)$

The maximum wheel speeds for which there is no sliding can be found from the smaller of Eq. (35) and the dominant order expression $ω∼vD/h$Display Formula

(37)$ψ˙+max∼|ω|smaller(|H|−hsgn(RG))/Rwψ˙−max∼|ω|smaller(|H|+hsgn(RG))/Rw$

###### Path Following.

Here, we show that the wheel sliding limits derived in Sec. 3.1 can be used to follow any path with reasonable accuracy without any feedback. Assume the robot must move on a given path Display Formula

(38)$y=y(x)$
in Cartesian coordinates. Note that a similar procedure can be used for polar coordinate. The slope of the curve and its radius of curvature are Display Formula
(39)$tan ϕ(x)=dy(x)dx ρ(x)=[1+(dy(x)dx)2]3/2d2y(x)dx2$

The speed along the curve is to be specified either as a known function of x, that is $v(x)$, or as a function of the radius of curvature, which is a known function of x, that is $v[ρ(x)]$ so that in either case Display Formula

(40)$v(x)=x˙2+y˙(x)2=1+(dy(x)dx)2x˙$

Hence, we can define the x component of the robot velocity as Display Formula

(41)$x˙=vx(x)=v(x)1+(dy(x)dx)2$

Note that since $v(x)$ and y(x) are known as a function of x, the parametric equations of the curve traversed with the specified speed can be found by integrating Eq. (41) with a suitable initial condition, $x(0)=x0$ to find x(t), and substituting the result into Eq. (38) to obtain $y(t)=y[x(t)]$. Also, the y velocity component can be obtained by differentiating $y(t)=y[x(t)]$ as Display Formula

(42)$y˙=y(x)dxvx(x)=vy(x)$

The velocity components in the robot body-fixed reference frame $eu−ev, vu,vv$, are given by means of the transformation Display Formula

(43)$vu(x,θ)=y˙ sin θ+y˙ cos θ=v(x)cos(θ−ϕ(x))vv(x,θ)=y˙ cos θ−y˙ sin θ=v(x)sin(θ−ϕ(x))$

Taking the time derivatives of Eq. (43) and using Eq. (19), we can derive the acceleration components in the robot body-fixed reference frame, $v˙u,v˙v$, to dominant order as Display Formula

(44)$v˙u(x,θ)=[dvy(x)dxsin θ+dvx(x)dxcos θ]vx(x)+vuθ˙=[dvy(x)dxsin θ+dvx(x)dxcos θ]vx(x)+vuvv/d+O(ϵ)v˙v(x,θ)=[dvy(x)dxcos θ−dvx(x)dxsin θ]vx(x)−vuθ˙=[dvy(x)dxcos θ−dvx(x)dxsin θ]vx(x)−vuvv/d+O(ϵ)$

where from Eqs. (41) and (42), we have Display Formula

(45)$dvx(x)dx=dv(x)dx1+(dy(x)dx)2−v(x)dy(x)dxρ(x)dvy(x)dx=d2y(x)dx2vx(x)+dy(x)dxdvx(x)dx$

We are now ready to use the three equations in Eqs. (18) and (19) to solve for $θ(t)$ and the two wheel speeds $v±(t)$ with first-order correction terms. Equation (19) can be written as a first-order differential equation Display Formula

(46)$θ˙=vv(x,θ)d+ϵω1(x,θ)+O(ϵ2)$

where Display Formula

(47)$ω1(x,θ)=[dvydxcos θ−dvxdxsin θ]vxd+2ρwvuvvd2$
Then integrating Eq. (46) together with Eq. (41) and initial conditions $θ(0)=θ0$ and $x(0)=x0$, we obtain the functions $θ(t)$ and x(t). Finally, using Eq. (18), we find the wheel speeds $v±$ as functions of time that produce the prescribed curve with the prescribed speed as Display Formula
(48)$v±(t)=−Rwψ˙±=vu[x(t),θ(t)]∓hdvv[x(t),θ(t)] +ϵ(au0∓am0∓hav0/d)+O(ϵ2)$

with

$au0=Iav˙u[x(t),θ(t)]−vv[x(t),θ(t)]2d+O(ϵ)am0=[k2+d2]v˙v[x(t),θ(t)]+dvu[x(t),θ(t)]vv[x(t),θ(t)]dh+O(ϵ)av0=v˙v[x(t),θ(t)]+Iavu[x(t),θ(t)]vv[x(t),θ(t)]d+O(ϵ)$

Note that in addition to Eq. (48), we may also calculate the wheel speeds using Eqs. (35)(37). This is achieved by assuming the instantaneous circle radius RG is equal to the radius of curvature $ρ(x)$ calculated in Eq. (39) for the path y(x) at each instant of time.

## Robot Sliding Experiment

In order to verify the theoretical results, we performed circular motion experiments at various radii and compared the actual sliding limits with the predicted values. Moreover, we applied the methodology presented in Sec. 3.2 for the robot to follow a sinusoidal path. The mobile robot is differential drive where each wheel speed is maintained using a local proportional-integral controller. The maximum reliable wheel speed achieved by the proportional-integral controller is 250 rpm. Other relevant model parameters are

$h¯=0.0988 m, d¯=0.055 m, l¯=0.1133 m, R¯w=0.046 m, m¯=1.38 kg, m¯w=0.0312 kg$

The characteristic length and time scale used for nondimensionalization are defined as $R¯c=h¯$ and $t¯c=h¯/g$ where $g=9.807 m/s2$ is acceleration due to gravity. Also, the small parameter for the asymptotic solution is selected as $ϵ=0.01$.

The experiments were performed in a laboratory with a $4.5 m×4.0 m×3.0 m$ dedicated space for quadrotor and mobile robot operations and equipped with a Vicon motion capture system (Centennial, CO) capable of submillimeter three-dimensional localization at 250 frames/s. There are six 1.3 MP infrared cameras with 1280 × 1024 resolution located along the top perimeter of the lab. The four corner cameras are oriented toward the center of the floor for better ground vehicle motion capture. The camera system is wired to a computer with a 3.7 GHz Intel Xeon processor. The cameras capture four infrared light-emitting diode balls located on the top surface of the robot and provide position and orientation data to the software implemented in matlab/simulink version 2016a. This way, we can capture the motion data and observe when the robot deviates from the predicated circular or sinusoidal motion with no slipping.

###### Friction Coefficient Estimation.

Our theoretical sliding prediction requires an estimate of the static friction coefficient between the mobile robot and the laboratory floor. Hence, we performed simple static friction experiments using a force gauge to measure the force before the robot starts moving on the floor. Since the robot wheels have removable tires, we performed two sets of experiments, one set with tires and another set without tires. The results of various trials are shown in Fig. 2. The friction coefficient with tires has a mean value of 0.387 and standard deviation of 0.034 while the friction coefficient without tires has a mean value of 0.227 and standard deviation of 0.020.

###### Circular Motion Experiments.

The theoretical relationship between wheel speed ratio and the corresponding radius of the circular motion is given by Eqs. (30) and (36). To verify this, we performed experiments with various wheel speed ratios while avoiding the sliding limits. These experiments were performed using the wheels with and without tires. The results are shown in Fig. 3 where the theoretical values perfectly match the experimental ones in both cases.

To verify the predicted sliding limits as determined from Eqs. (35) and (37), we performed experiments at various speeds and wheel ratios. Given a radius and thus a wheel ratio, we performed experiments at slow speeds and then repeated them by slightly increasing the speeds until sliding was visible as observed by the overhead cameras.

Figure 4 shows the theoretical inner (left) and outer (right) wheel sliding speed limits based on the experimentally derived static friction coefficients using wheels with and without tires. The solid lines indicate the inside (left) and the dashed line the outside (right) wheel theoretical slip speed limits. The sliding wheel speed differences (and thus ratios) are larger than those required to achieve circular motion of various radii, as shown in Fig. 3. Hence, it is expected that the inner wheel will always slide first. This is confirmed by the experimentally derived wheel speeds shown with markers “$*$” for the inner wheel and “+” for the outer wheel. In the experiments, we noted the lowest wheel speeds when sliding became visible, which corresponded to at least 1% error in the radius of each circle. Therefore, the experiments verify that the theory very accurately predicts the inner wheel sliding limits as indicated by “$*$.” The “+” markers only indicate the outer wheel speeds corresponding to the inner wheel sliding limits. We did not attempt to experimentally verify the outer wheel sliding limits since our objective was only to determine when sliding begins.

Figure 5 shows the mobile robot going through several revolutions of a sample circular motion of radius 0.3 m, as captured by the camera system. In this case, the mobile robot wheels are with tires. In the top graph, the outer (right) and inner (left) wheel speeds are 200 and 100 rpm, respectively. While in the bottom graph, the outer (right) and inner (left) wheel speeds are 220 and 110 rpm, respectively. Clearly, the circular motion is accurate in the top graph while the bottom graph demonstrates significant deviation from the circle as a result of slipping. Thus, we can conclude that at least one of the two wheels speeds has exceeded its sliding limit. This limit has been accurately predicted in the top graph of Fig. 4 where the inside wheel slips at about 105 rpm.

###### Path Following Experiments.

In addition to improving closed-loop control, wheel sliding limits can be used for accurate short-term trajectory tracking in the absence of position feedback. Moreover, the path following equations derived in Sec. 3.2 can be applied to any twice differentiable path curve. Hence, curves with discontinuous derivatives must be approximated to smooth out the edges, which is normally required even for closed-loop control methods. Here, we selected a sinusoidal path for the robot to follow in order to evaluate the performance of the methodology for paths with continuously varying radius of curvature and switching between convex and concave curves. Such paths are expected to increase the probability of sliding due to continuous acceleration/deceleration of wheels. The sinusoidal path is defined as Display Formula

(49)$y(x)=a sin λx$
where a and λ are the amplitude and wavelength of the sinusoidal path. The velocity profile $v(x)$ must be compatible with the path defined in Eq. (49). Hence, we define it as
$v(x)=vmax−vmin2cos 2λx−vmax+vmin2$

where Vmin is the minimum robot speed while traversing the path, which occurs at or near the peaks, where the radius of curvature is a minimum. The maximum speed of the robot on the path, Vmax, occurs at or near the maximum radius of curvature. However, since the radius of curvature approaches infinity where the path crosses the x-axis, we need to define Vmax to be less than or equal to the robot's maximum speed as determined by the motors rotating the wheels. Note that Vmax and Vmin may be determined based on the wheel sliding limits using Eqs. (35)(37). Moreover, Fig. 4 shows that the inside wheel always slips first. Thus, Vmax and Vmin are determined by the sliding limits of the inside wheel.

The minimum radius of curvature for the path is $ρmin=(1/aλ2)$ occurring at $(dy(x)/dx)=0$. For the current experiments, we choose a = 0.25 m and λ = 4 rad/m resulting in $ρmin=0.25$ m. Using the constant speed results for the robot with tires (Fig. 4), the maximum inner and outer wheel speeds for a circle of 0.25 m radius are 80 rpm and 190 rpm, respectively. Since the maximum radius of curvature is $∞$, wheel speeds are set to their maximum reliable value of 250 rpm. The wheel speed profiles for the selected sinusoid as determined by Eq. (48) and by using Eqs. (35)(37) are presented in Fig. 6 subject to maximum of 250 rpm. It can be seen that both methods produce similar profiles. The main difference is that the ones produced by Eq. (48) are smooth but slightly violate the maximum allowed wheel speeds applied only at the initial point.

The speeds corresponding to minimum and maximum radii of curvature can then be calculated as Vmin = 0.684 m/s and Vmax = 1.224 m/s, respectively. However, these values correspond to constant speed circular motion on a homogeneous surface. Since the trajectory tracking process is completely open loop, position errors are expected to grow due to various reasons including variations in surface friction and the expected magnitude of acceleration/deceleration on a given path. Thus, the speed profile must be scaled down since the limits were derived based on constant speed while the robot will continuously change speed on the sinusoidal path.

We scaled down Vmin and Vmax values for the experiments by various factors but only present those ranging from 0.3 to 0.6 as sample representatives. The position tracking error plots and orientation angle errors for scale factors of 0.3 and 0.6 are presented in Figs. 7 and 8, respectively. Note that the position tracking errors and the angle errors are the difference between those predicted by Eqs. (41), (46), and (38) and those experimentally measured by the camera system. Figure 9 shows the paths followed when using scale factors of 0.3, 0.4, 0.5, and 0.6 compared with the desired reference sinusoidal path. It is clear that the tracking errors increase with speed and accumulate with time. Thus, the specific scaling value should be decided based on expected robot accelerations/decelerations for a given path and on the required precision and expected duration of open-loop control.

Another interesting observation is that the tracking errors increase significantly at each peak where the radius of curvature is minimum and the robot is making a turn, particularly at higher speeds. Therefore, scaling down the robot speed throughout the trajectory may not be necessary and the robot should instead be slowed down more significantly where radius of curvature is small (i.e., approaching d). In general, the method is not limited to any path or any specific speed. However, its accuracy is limited with respect to rapidly changing speed profiles, i.e., large accelerations/decelerations of the mobile robot.

###### Practical Implementation and Application.

Practical implementation of the method is presented here based on the experimental results. Initially, the friction coefficient must be estimated either a priori or in real time through slip detection. Next, a speed scaling factor should be selected based on the demands of the trajectory to be followed. If sharp changes in speed (i.e., high accelerations) are required, then the overall speed profile must be scaled down. The magnitude of scaling also depends on the duration of expected trajectory tracking without any feedback and the required precision. The most important application of the method is when GPS or other absolute position feedback may not be available for a limited time or when feedback is only available at very low rate. An example is when onboard vision is used as feedback for obstacle avoidance and path planning. The processing of images in such cases may be much slower than would be required to accurately track a trajectory. In such cases, the method can then be applied to follow the trajectory more precisely in between each available feedback.

## Conclusions

A method based on asymptotic analysis was presented that predicts wheel sliding speed limits for mobile robots in terms of a known static friction coefficient. We performed experiments that verified the theoretically predicted wheel sliding speeds for constant speed circular motions of various radii. We also showed that the inner wheel in a circular motion will always slide first. Using these results, we showed that the wheel speed profiles provided by our method can accurately guide the robot through a sinusoidal path of continuously changing curvature though some scaling may be necessary due to slipping at the peaks. Thus, the predictive model can aid with both open-loop and closed-loop controllers with slow sample rates to achieve more precise mobile robot navigation particularly in absence of absolute position measurements.

## References

Kolmanovsky, I. , and McClamroch, N. , 1995, “ Developments in Nonholonomic Control Problems,” IEEE Control Syst. Mag., 15(6), pp. 20–36.
Fierro, R. , and Lewis, F. , 1997, “ Control of a Nonholonomic Mobile Robot: Backstepping Kinematics Into Dynamics,” J. Rob. Syst., 14(3), pp. 149–163.
Yang, J.-M. , and Kim, J.-H. , 1999, “ Sliding Mode Control for Trajectory Tracking of Nonholonomic Wheeled Mobile Robots,” IEEE Trans. Rob. Autom., 15(3), pp. 578–587.
Astolfi, A. , 1999, “ Exponential Stabilization of a Wheeled Mobile Robot Via Discontinuous Control,” ASME J. Dyn. Syst., Meas., Control, 121(1), pp. 121–126.
Klančar, G. , Matko, D. , and BlaŽič, S. , 2005, “ Mobile Robot Control on a Reference Path,” Mediterranean Conference on Control and Automation, pp. 1343–1348.
Ghasemi, M. , Nersesov, S. , and Clayton, G. , 2014, “ Finite-Time Tracking Using Sliding Mode Control,” J. Franklin Inst., 351(5), pp. 2966–2990.
Raimúndez, C. , Villaverde, A. , and Barreiroz, A. , 2014, “ Adaptive Tracking in Mobile Robots With Input-Output Linearization,” ASME J. Dyn. Syst., Meas., Control, 136(5), p. 054503.
Ghasemi, M. , Nersesov, S. , Clayton, G. , and Ashrafiuon, H. , 2014, “ Sliding Mode Coordination Control for Multiagent Systems With Underactuated Agent Dynamics,” Int. J. Control, 87(12), pp. 2615–2633.
Luviano-Juárez, A. , Cortés-Romero, J. , and Sira-Ramírez, H. , 2015, “ Trajectory Tracking Control of a Mobile Robot Through a Flatness-Based Exact Feedforward Linearization Scheme,” ASME J. Dyn. Syst., Meas., Control, 137(5), p. 051001.
Durrant-Whyte, H. , and Baily, T. , 2006, “ Simultaneous Localization and Mapping (SLAM)—Part I,” IEEE Rob. Autom. Mag., 13(3), pp. 99–108.
Baily, T. , and Durrant-Whyte, H. , 2006, “ Simultaneous Localization and Mapping (SLAM)—Part II,” IEEE Rob. Autom. Mag., 13(3), pp. 108–117.
Kia, S. , Rounds, S. , and Martinez, S. , 2016, “ Cooperative Localization for Mobile Agents: A Recursive Decentralized Algorithm Based on Kalman-Filter Decoupling,” IEEE Control Syst. Mag., 36(2), pp. 86–101.
Ward, C. C. , and Iagnemma, K. , 2008, “ A Dynamic-Model-Based Wheel Slip Detector for Mobile Robots on Outdoor Terrain,” IEEE Trans. Rob., 24(4), pp. 821–831.
Iagnemma, K. , Kang, S. , Shibly, H. , and Dubowsky, S. , 2004, “ Online Terrain Parameter Estimation for Wheeled Mobile Robots With Application to Planetary Rovers,” IEEE Trans. Rob., 20(5), pp. 921–927.
Balakrishna, R. , and Ghosal, A. , 1995, “ Modeling of Slip for Wheeled Mobile Robots,” IEEE Trans. Rob. Autom., 11(1), pp. 126–132.
Wang, D. , and Low, C. B. , 2008, “ Modeling and Analysis of Skidding and Slipping in Wheeled Mobile Robots: Control Design Perspective,” IEEE Trans. Rob., 24(3), pp. 676–687.
Whitman, A. , Clayton, G. , and Ashrafiuon, H. , 2017, “ Asymptotic Inverse Solution and Trajectory Planning for Open-Loop Control of Mobile Robots,” ASME J. Dyn. Syst., Meas., Control, 17(5), p. 051004.
Whitman, A. , Ashrafiuon, H. , and Muske, K. , 2011, “ Analytical Stability Analysis of Surface Vessel Trajectories for a Control-Oriented Model,” ASME J. Comput. Nonlinear Dyn., 6(3), p. 031010.
Song, T. , Xi, F. , Guo, S. , Tu, X. , and Li, X. , 2017, “ Slip Analysis for a Wheeled Mobile Manipulator,” ASME J. Dyn. Syst., Meas., Control, 140(2), p. 021005.
Ojeda, L. , Cruz, D. , Reina, G. , and Borenstein, J. , 2006, “ Current-Based Slippage Detection and Odometry Correction for Mobile Robots and Planetary Rovers,” IEEE Trans. Rob., 22(2), pp. 366–378.
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## References

Kolmanovsky, I. , and McClamroch, N. , 1995, “ Developments in Nonholonomic Control Problems,” IEEE Control Syst. Mag., 15(6), pp. 20–36.
Fierro, R. , and Lewis, F. , 1997, “ Control of a Nonholonomic Mobile Robot: Backstepping Kinematics Into Dynamics,” J. Rob. Syst., 14(3), pp. 149–163.
Yang, J.-M. , and Kim, J.-H. , 1999, “ Sliding Mode Control for Trajectory Tracking of Nonholonomic Wheeled Mobile Robots,” IEEE Trans. Rob. Autom., 15(3), pp. 578–587.
Astolfi, A. , 1999, “ Exponential Stabilization of a Wheeled Mobile Robot Via Discontinuous Control,” ASME J. Dyn. Syst., Meas., Control, 121(1), pp. 121–126.
Klančar, G. , Matko, D. , and BlaŽič, S. , 2005, “ Mobile Robot Control on a Reference Path,” Mediterranean Conference on Control and Automation, pp. 1343–1348.
Ghasemi, M. , Nersesov, S. , and Clayton, G. , 2014, “ Finite-Time Tracking Using Sliding Mode Control,” J. Franklin Inst., 351(5), pp. 2966–2990.
Raimúndez, C. , Villaverde, A. , and Barreiroz, A. , 2014, “ Adaptive Tracking in Mobile Robots With Input-Output Linearization,” ASME J. Dyn. Syst., Meas., Control, 136(5), p. 054503.
Ghasemi, M. , Nersesov, S. , Clayton, G. , and Ashrafiuon, H. , 2014, “ Sliding Mode Coordination Control for Multiagent Systems With Underactuated Agent Dynamics,” Int. J. Control, 87(12), pp. 2615–2633.
Luviano-Juárez, A. , Cortés-Romero, J. , and Sira-Ramírez, H. , 2015, “ Trajectory Tracking Control of a Mobile Robot Through a Flatness-Based Exact Feedforward Linearization Scheme,” ASME J. Dyn. Syst., Meas., Control, 137(5), p. 051001.
Durrant-Whyte, H. , and Baily, T. , 2006, “ Simultaneous Localization and Mapping (SLAM)—Part I,” IEEE Rob. Autom. Mag., 13(3), pp. 99–108.
Baily, T. , and Durrant-Whyte, H. , 2006, “ Simultaneous Localization and Mapping (SLAM)—Part II,” IEEE Rob. Autom. Mag., 13(3), pp. 108–117.
Kia, S. , Rounds, S. , and Martinez, S. , 2016, “ Cooperative Localization for Mobile Agents: A Recursive Decentralized Algorithm Based on Kalman-Filter Decoupling,” IEEE Control Syst. Mag., 36(2), pp. 86–101.
Ward, C. C. , and Iagnemma, K. , 2008, “ A Dynamic-Model-Based Wheel Slip Detector for Mobile Robots on Outdoor Terrain,” IEEE Trans. Rob., 24(4), pp. 821–831.
Iagnemma, K. , Kang, S. , Shibly, H. , and Dubowsky, S. , 2004, “ Online Terrain Parameter Estimation for Wheeled Mobile Robots With Application to Planetary Rovers,” IEEE Trans. Rob., 20(5), pp. 921–927.
Balakrishna, R. , and Ghosal, A. , 1995, “ Modeling of Slip for Wheeled Mobile Robots,” IEEE Trans. Rob. Autom., 11(1), pp. 126–132.
Wang, D. , and Low, C. B. , 2008, “ Modeling and Analysis of Skidding and Slipping in Wheeled Mobile Robots: Control Design Perspective,” IEEE Trans. Rob., 24(3), pp. 676–687.
Whitman, A. , Clayton, G. , and Ashrafiuon, H. , 2017, “ Asymptotic Inverse Solution and Trajectory Planning for Open-Loop Control of Mobile Robots,” ASME J. Dyn. Syst., Meas., Control, 17(5), p. 051004.
Whitman, A. , Ashrafiuon, H. , and Muske, K. , 2011, “ Analytical Stability Analysis of Surface Vessel Trajectories for a Control-Oriented Model,” ASME J. Comput. Nonlinear Dyn., 6(3), p. 031010.
Song, T. , Xi, F. , Guo, S. , Tu, X. , and Li, X. , 2017, “ Slip Analysis for a Wheeled Mobile Manipulator,” ASME J. Dyn. Syst., Meas., Control, 140(2), p. 021005.
Ojeda, L. , Cruz, D. , Reina, G. , and Borenstein, J. , 2006, “ Current-Based Slippage Detection and Odometry Correction for Mobile Robots and Planetary Rovers,” IEEE Trans. Rob., 22(2), pp. 366–378.

## Figures

Fig. 1

The mobile robot and its model

Fig. 2

Static friction coefficient test results with and without tires

Fig. 3

Theoretical and experimental wheel speed ratios as a function of circular path radii using wheels with tires (top) and without tires (bottom)

Fig. 4

Theoretical and experimental slipping speed limits as a function of circular path radii using wheels with tires (top) and without tires (bottom). The solid lines indicate the inside (left) and the dashed line the outside (right) wheel theoretical slip speed limits.

Fig. 5

Mobile robot using wheels with tires achieving circular paths of radius 0.3 m using wheel speeds of ψ˙±=200,100 rpm (top) and ψ˙±=220,110 rpm (bottom)

Fig. 6

Sinusoidal path wheel speed profiles as determined by Eq. 48 (top) and by using Eqs. (35)(37) (bottom) subject to maximum of 250 rpm.

Fig. 7

Sinusoidal trajectory position tracking errors (top) and orientation error (bottom) with a speed scale factor of 0.3

Fig. 8

Sinusoidal trajectory position tracking errors (top) and orientation error (bottom) with a speed scale factor of 0.6

Fig. 9

The paths achieved by the robot using various speed scale factors

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