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Proceedings of the ASME 2016 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference

IDETC 2016 August 21-24, 2016, Charlotte, North Carolina

IDETC2016-60303

DRAFT: KINEMATIC ANALYSIS OF A SKID-STEER MOBILE ROBOT OPERATING ON NON-PLANAR SURFACES

Joshua Qualls

Tennessee Technological University Cookeville, Tn. USA

Stephen Canfield Tennessee Technological University

Cookeville, Tn. USA

Alexander Shibakov Tennessee Technological University

Cookeville, Tn. USA

ABSTRACT:

Mobile robotic systems are advancing manufacturing operations

that generally exhibit less automation. Fields commonly

experiencing a lesser degree of automation include shipbuilding,

pipe inspection, and construction applications on non-planar

surfaces. The mobile robotic systems operating in such

environments are typically subjected to motion along non-planar

surfaces and often assume climbing configurations when

completing the desired tasks. These tasks are usually considered

planar in nature, however, in practice the surface on which the

tasks are completed is generally non-planar. The surfaces are

commonly modeled as geometric shapes such as spheres or

cylinders allowing for accurate surface representations. A

common non-planar task is the welding of ship hulls. The

process consists of welding segments together and is normally

completed by a skilled laborer due to the curvature of the surface.

Through the advancement of kinematic modeling of mobile

robotic platforms there exists the ability to develop future robotic

platforms with the potential to perform operations, in a task

efficient manner, on non-planar surfaces. While the majority of

kinematic models presented for mobile robots assume operation

on planar surfaces, there have been a number of studies that

consider the kinematic behavior on non-planar surfaces. This

past work generally takes one of two approaches, either

developing assumed modifications to the existing kinematic

constraints as algebraic equations, or making use of a set of

differential equations describing the instantaneous motion of the

contact point between non-planar surfaces. The second approach

is more general and has been shown to be applicable to both

general and specific terrains. However, the previous work has

focused solely on differential-steer platforms with a passive

castor. Conversely, Skid-Steer mobile robots provide simple,

robust platforms that have several features making them well

suited to manufacturing tasks. This paper will present a

kinematic model for an SSMR operating on a non-planar,

nominally cylindrical surface. This method is readily generalized

to other surface geometries. The method is based on a kinematic

model consisting of a set of differential-algebraic equations with

the differential equations derived from Montana’s contact

equations and the algebraic equations arising from an

instantaneous model of the SSMR as a hybrid mechanism

consisting four, five-dof serial chains. The method is

demonstrated over several applications and validated through

comparison with a prototype system operating on a cylindrical

tank. A discussion of the resulting approach and how this could

be used in as a tool to guide the modeling and fabrication of

future mobile manufacturing robot platforms is included.

INTRODUCTION

As mobile robots gain interest in various industries their

incorporation into manufacturing applications is increasing in

areas such as mobile inspection, welding, and painting, [1].

Research focusing on applications within these areas has

typically been driven by the need to control the desired path in

order to complete the required task at hand. The advanced

control strategies needed to accommodate such motion has

greatly improved in part due to continued research in the area of

mobile kinematics and dynamics. This research has allowed for

the development of robust models to accurately predict the

behavior of the robotic platform as it navigates along a desired

path. Typically, robotic topologies incorporate wheels enabling

mobility, therefore rolling motion has been widely studied [2-3].

The primary constraints for this rolling motion is provided by the

contact point between the wheels and the surface.

Tracked mobile robots are also widely studied in the literature

[4-7] and make use of an approach similar in nature to the rolling

motion of wheels, [4]. The models for wheeled vehicles rely on

the pure roll and no slip assumptions, which is valid when

dynamic effects are minimal, however, in the case of tracked

vehicles it is accepted that slipping occurs between the tracks and

terrain, therefore the pure roll and no slip assumptions are not

valid, [8-10]. The consideration of slip has been focused on by

many works, [11-12], and is often incorporated into the

kinematic models for the differential, skid, and omni-directional

robot platforms, [12-14]. While these techniques are quite

useful, their main limitation is the surface of operation. Typically

mobile robot operation is assumed to occur on a planar surface

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when in reality the desired tasks are usually carried out on non-

planar surfaces. Consider a robot performing a seam weld on a

storage tank or a ship hull. The typical welding path is

constructed by segments of linear spines called gaits and are

often found on surfaces of varying curvature; in order for the

robot to complete the desired task along the path adequate

information about the path and robot location must be provide.

The motion of the robot is dependent on the surface through the

knowledge that the kinematics of vehicles are dependent on the

terrain, [14-15], thus in order to provide accurate information on

the robots position and orientation effects of the non-planar

surface must be accounted for in the kinematic model. This

accommodation of non-planar surface conditions has been

studied by previous researchers, [16-18], with a kinematic model

of contact provided by Montana, [19]. This model incorporates a

set of contact equations, constructed from a set of five first order

ordinary differential equations, and used to describe the motion

of the contact point between two surfaces. These contact

equations have been used in areas ranging from mobile robot

kinematics to the kinematics of grasping objects, [16-18, 20],

and hold valid for a variety of surfaces that can be characterized.

This paper follows the work presented in [18, 21, 22-23].

However, the previous works, [16, 21] employed a strategy used

to investigate wheeled robot behavior over an arbitrary terrain

without slip. It is apparent however that slip is necessary in order

to achieve certain degrees of motion, as is the case with skid steer

platforms, thus requiring a more robust model with the capability

to estimate the path of the mobile robot when subjected to

slipping conditions. The proposed formulation satisfies the same

set of constraints as in [16,21] although instead assuming the

translational motion of the contact points between the wheels and

the surface are zero, the current model allows for the

incorporation of slip. The amount of slip being experienced is

then determined by the location of the instant centers as the

mobile robot navigates along the climbing surface. The primary

context of this work is for mobile manufacturing in which

manufacturing tasks are performed with mobile manipulators.

Consider a mobile manufacturing robot, such as that

described in [1] and shown for example in Fig. 1. In this

example, the robot is operating on a cylindrical tank performing

inspection processes for pitting or cracking. The robot is driven,

either through remote operation or with some closed-loop

control, to follow a specific path defined by the parameters of the

specific inspection task. This maneuver requires the robot to

collect adequate data about the surface condition and relay that

information to observers. This task represents a large grouping

of manufacturing tasks that are ideally suited to mechanization

through mobile robots. The primary objective of this paper is to

develop a kinematic model of the robot system when operating

on non-planar navigation that is suitable for skid-steer mobile

robots (SSMR). This model will build directly on the approach

developed in [21] for differential steer mobile robot systems

operating on non-planar surfaces. The primary climbing surface

assumed in this paper is modeled as a spherical surface, however,

the approach can be applied to a variety of surfaces through

means of defining curvature, metric and torsion of the varying

surfaces.

Figure 1: Mobile Robot Performing Surface Inspection

APPROACH

A model is developed for the SSMR consisting of two tracks (left

and right) and operating on a non-planar surface. An equivalent

model of the SSMR is constructed as a multi-body system in

which the contact portion of each track and the chassis represent

bodies on the robot and are connected through a set of prismatic

joints lying longitudinally along the robot with its axis aligned

with the longitudinal direction of the robot and an additional

revolute joint between the right track and chassis (Fig. 3). The

contact patch of each tract couples to the surface with a higher

pair contact. The climbing surface is modeled as a sphere with

the inertial frame, {I}, located at the center.

Frame Assignments:

The motion generated by the robot along the climbing surface is

defined using a series of frames. The frame notation used here

follows the notation provided in [22].These frames represent the

robot local frame {r}, the contact frames for the spherical

climbing surface, {Si}, and each robot track, {Ti}, with i= L, R,

C. Figure 3 shows the inertial frame through the point of contact

while Figure 2 represents the robot kinematic model. The

contact point for each track is then determined through a set of

coincident frames. Two of these frames are defined according the

geometry of the contact surfaces following the method provided

in [17]; {Si} and {Ti}. The third frame, {0i} serves as the first in

a chain describing the robot kinematics. Frames {Si} and {Ti}

3

are constructed to form an outward normal Gaussian map for the

surface. Frame {Si} describes the climbing surface at the point

of contact of track i and is located relative to {I} through the

spherical coordinates 𝜃 and 𝜙 as,

𝑇𝑆𝑖𝐼 = [

𝑠𝜃𝑖𝑐𝜙𝑖 𝑠𝜃𝑖𝑠𝜙𝑖 𝑐𝜃𝑖 𝑅𝑐𝜃𝑖𝑠𝜙𝑖

𝑐𝜃𝑖𝑐𝜙𝑖 𝑐𝜃𝑖𝑠𝜙𝑖 −𝑠𝜃𝑖 𝑅𝑠𝜃𝑖𝑠𝜙𝑖

−𝑠𝜙𝑖 𝑐𝜙𝑖 0 𝑅𝑐𝜃𝑖

0 0 0 1

] (1)

Where 𝜃𝑖 and, 𝜙i represent rotations about the global Z and X

axes and R is the radius of the sphere. Frame {Ti} describes the

contact point of the track surface and is located relative to {Si}

by a rotation of ψ about the zS axis with zt = -zs. Finally, frame

{0i} is defined as a fixed rotation relative to {Ti} to facilitate

D&H frame assignment;

𝑅𝑂𝑖𝑇𝑖 = [

0 0 1−1 0 00 −1 0

] (2)

Mobile Robotic Kinematics:

An instantaneous model of the robot is shown in Figure 2 as a

schematic in a two-dimensional view with the length and width

of the robot is denoted as l and 2b respectively. Figure 2 also

represents the system in 3D as a parallel mechanism constructed

from three serial chains, similar to that seen in, [21]. Note that

an additional chain is added to the model to add a single point of

stability to the model that assumes only one contact point for

each track. In practice, deflection in the track will make contact

at more than one point. The contact point for each track is

assumed a higher-pair point contact that can experience a

combination of rotation and slip relative to the surface. The slip

for each wheel is calculated using the kinematic characterization

procedure demonstrated in [1] for SSMR’s in climbing

configurations. The slip is represented through determined

location of the instant center for each track contact patch relative

to the SSMR chassis. The locations of these instant centers are

assumed to remain constant over the robot motion. The slip

motion applies to the contact point and is superimposed on

rotations seen in three serial chains making up the closed-chain

mechanism model. Coupling the slip and rotation treats the robot

as an instantaneous holonomic system that can be used to

evaluate the remaining velocity components. This yields three

homogenous transformations, one associated with each wheel

chain as, T0Lr, T0R

r, and T0Cr and three Jacobian matrices relating

the velocity of the robot chassis to the configuration state

velocity for each track as:

𝐯𝑟 = 𝐉𝐿�̇�𝑳 = 𝐑𝟔0𝑅0𝐿 𝐉𝑹�̇�𝑅 = 𝐑𝟔0𝐶

0𝐿 𝐉𝐶�̇�𝐶 (3)

Y

Z

X

{I}

ɸ

{r}

Track{T}

ψ

x0

ys

z0

Xt

Zt

xs Y

t

Surface {S}

y0 z

s

Figure 3: Spherical Surface and Contact Frames

X

Y

Yc

Xc

l

x y

V

x0

2b

z0

x0

Figure 2: Spherical Surface and Contact Frames

{R} {L} {C}

{r}

4

Where qi is the vector of joint parameters for each track chain,

i=L, R, C and 𝐑𝟔0𝑅0𝐿 = [

𝐑0𝑅0𝐿 𝟎

𝟎 𝐑0𝑅0𝐿 ], 𝐑𝟔0𝐶

0𝐿 = [𝐑0𝐶

0𝐿 𝟎

𝟎 𝐑0𝐶0𝐿 ] , with

frames {0R} and {0C} projected onto {0L} by the adjoint

transformation operators 𝐑0𝐿0𝑅

, 𝐑0𝐶0𝑅. The velocity vector 𝑣𝑟 is

unknown. The approach augments this vector with the

projections of the configuration velocities onto the left track

frame. Row reduction then reveals these matrices in upper

diagonal form and gives a toral of four constraints used to first

define the remaining velocity terms in 𝑣𝑟 and then the

configuration space velocity. Once complete, the slip velocity is

added through superposition at each frame using the method

shown in [1]. These configuration velocities then define the

relative velocity between the track contact points and the

climbing surface, and consist of a combination of angular and

linear velocity terms.

Contact Equations:

The wheel contact point locations are now propagated forward

in time through a set of differential equations describing the

motion of contact between the robot tracks and climbing surface,

using the method demonstrated in [19]. The contacting surfaces

of the track and the spherical surface are characterized through a

set of measures, metric M, curvature K and torsion T can be

found at each location u on the surface using the formulas given

in [17]. In this application, the track is represented by a planar

surface yielding Metric, curvature and torsion arrays as;

𝐌𝒕 = [1 00 1

], 𝐊𝒕 = [0 00 0

], 𝐓𝒕 = [00

]. (4)

The climbing surface is modeled as a sphere yielding;

𝐌𝑺 = [R 00 𝑅

], 𝐊𝑺 = [0 00 0

], 𝐓𝑺 = [00

]. (5)

The contact equations are given for the motion of the contact

point on track as �̇�𝑡 = [�̇�𝑡 �̇�𝑡]𝑇, and the motion of the contact

point on the climbing surface, �̇�𝑠 = [�̇�𝑠 �̇�𝑠]𝑇 are found using

the contact equations given in [17]. Here the equations represent

specifically the contact between the planar tracks and spherical

climbing surface as

�̇�𝑡 = (𝑲�̃�)−1 ([−ω𝑦

ω𝑥] − 𝑲�̃� [

𝑣𝑥

𝑣𝑦]) (6)

�̇�𝑠 = 𝑅𝜓(𝑲�̃�)−1 ([−ω𝑦

ω𝑥]) (7)

�̇� = ω𝑧 + 𝐓𝑠𝐌𝑠�̇�𝑠 (8)

where [�̇�𝑡 �̇�𝑡]𝑇 = [�̇�𝑇𝑖 �̇�𝑇𝑖]𝑇 , [�̇�𝑠 �̇�𝑠]𝑇 = [�̇�𝑆𝑖 �̇�𝑆𝑖]

𝑇 and �̇�

is the rotation between the track and surface frames about the

common normal. 𝑅𝜓 = 𝑅𝑇𝑖𝑆𝑖 is the orientation of the track frame

projected on the surface frame. The curvature of surface relative

to the track at the point of contact, 𝐾2̃, is provided as,

𝑲�̃� = 𝑹𝜓𝑲𝑠𝑹𝜓 (9)

with [−𝜔𝑦 𝜔𝑥]𝑇 = [−𝜔𝑦𝑇𝑖 𝜔𝑥𝑇𝑖]𝑇 and the relative rotational

velocities as [−𝑣𝑥 𝑣𝑦]𝑇 = [𝑣𝑥𝑇𝑖 𝑣𝑦𝑇𝑖]𝑇. Equations 6-8

represent 5 equations each for each track and similar equations

for the caster given in [21], i=L, R, C or 15 equations total to

solve for the contact point velocities along the tracks and

climbing surface. �̇�𝑤 , �̇�𝑠 are mapped into the contact parameters

as:

�̇�𝑖 = �̇�𝑠𝑖(1)

𝑅𝑐 (10)

And

�̇�𝑖 = �̇�𝑠𝑖(2)

𝑅𝑐 (11)

IMPLEMENTATION

The model presented above is solved as an initial value problem

with the initial sate of the robot given as well as a desired motion

in terms of longitudinal and angular velocity defined in the local

robot frame. The initial conditions define initial configuration of

the platform as well as the location of the SSMR contact points

on the spherical surface. The robot then begins motion with the

defined velocities, these can be constant or changing in time. At

each time step, the configuration velocities are first found along

with slipping velocity. This gives the set of 𝜔𝑥,𝑦,𝑧 , 𝑣𝑥,𝑦for each

contact point needed to solve the contact equations. The

numerical integrator function ODE45 is then implemented to

solve the differential equations associated with contact leading

to a new locations of the track contact on the climbing surface.

The robot configuration is then updated to satisfy the kinematic

constraints contained in the model and the process is then

repeated to simulate motion along a path.

RESULTS

This section presents a series of maneuvers performed by a

SSMR on a spherical surface using the model defined in the

previous sections. The maneuvers are considered typical in

manufacturing environments and are divided into three cases.

The first case considers a single maneuver performed on multiple

spherical surfaces of varying radius. The second case considers

multiple maneuvers on a single spherical surface of constant

radius with the addition of translation velocities (slip) of the

tracks contact point. The third case presents a single maneuver

on a single spherical surface in which the track instant centers

are located outside of the robot chassis to model slipping.

Case 1: Robot Maneuvers on Spheres of Varying Radius:

The first case considers a single maneuver performed on multiple

spherical surfaces of varying radius. The first example in case 1

is selected for this maneuver to give a contractible closed-path

on a spherical surface. This maneuver is performed on spheres

of radius varying from Rc = 1e5, 1e3, and 1e2 units. The results

are shown in Figures 4-6 which represent locally planar

conditions resulting in a purely planar circular path. Figure 6

5

displays that as Rc decreases the paths remain closed and become

increasingly non-planar.

Case 2: Robot maneuvers on single sphere with prescribed

slipping:

The second case considers two different maneuvers of the robot

on a single sphere. Each maneuver is defined as set of constant,

open-loop kinematic commands defined in the robot frame on a

spherical surface of radius Rc=1e3. In the first example the robot

operates at a constant linear and angular velocity and zero

translation velocity, (no slip) while the second incorporates

varying amounts of prescribed translation velocities. These

results show the paths of the left track contact patch on the

spherical surface. Figure 5 shows the contact patches move in

closed paths which generally demonstrates the robustness of the

numerical integration method over an extended period of travel.

Further, the results show that when a robot operates on a surface

that can be considered locally planar the addition of translation

velocities does indeed affect the path of travel of the SSMR.

Case 3: Slip varying with Instant Center Location

The third case considers the effect of slip when subject to varying

instant center locations for the left and right track. Each

maneuver is defined with a constant forward velocity and

angular velocity. The translation velocities of each track are then

defined by prescribing a distance from the center of the robot

chassis out to the instant centers of the left and right tracks

respectively. The translation velocity in the x direction, 𝑣𝑥=

𝜔(𝑦𝑟𝑖𝑐 − 𝑏) with 𝑦𝑟𝑖𝑐 = 𝑥𝑏 and in the y direction, 𝑣𝑦 = 0 are the

slip velocities. The maneuvers are performed on a spherical

surface, Rc=1e3. These results are shown as the path of the left

track contact location on the spherical surface. It can be seen

from figure that as the value of the instant center location

increases a more significant amount of slip is experienced by the

Figure 4: Robot Path on Sphere of Radius 1e5

Figure 5: Robot Path on Sphere of Radius 1e3

Figure 6: Robot Path on Sphere of Radius 1e2

Figure 7: Robot Traveling with various Translational

Velocities

6

vehicle and when the instant centers are located in the center of

the tracks, no slipping is incurred.

DISCUSSION AND CONCLUSIONS

The work presented in this paper is a method used to determine

the trajectory of a skid steer mobile robot using a geometric

representation of the contacting surfaces, i.e. tracks and spherical

climbing surface. The model discussed above is presently

applied to a spherical surface which yields a varying degree of

curvature based on the radius of the sphere. The ability to vary

the degree of curvature lends itself to more accurately represent

a non-planar surface such as those seen in manufacturing

environments. The main goal of the paper is to provide a

preliminary model used to evaluate the effect of slip on the

motion of a skid steer mobile robot when navigating along an

uneven surface.

Some discussion of the method follows. The current model is

constructed with a mobile robot model resembling a skid steer

platform. The assuming a holonomic-form of the mobile robot as

shown in Figure 3, and by applying the kinematic constraints,

Equation 3, are able to solve for a viable robot velocity (vr). The

integration of the contact equations yields a new contact point

location on the surface and track as well as yields a updated

orientation of the track frames relative the spherical surface

frame. This progresses forward until either the desired task is

completed or integration time has expired.

This formulation satisfies the same set of constraints as the one

by [20,21], however instead of assuming that the motion of the

robot platform is unaffected by slipping conditions, 𝑣𝑥 = 𝑣𝑦 =

0. The current paper incorporates the effects of slip, and assumes

this effects the platform motion, which is inherently desired

when modeling the motion of the skid steer platform. It can be

seen in [19] and from Equation 6-8, since the tracks are

represented as a planar surface, the track surfaces are

characterized with M=I, K=0, T=0; thus resulting in the negation

of any effects of slip that would occur on the climbing surface.

In order to determine the effect of slipping on the climbing

surface the authors projected any slipping experienced on the

tracks onto the climbing surface. The slip included in the

discussed model was incorporated by either 1.) prescribing the

translational velocities themselves, Figure 7, or 2.) calculating

the translational velocities based on the location of the instant

centers of the left and right tracks, Figure 8.

The results provided demonstrate the skid steer platform

operating on a spherical surface and oriented in a climbing

configuration. The radius of the spherical surface (𝑅𝑠) varies in

size from 5 to 500 times larger than the width of the robot. It

must be noted that surface imperfections such as pitting, divots,

or weld seams have been ignored due to the relative small

dimension when compared to the size of the climbing surface. It

can be seen from Figures 4-6 that when provided in a constant

linear and angular velocity the resulting tracks are closed

contractable paths. Figures 4-5 detail that when the radius of the

sphere, (𝑅𝑠), is much greater than the dimension of the robot the

path is considered locally planar thus resulting in a purely

circular path. However, when the sphere radius, (𝑅𝑠), approaches

the dimensions of the robot the path is no longer a pure circle.

Figures 7-8 also provide information on the effects 𝑣𝑥 , 𝑣𝑦 , and

the left and right track instant centers have on the path of the

mobile robot platform. It can be seen from Figure 8, when the

instant center is located a length b from the chassis the effect of

slipping is minimized, and when instant center distance increases

the resulting amount of slip experienced increases.

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