1 Copyright © 2013 by ASME

Proceedings of the ASME 2013 International Design Engineering Technical Conference & Computers and Information in Engineering Conference

IDETC/CIE 2013 August 4-7, 2013, Portland, Oregon, USA

DETC2013-12401

KINEMATIC MODELING, ANALYSIS AND CONTROL OF HIGHLY RECONFIGURABLE ARTICULATED WHEELED VEHICLES

ABSTRACT

The Articulated Wheeled Vehicle (AWV) paradigm

examines a class of wheeled vehicles where the chassis is

connected via articulated chains to a set of ground-contact

wheels. Actively- or passively- controlled articulations can help

alter wheel placement with respect to chassis during locomotion,

endowing the vehicle with significant reconfigurability and

redundancy. The ensuing ‘leg-wheeled’ systems exploit these

capabilities to realize significant advantages (improved stability,

obstacle surmounting capability, enhanced robustness) over both

traditional wheeled-and/or legged-systems in a range of uneven-

terrain locomotion applications. In our previous work, we

exploited the reconfiguration capabilities of a planar AWR to

achieve internal shape regulation, secondary to a trajectory-

following task. In this work, we extend these capabilities to the

full 3D case – in order to utilize the full potential of kinematic-

and actuation- redundancy to enhance rough-terrain locomotion.

INTRODUCTION Articulated Wheeled Vehicles (AWVs) consist of a principal

vehicle-chassis connected to a set of wheels with ground contact

via actively- or passively-articulated chains. The presence of the

so-called ‘articulated leg-wheel’ chain endows reconfigurability,

by allowing relocation of the wheel with respect to the chassis.

The AWV paradigm offers immense possibilities for

enhanced locomotion-performance of autonomous mobile

robots while assuring the reliability of the system. For instance,

the articulated-suspensions permit the vehicle to change the

location of center of mass by adjusting the linkages/joints, so as

to avoid the rollover when passing the uneven terrain [1, 2].

Similarly, the vehicle can reduce its footprint to pass narrow

doorways or expand the wheelbase when stability is required.

The resulting class of Leg-Wheel Vehicles (as the AWVs are also

called) have many potential applications (ranging from planetary

exploration [3, 4], agriculture [5, 6] to military and rescue

operations) due to their advantages over traditional wheeled

systems.

Figure 1: Schematic of Leg-Wheel Vehicles.

In most applications, the wheels of an AWR are considered

to be rigid discs with a single point-of-contact with the terrain.

The wheel velocities are governed by a set of non-holonomic

constraints, which permit rolling along the disk-plane without

allowing lateral side-slip. However, potentially, these velocity-

level constraints can be violated, but results in slipping and

skidding. Minimization of slipping and skidding is usually

desired (both from the perspective of reducing the energy

dissipation and improving measurement uncertainty) and can be

achieved by adding intermediate articulations. Additionally, the

articulations and actuation within the leg-wheel chain, serves to

Aliakbar Alamdari Mechanical and Aerospace

Engineering,

SUNY at Buffalo Buffalo, NY, 14260

aalamdar@buffalo.edu

Xiaobo Zhou Mechanical and Aerospace

Engineering,

SUNY at Buffalo Buffalo, NY, 14260

xzhou9@buffalo.edu

Venkat N. Krovi Mechanical and Aerospace

Engineering, SUNY at Buffalo

Buffalo, NY, 14260 vkrovi@buffalo.edu

2 Copyright © 2013 by ASME

redirects the motion and forces from axle prior to their reaching

chassis serving as an effective suspension. Thus, the choices for

the topology, dimension and location of articulations (and

subsequent actuation) affects the overall performance of the

vehicle.

Proper design and control schemes for the articulated-leg-

wheel system are needed to achieve proper coordination of the

rolling and steering of the wheels and serves to motivate our

efforts. There is a clear need to quantitatively examine the role

of topologies, dimensions, and configurations of wheels-

articulation within the individual sub-chain as well as the overall

attachment of the sub-chains to the central chassis. In our

previous work [7, 8], we explored the role of that design

selection process by developing systematic analysis and

evaluating performance of the planar version AWR. In this paper,

we extend it into the full 3D case, focusing on exploiting the

kinematic- and actuation-redundancy for reconfigurability of the

system while enhancing the stability of the vehicle in rough

terrain locomotion.

The rest of the paper is organized as follows: In Section 2

we briefly survey the background and literature on other AWRs.

Section 3 briefly examines the development of a computational

approach for twist-based kinematic modeling followed by the

kinematic modeling of a Wheel Leg Actively Articulated Vehicle

(WLAAV) as a concrete case study in Section 4. In Section 5, a

kinematic control scheme is developed for realizing a primary

trajectory-following task while secondarily ensuring constant

roll-pitch by the WLAAV chassis. Finally, Section 6 studies the

performance via simulations followed by some remarks on

ongoing work in Section 7.

BACKGROUND The main research in passive AWRs concerns designing

suspension mechanism to negotiate with uneven terrain. They

change their configuration according terrain topology. Passive

AWRs such as Shrimp [4], Sojourner [10] and Nomad [11] are

designed to have fewer DOF such that the weight of the system

can be supported by the structure. The main advantages of

passive AWRs are in terms of power consumption, payload

capacity and controller design.

Actively articulated vehicles enhance the mobility of the

robots to obtain better performance such as stability and traction,

as demonstrated by Sample Return Rovers (SRR) [12], Athlete

[13], Workpartner [14], Hylos [15] and Azimut [16]. On the other

hand, more actuators, extra weight and control complexity will

be added to the system.

Kinematic modeling of ordinary wheeled mobile robots

[17] as well as classification scheme [18] have been explored.

Tarokh et al. [19] explored kinematic modeling, analysis and

balance control for high mobility wheeled rover traversing

bumpy terrain and illustrated using the highly articulated multi-

task rover (MTR) example.

Similarly, Grand et al. [20] proposed a general kinetostatic

formulation for articulated wheeled rovers moving on uneven

terrain. Their method was applied to the motion control of

wheeled-legged rover based on the decoupling of the posture and

trajectory parameters.

The estimation of wheel terrain contact angles based on

extended Kalman filter for improving ground traction and to

reduce power consumption was investigated in [21]. Two

innovative approaches were presented in [22] for wheeled-

legged vehicle, by defining adhesion and stability coefficients as

traversability indicators. The authors also propose an innovative

optimal force distribution method to make the front and rear

wheel adhesion coefficients equal. Jarrault et al. [23] explored

the contact stability optimization of robust obstacles passage for

high mobility wheeled legged robot. Their optimization

algorithm used both kinematic redundancies to modify the

position of center of mass of rover and distributing of contact

forces, and the actuation redundancy to improve the frictional

contacts.

In our own work, we focus on developing a systematic and

general-purpose modeling, analysis and operational framework,

suitable for both the design and control of such articulated

wheeled vehicles. We build upon our previous work [7, 8] and

focus on extending it to aid the design, analysis and control of a

fully 3D vehicle traversing significantly rough uneven terrain.

COMPUTATIONAL TWIST-BASED KINEMATIC MODELING

Figure 2 depicts a general model of the wheeled-legged

actively articulated vehicles (WLAAV). We define the inertial

frame of reference {F} = (Of,X,Y,Z), the robot platform has a

frame {B}= (Ob,bx,by,bz) attached to the body at the center-of-

mass of platform Ob. This allows us to easily examine both

translational , ,b b bx y z and rotational velocities , ,b b b of

the frame {B} with respect to the inertial frame ( b is the roll,

b is pitch, and b is yaw angle).

A general vehicle chassis is assumed to possess ‘n’ leg-

wheel branches, each with an arbitrary number of articulations

and ending with disk-wheel. The {W}= (Ow, wx,wy,wz),

coordinate frame of the wheel, is attached to the wheel axle, and B

WA is the homogenous transformation that allows coordinates

in the {W} to be expressed in the {B} frame. In our modeling,

each wheel is assumed to be represented by a rigid disc in contact

with a non-deformable terrain, i.e. wheel-terrain contact

mechanics are considered in this simplified form in order to

facilitate development of the control scheme.

3 Copyright © 2013 by ASME

Figure 2: General articulated wheel robot.

The last frame in each leg-wheel chain is the wheel terrain

contact frame {C}= (Oc,Cx,Cy,Cz), with its x-axis is tangent to

the terrain at the point of contact and y-axis is normal to the

terrain.

x z zC C W and y x zC C C

is contact angle defined as the angle between xC and zW

(where xC always lies in the wheel plane). This angle can be

measured using force sensor on the wheel axle, or could be

estimated by Kalman filtering [12]. This angle is considered

constant and equal to zero for flat surface moving. Following the

formulation conventions from Murray et al. [24], the twist matrix

of frame B w.r.t frame A (and expressed in frame A) is given by:

1[ ] A

A A A

B B BT A A (1)

and can re-express the twist matrices/vectors in any convenient

frame of reference via the Adjoint transformation. The velocity

of contact frame in frame F can be obtained by:

1 ...F F F F F

F B m W F

B W C CV V V V V (2)

The Adjoint transformationC

F can be easily written by given

the homogenous transformation ( , )C C C

F F FA R r .

ˆ

0

C F O C

F FC

F C

F

R r R

R

(3)

ˆF Or is a 3 3 skew symmetric matrix, and allows for twist

vectors to be transformed from frame {F} to frame {C}. Using

the chain rule for homogenous transformations (and

differentiating it), one can express the twist of the contact point

respect to the fixed frame in the contact frame of reference as: 1

1 1 ...C B

F C F C B

C B B

W CC m W

W W C

V V V

V V

(4)

Individual joint twist can now be re-expressed as:

1 1 16 6 6 1

C B CB C B B

BV V t (5)

Allowing equation (4) to be written in compact form as:

C F

F C F

C F CV V Bq (6)

where, 1 2

2 3[ ... ]C C C C

m W

W CB t t t t is twist

assembled matrix and is a vector consists of m joint variables

through the chain and wheel rotational velocity . Various

contact constraints can be easily imposed in the contact frame of

reference. E.g. If pure rolling condition is assumed to be true, the

constraints at the contact point can be represented as:

4 6 6 1[0 0 0 0]

CT F T

CS V (7)

where 4 4

6 4

2 2

[ ]0

IS

is a wrench basis selector matrix. The

matrix S represents the direction where force can be exerted, and

selects the first four rows of twist vector and restricts the

translation motion at the contact point. Substituting Eq. (7) into

(6), the constraint conditions for pure rolling of wheel of

branches can be expressed as:

i

BT C F

B B

P

S V Bq (8)

or

[ ] B

F

i BP V Aq (9)

Assembling of all constraints for all legs of AWV, we could

assemble the kinematics equation from equation (9).

1

2 1

.

.

0

0

BF

B

n

n

P

P A

V q

A

P

(10)

This kinematic model is very useful for a variety of reasons:

First, it forms the basis for the kinematic control of the AWV

while avoiding the slip. Note that the contact velocity C

F

BV

has been eliminated from the equation. Second, the static model

can also be easily extracted, which now relates the contact force

act on the wheel to the torques exerted at the joints and the total

other wrenches applied to the robot.

CASE-STUDY: THE WHEELED-LEGGED ACTIVELY ARTICULATED VEHICLE (WLAAV)

4 Copyright © 2013 by ASME

Figure 3: Kinematic modeling of the WLAAV.

The WLAAV has three leg-wheel sub-chains attached

symmetrically to a triangular platform. Each leg wheel system

features a spatial pantograph mechanism and has five DOF. Four

of these articulations of spatial pantograph have motors attached.

We now perform the kinematic analysis of a highly mobile

maneuverable and reconfigurable wheeled legged actively

articulated vehicle shown in Figure 3. One motor powers the

wheel; one motor for steering of caster and one motor for

changing the elevation of the link (a2) with respect to the

platform and another one attached to the link (a1) where can

change the geometry of the leg.

The linear spring/damper in each leg plays an important

role and acts like a shock absorber and from ground and platform

weight carrier in a parallelogram consisting of four bars. In this

part, we assign coordinate frames, as shown in Figure 4, and set

up the homogenous transformation between the various frames

of references.

2

2

3

1 0

1 0

0 0 2 0

0 0 2 0

( ) in( ) 0 ( ) ( )

in( ) ( ) 0 ( ) ( ),

0 0 1 0

0 0 0 1

( ) 0 ( ) ( )

( ) 0 ( ) ( )

0 1 0 0

0 0 0 1

B

C

C

C

Cos S a Cos a Cos

S Cos a Sin a SinA

Cos Sin a Cos

Sin Cos a SinA

4

4

1 0 3 0

1 1 3 03

( ) ( ) 0 ( )

( ) ( ) 0 ( ),

0 0 1 0

0 0 0 1

0 1 0

( ) 0 ( ) ( )

( ) 0 ( ) ( )

0 0 0 1

C

C

c c cC

A

c c c

Cos Sin a Cos

Sin Cos a SinA

m

Cos Sin nCosA

Sin Cos nSin

( ) 0 ( ) ( )

( ) 0 ( ) ( )

0 1 0 0

0 0 0 1

A

C

Cos Sin rCos

Sin Cos rSinA

(11)

Figure 4: Parameters of wheel-leg.

Twist vectors expressed in local frames are found as:

[ , , , , , ]B

F T

B b b b b b bV x y z (12)

2

2 1[0 0 0 0 ]C

B T

CV a 3

2

3

43

4

4

2 0 0

3 1 1

[0 0 0 0]

[0 0 0 0 ]

[0 0 0 0]

[ 0 0 0 0]

CC T

C

CC T

C

AC T

A c c

CA T

C

V a

V a

V n

V r

Contact twist expressed in contact frame is: 32

2

2 2 2 3

43 4

3 4

CCC BCF C F C B C

C B B C C C C

C A CC CC C A

C C A A C

V V V V

V V V

(13)

Applying the non-holonomic constraints to the system using

equation (7) we get:

1

2 1

.

.

0

0

BF

B

n

n

P

P A

V q

A

P

(14)

Constraints at contact points restrict the motion of the

components of the system such that the cooperation of the

actuators is not straight forward and redundancy in the system

further complicates the control. One can systematically deal with

the overall effect of all constraints on the motion by using our

twist based modeling approach, and then a controller could be

designed by putting the kinematic model into different forms.

KINEMATIC CONTROL OF WLAAV We have formed the kinematic equations of WLAAV in the

previous part. The purpose of this spatial articulated vehicle

design is to enable the robot to change the arrangement of its legs

while moving around on uneven terrain.

Thus there are two control tasks, the first one is path

following in Cartesian space (x, y, z), and the second one is

5 Copyright © 2013 by ASME

posture regulation for controlling the platform’s Pitch, Roll and

Yaw. The robot has 15 internal articulations, with 12 actuated

ones, and the rest passive. Three joints are activated whenever

the changing of elevation of the chassis is required; otherwise

these three joints are turned off. The robot possesses both

kinematic redundancy and actuation redundancy that needs to be

resolved.

The basic control law we will use is close loop resolved

motion rate control (RMRC). In RMRC, Jacobian matrix is used

to map the desired velocity in task space to the velocity in joint

space. And certain velocity controllers are implemented to track

the joint velocity signal. The task space variables are (

, , , , ,b b b b b bx y z ), and controlled variable are ( 1 1 1 1

0, , , ,c

2 2 2 2 3 3 3 3

0 0, , , , , , ,c c ). In this control scheme, we will use

the augmented kinematics method to resolve the redundancy.

Note that bX is desired base motion expressed in inertial

frame, in order to use the inverse kinematic model we transform

into body frame by a rotational matrix.

, , ,

0 0 1 0 0

0 0 1 0 0

0 0 1 0 0

b b b

B

F z y z

b b b b

b b b b

b b b b

R R R R

C S C S

S C C S

S C S C

(15)

This control method is a more general method that could be

potentially used for other AWRs. As there are two control tasks

and the path following part has the first priority, we could use

pseudo inverse and potential function techniques to resolve

redundancy. Recall that the kinematic equation is:

bJX Aq (16)

We first find desired base motion d

bX .Then, to resolve the

kinematic redundancy; we use the following control scheme. # #( )[ ( )]

ph

d d

b b b

q q

q J A X K X X I J J z (17)

where # 1( ) T TJ J JJ is pseudo inverse of J. The overall control

scheme is shown in Figure 5. The particular solution ( pq )

enables the AWV to achieve the primary task (path following by

chassis). The homogeneous part ( hq ) now serves to achieve the

secondary task (regulation of leg configuration).

Figure 5: Redundancy resolution control scheme.

SIMULATION RESULTS The simulation is done in MATLAB with the WLAAV

parameters shown in Table 1. Note that all three leg-wheel sub-

chains are assumed to be identical and symmetrically located

around the chassis.

Table 1: Simulation parameters of the WLAAV

0 160a mm 45m mm

1 35a mm 30n mm

2 110a mm 30r mm

3 110a mm 0 0 0

1 2 30 , 120 , 240

In this simulation, the desired vehicle path and velocity are given

and applied to the system and are the primary requirement while

the posture controller is the secondary criterion. The path

following task is to drive the vehicle to track a circular path with

center at (0, 0) and radius 1000mm. the time trajectories are

given by ( )d

bX Rcos t a and ( )d

bY R sin t b where R is the

radius and ‘a’ and ‘b’ are the coordinates of the center of the

circle. This sinusoidal trajectory will effectively change the

configuration of the legs and wheels. Furthermore, the desired

path is changing sinusoidal in z direction 5 (5 )d

bZ sin t . It is

noted that the leg angles change in such a way to balance the

vehicle.

The control performance of control scheme is shown in

Figure 6. We could see clearly that the vehicle could follow the

desired path smoothly. This result indicates the capability of the

WLAAV for various applications that require the configuration

to change its shape to avoid obstacles and improve stability. The

redundancy problem in the system has been resolved.The error

between the actual and desired paths is shown in the figure which

can be further reduced by using a tuned PID controller.

Figure 6: Simulation for path following.

-1000-500

0500

1000

-1000

-500

0

500

1000

-50

0

50

X (mm)

Trajectory Tracking in Sinusoidal Circular Path

Y (mm)

Z (

mm

)

Desired Path

Actual path

6 Copyright © 2013 by ASME

DISCUSSION The reconfigurability and redundancy of leg-wheel

articulated vehicles creates significant benefits but these

capabilities need to be done by careful modeling, analysis and

control. In this paper, we developed a generic twist-based

kinematic modeling methodology for spatial articulated wheeled

vehicles to help systematize the modeling of such complicated

systems for following analysis and control. The model is a very

general one and could be applied for any articulated system with

active suspensions. The model is used to control the posture of

the robot and its static stability. Results show the validity of the

model and the feasibility of this approach. The main feature of

the work is its generality, e.g. dealing with both active (actuated)

and passive (compliant) joints and linkages. This framework was

deployed in the case-study of a wheeled actively articulated

reconfigurable vehicle capable of such reconfiguration. The

kinematic control scheme are deployed and the results are

studied to solve the issues of maintaining kinematic consistency

of the constraints and resolving the redundancies inherent in such

articulated wheeled robots.

ACKNOWLEDGMENTS This work was supported in part by the National Science

Foundation Award CNS-1314484.

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