SELECTION OF DIFFERENT PATHS FOR DOUBLY

NONHOLONOMIC MOBILE MANIPULATORS

Alicja Mazur and El˙zbieta Roszkowska

Institute of Computer Engineering, Control and Robotics,

Wrocław University of Technology, ul. Janiszewskiego 11/17, 50-372 Wrocław, Poland

Keywords:

Mobile manipulator, Nonholonomic constraints, Path following, Frenet parametrization.

Abstract:

This paper describes a problem of designing control laws preserving a motion along desired path for doubly

nonholonomic mobile manipulators. The doubly nonholonomic mobile manipulator is composed of a mobile

platform moving without slipping effect between wheels and the surface (i.e. nonholonomic platform) and an

onboard manipulator equipped with special nonholonomic gears, designed by Nakamura, Chung and Sørdalen.

A task for any nonholonomic subsystem (i.e. nonholonomic platform or nonholonomic manipulator) is to

follow a desired path – some geometric curve parameterized by curvilinear distance from selected point. A

description of the nonholonomic subsystem relative to the desired path using so-called Frenet parametrization

is a basis for formulating the path following problem and designing a kinematic control algorithm.

1 INTRODUCTION

A mobile manipulator is deﬁned as a robotic system

which consists of a mobile platform, equipped with

non–deformable wheels, and a manipulator mounted

on the platform.

The problem with a path deﬁnition for the end-

effector of the mobile manipulator is that behavior

of the subsystems is unpredictable, because the same

path deﬁned in global coordinates can be executed by

separate subsystems or by both of them. Sometimes,

it is important to move the platform and simultane-

ously unload a payload: such a task is deﬁned rela-

tive to the base of the manipulator mounted on the

platform; deﬁnition relative to the end-effector is ill-

conditioned. In such situation the decomposition of

the task into tasks deﬁned separately for both subsys-

tems is more natural and convenient (Mazur and Sza-

kiel, 2009).

In the paper we assume that a desired task can be

decomposed into two separate subtasks deﬁned for

each subsystem independently: the end-effector has

to follow a desired geometric path described relative

to the base of manipulator (i.e. relative to the plat-

form) and the task of the platform is to follow a de-

sired curve lying on a plane. Such a formulation of

the task makes possible successive unloading of pay-

load transported by the mobile manipulator during the

control process.

P(s)

Figure 1: Decomposition of a task for doubly nonholo-

nomic mobile manipulator: P(s) – desired path for a non-

holonomic platform, Π(s) – desired path for a nonholo-

nomic manipulator.

Taking into account the type of components mo-

bility of mobile manipulators, there are 4 possible

conﬁgurations: type (h,h) – both the platform and

the manipulator holonomic, type (h,nh) – a holo-

nomic platform with a nonholonomic manipulator,

type (nh, h) – a nonholonomic platform with a holo-

nomic manipulator, and ﬁnally type (nh,nh) – both

the platform and the manipulator nonholonomic. The

notion doubly nonholonomic manipulator was intro-

duced in (Tcho´n et al., 2004) for the type (nh,nh).

Mazur A. and Roszkowska E. (2010).

SELECTION OF DIFFERENT PATHS FOR DOUBLY NONHOLONOMIC MOBILE MANIPULATORS.

In Proceedings of the 7th International Conference on Informatics in Control, Automation and Robotics, pages 15-21

 SciTePress

2 MATHEMATICAL MODELS OF

DOUBLY NONHOLONOMIC

MOBILE MANIPULATOR

In this paper we restrict our considerations to mobile

manipulators of (nh,nh) type, i.e. to doubly nonholo-

nomic objects. We will only discuss the constraints

occurring in the motion of both subsystems. Non-

holonomic constraints appearing in the motion of me-

chanical systems come from different sources. Very

often they come from an assumption that a motion of

the system can be treated as pure rolling of compo-

nents, without slippage effect. We have taken such as-

sumption in description of constrained motion of the

considered mobile manipulator.

2.1 Nonholonomic Constraints for

Wheeled Mobile Platform

Motion of the mobile platform can be described by

generalized coordinates q

∈ R

and generalized ve-

locities ˙q

∈R

. The wheeled mobile platform should

move without slippage of its wheels. It is equiva-

lent to an assumption that the momentary velocity at

the contact point between each wheel and the motion

plane is equal to zero. This assumption implies the

existence of l (l < n) independent nonholonomic con-

straints expressed in Pfafﬁan form

A(q

) ˙q

= 0, (1)

where A(q

) is a full rank matrix of (l×n) size. Since

due to (1) the platform velocity is in a null space of

A(q

), it is always possible to ﬁnd a vector of special

auxiliary velocities η ∈ R

, m = n−l, such that

˙q

= G(q

)η, (2)

where G is an n ×m full rank matrix satisfying the

relationship AG = 0. We will call the equation (2) the

kinematics of the nonholonomic mobile platform.

2.2 Nonholonomic Constraints for

Manipulator

A rigid manipulator can be a holonomic or a non-

holonomic system – it depends on construction of its

drives. In (Nakamura et al., 2001; Chung, 2004) the

authors have presented a new nonholonomic mechan-

ical gear, which could transmit velocities from the

inputs to many passive joints, see Figures 2-3. The

prototype of 4-link manipulator with such gears has

been developed in last 1990th years. Similar non-

holonomic 3-link manipulator is under construction at

Pozna´n University of Technology, Poland (Michałek

and Kozłowski, 2004). The nonholonomic constraints

in the gear appear by assumption on rolling contact

without slippage between balls of gear and wheels in

the robot joints.

Figure 2: Schematic of the nonholonomic gear.



1,1

2,1



Figure 3: Nonholonomic gear seen from above.

The basic components of the gear presented in

Figure 2 are a ball and three wheels – an input wheel

IW and two output wheels OW

and OW

. The veloc-

ity constraints of the ball are only due to point contact

with the wheels. The input wheel IW is mounted in

the ﬁrst joint, the output wheels are mounted in the

next joint. The wheel IW rotates around the ﬁxed

axis α

with an angular velocity u

, which plays the

role of a control input. The rotating input wheel IW

makes the ball rotate. The wheel OW

rotates around

an axis α

, which makes with the axis of the input

wheel a joint angle θ

. The angular velocity

= u

is the second control input for the manipulator with

nonholonomic gears.

Nonholonomic constraints (the kinematics) of n-

pendulum have the form

= u

, (3)

= a

sinθ

i−1

i−2

∏

j=1

cosθ

, i ∈{2, ... ,n}, (4)

with positive coefﬁcients a

depending on gear ratios.

Hypothetical manipulator with 3 links and nonholo-

nomic gears has been presented in Figure 4.

Figure 4: Schematic of 3-link nonholonomic manipulator.

It is worth to emphasize that only two inputs u

and u

are able to control many joints of manipulator

equipped with gears designed by Nakamura, Chung

and Sørdalen.

3 DESCRIPTION OF

NONHOLONOMIC SYSTEM

RELATIVE TO A GIVEN PATH

For nonholonomic systems whose workspace is pla-

nar, it is possible to describe state variables relative to

global inertial frame as well as to a given path (Mazur,

2004), see Figure 5.

M’

Figure 5: Illustration of path following problem for the non-

holonomic platform.

The path P is characterized by a curvature c(s),

which is the inversion of the radius of the circle tan-

gent to the path at a point characterized by the param-

eter s. Consider a moving point M and the associated

Frenet frame deﬁned on the curve P by the normal

and tangent unit vectors x

and

. The point M is

the mass center of a mobile platform and M’ is the

orthogonal projection of the point M on the path P.

The point M’ exists and is uniquely deﬁned if

the following conditions are satisﬁed (Fradkov et al.,

1999):

• The curvature c(s) is not bigger than 1/r

min

> 0.

• If the distance between the path P and the point M

is smaller than r

min

, there is a unique point on P

denoted by M’.

The coordinates of the point M relative to the Frenet

frame are (0, l) and relative to the basic frame X

are

equal to (x,y), where l is the distance between M and

M’. A curvilinear abscissa of M’ is equal to s, where

s is a distance along the path from some arbitrarily

chosen point.

If we want to express the position of the point M

not in coordinates (x, y) relative to inertial frame, but

relative to the given path P, we should use some geo-

metric relationships, (Mazur, 2004),

l = (−sinθ

cosθ

)



˙x

˙y



, (5)

˙s =

(cosθ

sinθ

)

1−c(s)l



˙x

˙y



, (6)

where ˙x and ˙y are deﬁned by nonholonomic con-

straints for the system (wheeled mobile platform or

nonholonomic manipulator) and θ

is a desired orien-

tation at the point M’ on the path.

3.1 Path Following with Desired

Orientation

3.1.1 Nonholonomic Mobile Platform

Posture of the mobile platform is deﬁned not only by

the position of the mass center, but by the orientation,

too. For this reason, it is necessary to deﬁne the ori-

entation tracking error equal to

θ = θ−θ

. Moreover,

at the point M’ the desired orientation of the platform

fulﬁlls a condition, (Samson, 1995),

= c(s)˙s. (7)

Then the coordinates

ξ = (l,

θ,s)

(8)

i.e. the Frenet coordinates (l,s) and orientation track-

ing error

θ, constitute path following errors for non-

holonomic mobile platform. It is worth to mention

that Frenet parametrization is valid only locally, near

the desired path.

As we mentioned earlier,out of Frenet coordinates

it is enough to consider only l and

θ. Due to expres-

sions (5) and (7), Frenet variables for mobile platform

of (2,0) class described by nonholonomic constraints





˙x

˙y









cosθ 0

sinθ 0

0 1









= G(q

)η (9)

can be deﬁned as follows

(

l = vsin

θ,

θ =

θ−

= ω−vcos

c(s)

1−c(s)l

= w,

(10)

where w is a new control input for the second equa-

tion.

For the system (10) we can use many control

laws, e.g. algorithm introduced in (Samson and Ait-

Abderrahim, 1991),

(

v = const

w = −k

sin

−k

θ, k

> 0,

(11)

which is asymptotically stable. It can be shown using

the following Lyapunov-like function

V(l,

θ) = k

(12)

and Barbalat lemma.

Path following with a desired orientation is very

important for mobile systems, especially for mobile

manipulators. It comes from the fact that it would

be impossible to unload a payload if the platform had

wrong orientation, i.e. it would be in a right place but

back-oriented.

3.1.2 Nonholonomic Manipulator

For a nonholonomic manipulator it is possible to fol-

low along the desired path with prescribed orienta-

tion. However, this issue has a drawback. Namely,

nonholonomic manipulator has only two control in-

puts, therefore it is impossible to have the mentioned

three parameters (l, s,

θ) under control. In such a case

many authors decide to regulate only two tracking er-

rors (l,

θ) to zero and they omit the differential equa-

tion for ˙s, because it does not matter at which point

s of the desired path the mobile platform enters the

desired curve P(s), see (Fradkov et al., 1999) for de-

tails. Such a case of the path following problem we

will call the asymptotic path following.

The Frenet parametrization can be evoked once

again in the problem of path following for the pla-

nar manipulator with nonholonomic gears moving on

the XZ surface.The role of the point M in Figure 5

plays a point at the end of a gripper. The orientation

of the end-effectorθ

is an rotation angle of the frame

associated with the gripper around –Y

axis, which is

located in the base of the manipulator. It means that

the orientation of the end-effector in the planar non-

holonomic n-pendulum is then equal to

∑

i=1

In the considered planar nonholonomic manipulator

lying in XZ-plane, relationships between velocity of

the working point M expressed in Cartesian and curvi-

linear coordinates have the form

= (−sinθ

cosθ

)



˙x

˙z



, (13)

˙s =

(cosθ

sinθ

)

1−c(s)l



˙x

˙z



, (14)

where l

denotes distance between the point M and

the path Π(s), and θ

is the orientation of the Frenet

frame in the point M’. Subscripts were introduced to

distinguish Frenet variables for both subsystems of

the (nh,nh) mobile manipulator.

Coordinates of the end-effector in the n-pendulum

relative to its base are equal to







x =

∑

i=1

cos



∑

j=1



z =

∑

i=1

sin



∑

j=1



(15)

Substituting time derivativesof variables (15), we ob-

tain the following equations

∑

i=1

cos

−

∑

j=1

∑

k=1

, (16)

−c(s) ˙s =

∑

i=1

−

c(s)

1−c(s)l

∑

i=1

sin

−

∑

j=1

∑

k=1

. (17)

Using the kinematics of the nonholonomic manipula-

tor given by (3)-(4), the equations (16) and (17) can

be expressed in the matrix form as follows





= H(q

,ξ

)













= H(q

,ξ

)u = K

,ξ

)u. (18)

Matrix K

,ξ

) fulﬁlls the regularity condition (i.e.

it is invertible) if some conﬁgurations, which imply

the matrix singularity, are excluded from a set of pos-

sibly achieved poses of the nonholonomic manipula-

tor.

For nonholonomic 3-pendulum matrix K

,ξ

)

has the form

,ξ

) =



l11

l12

l21

l22



with elements deﬁned below

l11

∑

i=1

cos(θ

−

∑

j=1

l12

= a

∑

i=2

cos(θ

−

∑

j=1

) +

cos(θ

−

∑

j=1

l21

= 1−

c(s)

1−c(s)l

∑

i=1

sin(θ

−

∑

j=1

l22

= a

[1−

c(s)

1−c(s)l

∑

i=2

sin(θ

−

∑

j=1

)]

[1−

c(s)

1−c(s)l

sin(θ

−

∑

j=1

)].

Note that Frenet transformation is valid only locally,

i.e. l

(0) < r

min

, where r

min

is an inversion of max-

imal curvature c

max

of the manipulator path Π(s),

therefore nominators of all fractions are well deﬁned.

In turn, the nonholonomic planar 3-pendulum cannot

achieve angles equal to θ

,θ

= 0,±π. Moreover,

singularities in K

matrix appear for sin(θ

−θ

) =

sin(θ

−θ

) = sin(θ

−θ

) = 0.

For the regular matrix K

, the following control

signals guaranteeing a convergence of tracking errors

to zero for pure kinematic constraints can be proposed

= −K

−1

,ξ

)Λξ

, Λ = Λ

> 0. (19)

It is easy to observe that the system (18) with closed-

loop of the feedback signal (19) has a form

+ Λξ

= 0,

i.e. it is asymptotically stable.

3.2 Path Following without Desired

Orientation

The manipulator with gears designed by Nakamura,

Chung and Sørdalen has two control inputs. It means

that only two parameters can be regulated during the

path following process. If we mean that the orien-

tation of the end-effector of such manipulator is not

very important, it is possible to control other Frenet

parameters, e.g. l

– distance error from the desired

path and curvilinear length s of the path.

In such a case the following differential equations

∑

i=1

cos

−

∑

j=1

∑

k=1

˙s =

1−c(s)l

∑

i=1

sin

−

∑

j=1

∑

k=1

have to be considered. Similarly to (16) and (17), us-

ing the kinematics of the nonholonomic manipulator

(3)–(4), these equations can be expressed in the ma-

trix form as follows



˙s



= H(q

,ξ

)













= H(q

,ξ

= K

,ξ

)u. (20)

Matrix K

,ξ

) fulﬁlls the regularity condition (i.e.

it is invertible) if some conﬁgurations, which imply

the matrix singularity, are excluded from a set of pos-

sibly achieved poses of the nonholonomic manipula-

tor.

For nonholonomic 3-pendulum matrix K

,ξ

)

has the form

,ξ

) =



s11

s12

s21

s22



with elements deﬁned below

s11

∑

i=1

cos(θ

−

∑

j=1

s12

= a

∑

i=2

cos(θ

−

∑

j=1

) +

cos(θ

−

∑

j=1

s21

c(s)

1−c(s)l

∑

i=1

sin(θ

−

∑

j=1

s22

c(s)

1−c(s)l

∑

i=2

sin(θ

−

∑

j=1

)

c(s)

1−c(s)l

sin(θ

−

∑

j=1

Singular conﬁgurationsof nonholonomic3-pendulum

for the matrix K

are equal to conﬁgurations

sin(θ

) = sin(θ

−θ

) = sin(θ

−θ

) = 0.

If the following control law is applied

= −K

−1

,ξ

)v, (21)

where v ∈ R

is a new input to the system (20), then

we obtain the decoupled and linearized control system

of the form



˙s







. (22)

Now it is possible to control each variable separately.

For instance, if we want to move along the desired

path, not only to converge to this path, it seems to be

a good idea to preserve ˙s 6= 0. Possible choice of the

control algorithm for the decoupled system (22) is

= −Λl

, v

= const, Λ > 0.

Such control algorithm guarantees the motion along

the geometrical curve with constant velocity and, si-

multaneously, the convergence of the distance track-

ing error l

to 0.

4 SIMULATIONS

As an object of a simulation study we have chosen a

planar vertical 3-pendulum with nonholonomic gears

mounted on a unicycle.

The desired path for the manipulator (a circle ) was

selected as

(s) = 0.25cos4s+ 1 [m],

(s) = −0.25sin4s+ 0.6 [m],

and the desired path for the mobile platform was a

straight line

P(s) : x(s) =

√

s [m], y(s) =

√

[m].

The initial conﬁguration of the manipulator was equal

to (θ

,θ

)(0)= (0,0.6732, −π/3) and initial pos-

ture of the platform was selected as (x,y,θ)(0) =

(0,2, 3π/4).

Tracking of the desired path for the mobile plat-

form has been presented in Figures 6–8. Parameters

of the Samson & Ait-Abderrahim algorithm were se-

lected as v = 1, k

= 0.1 and k

= 1.

Tracking of the desired path with prescribed orien-

tation for the nonholonomic manipulator 3R has been

presented in Figures 9–11.

Tracking of the desired path without preserved orien-

tation for the same manipulator has been presented in

Figures 12–13.

5 CONCLUSIONS

In the paper the problem of deﬁning the path for dou-

bly nonholonomic mobile manipulators has been con-

sidered. We have proposed a new approach to the path

0 5 10 15 20 25

x [m]

y [m]

Figure 6: Path tracking for the mobile platform – XY plot.

0 200 400 600

time [s]

l [m]

Figure 7: Path tracking for the mobile platform – distance

error l.

0 200 400 600

−0.5

0.5

1.5

time [s]

θ−θ

[rad]

Figure 8: Path tracking for the mobile platform – orientation

error

θ.

0 1 2 3 4 5 6 7 8 9 10

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

TIME [s]

Figure 9: Path tracking for the 3-pendulum: an error of x

coordinate.

0 1 2 3 4 5 6 7 8 9 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

TIME [s]

Figure 10: Path tracking for the 3-pendulum: an error of z

coordinate.

0 1 2 3 4 5 6 7 8 9 10

0.02

0.04

0.06

0.08

0.1

0.12

TIME [s]

Figure 11: Path tracking for the 3-pendulum: an error of

orientation

θ.

0 1 2 3 4 5 6 7 8 9 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

TIME [s]

Figure 12: Path tracking for the 3-pendulum – XZ plot.

as a geometric curve deﬁned with the orientation or

not. Path following problem with prescibed orienta-

tion is very important for mobile systems, especially

for mobile manipulators – it results from the fact that

it is impossible to realize a task, namely unload a pay-

load if the platform has wrong orientation during the

regulation process.

In turn, for nonholonomic manipulator the desired

path need not be deﬁned with orientation. In such a

case a new approach to the path following problem

has been presented in the paper. A new control algo-

rithm, guaranteeing not only asymptotic convergence

to the desired path but simultaneously the motion

0 1 2 3 4 5 6 7 8 9 10

0.02

0.04

0.06

0.08

0.1

0.12

TIME [s]

Figure 13: Path tracking for the 3-pendulum: a distance

error l

along the path with nonzero velocity, has been intro-

duced. It is possible to deﬁne another kinematic con-

trol algorithms with speciﬁc properties using Frenet

parametrization.

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