The subject of this work is modeling and classification of single-bodied wheeled mobile robots (WMRs). In the past, it was shown that the kinematics of each such robot can be modeled by one out of only five different generic models. However, the precise conditions under which a model is the proper description of the kinematic capabilities of a robot were not clear. These shortcomings are eliminated in this work, leading to a simple procedure for model selection. Additionally, a thorough analysis of the kinematic models and a classification of their singularities are presented.

WMR are typically developed with a specific application in mind so that the resulting design provides the level of mobility that is appropriate for the robot's operation. The design of the WMR implies its specific kinematics that is then used to derive and program the controller for the robot, which translates a desired movement into the appropriate actuation of the individual wheels (steering angles and rotational speeds). Once in operation, the control law and also higher level control layers, such as the path planner, will use the (inverse) kinematics implicitly through the implemented control algorithms. It is thus often impossible for such controllers to adapt their control laws once the kinematics of the drive changes significantly.

Such a change can occur in many different scenarios: one example is the case
of a fault in the drive – e.g. an impaired steering actuator – which
obviously has big influence on the kinematic capabilities of the robot. A
fault-tolerant controller for a wheeled robot could be able to adapt to this
new situation. Another example would be a mobile robot pushing a (passive)
roll container. If the container has less degrees of freedom (DoF) than the
robot, then the kinematic and dynamic model of the whole system need to be
considered by the controller. For control of modular WMR like those presented
in

Modeling of systems defined by wheel-like constraints is a well studied
field, and most of the pioneering work was conducted in the last two decades
of the last century. Early works are

The classification of WMR models found by

In this section, the classes of posture kinematic models of WMR are reviewed,
as they are developed in

A center-steerable standard wheel.

Let

Arrange the co-vectors

According to the definition of

The models of WMR presented so far have some special features which are not
addressed in popular publications like

Recall that

Example 1 shows that the dimension of the distribution

In the set

To the best of our knowledge, such a property was not mentioned in literature and shows a demand for a deeper understanding of the relation between robots and models.

Any robot which is at least capable of reaching any pose in the plane is able to fulfill most of the typical tasks of WMR. From a control engineering point of view, this formulation already sounds familiar, since accessibility is a well-known concept in the field of controllability analysis of nonlinear control systems.

Recall from Sect.

The set of poses accessible by motions in inv(

For sufficiency, we have to show that every robot with
rank(

rank(

Note that for rank

Note that for rank

How inv(

A robot with one fixed and one center-steerable wheel, that
would be degenerate in the sense of

There is a slight difference between the set of robots considered here, and
the set considered by

There is another difference in the approach of

We will therefore avoid using the notation of degeneracy and instead simply distinguish between wheel configurations that are controllable or not.

When talking about wheel (hardware) configurations and model classes, it is
impractical to use the same notation (

Controllable robots may have multiple structurally different controllable
wheel configurations. This fact was illustrated in Example 1, where the robot
was found to have two disjoint sets of controllable wheel configurations. The
property of the singular set

The relation of hardware designs and modes of operation.

Previous works

To resolve this ambiguity the following definition is made:

When a robot operates in a mode (

The hardware design-types that we introduce in Table

The robot type numbers were chosen to match the classification of two- and
three-wheeled robot classification from

The last row in Table

Exemplary, the following description shows how a robot with two or more
coaligned center-steerable standard wheels (HW-type Vb) can operate in
different modes. An example for a HW-type Vb robot is the snakeboard

a HW-type-Vb robot operates in mode (2, 0), if the axes
of all wheels are co-aligned. In such a configuration, the rank of

a robot is able to operate in this mode if the ICR is
defined by a unique intersection of the wheel axes. If these axes coincide,
then this condition is not satisfied and the robot is forced to operate in
mode (2, 0). This means that, in order to operate in mode (1, 2), a robot must
not reach a configuration that is consistent with the conditions of operating
in mode (2, 0). Thus, for HW-type-Vb robots,

mode (1, 1) can be enforced for a HW-type-Vb robot by introducing a virtual constraint that blocks the steering of exactly one wheel. The axis of this blocked wheel must not coincide with the wheel contact points of the other wheels, otherwise forcing mode (2, 0).

by driving a HW-type-Vb robot in this mode, the restrictions from the sliding constraints are neglected for all wheels.

when a (2, 1) model is used for a HW-type-Vb robot, the sliding constraints are neglected for all but one wheel.

For the virtually constrained modes respecting all sliding constraints, the
following conditions must be satisfied:

the number of DoF of the desired mode must be lower or equal to the number of DoF of the physical mode;

the DoN of the desired mode must be higher or equal to the DoN
of the physical mode (see also Table

The mappings from Table

Place the origin

The reason why a (2, 1) model is not as accurate as a (1, 2) model for a robot
operating in mode (1, 2) is found in Table

An intuitive interpretation of these arguments is found by considering
actuation: in a (2, 1) model, there is no state corresponding to

One could argue that also for a (1, 2) model there are unmodeled steering
actuators. If a robot has more than two center-steered wheels, then only two
of them appear in the model. However, since the ICR is 2-DoF, at most two
steering angles can be restricting the motion of the robot at a time. This
can be accounted for using a switching strategy

By singularities we mean configurations, at which sliding constraints become
linear dependent. Note that this definition is not bound to a rank loss of

Singularities of type A are those, where the wheel axes of all wheels of a
robot with coaligned center-steerable wheels (HW-type Vb) coincide. This
singularity was introduced in Example 1 as a model-switch, leading to a
hybrid WMR model. From a mathematical point of view, for hybrid systems, the
meaning of

The mathematical theory that allows to conclude about the properties of
existence and uniqueness is the theory of Carathèodory differential
equations and Lebesgue integration. A generalization of the notion of

there must exist a function

For a type Vb robot operating in mode (1, 2), the vector field

In the feedback-case, the situation is more challenging: suppose a reference
trajectory is designed in a way that requires the robot to stay in a singular
configuration for longer time. In the feedforward-case this does not
compromise the Carathèodory conditions, as the robot will still be in

An example for a singularity of type B for a robot with three cs-wheels that are not coaligned: no unique ICR is defined by wheels one and two, but one and three.

A singularity of type A is only present for robots with coaligned wheel
contact points (HW-type Vb). When the wheel contact points are not on a
straight line (HW-type Va), then one or more (but not all) wheel axes may be
coaligned. Such a situation is a singularity of type B, shown in
Fig.

A robot is in a type C singularity when the ICR coincides with the contact
point of a center-steerable standard wheel, see Fig.

The previous sections tried to require a minimum of pre-knowledge in
differential geometry to make this article easily accessible to a broader
audience. A crash-course for control engineers is found in

An example for a singularity of type C for a robot with three cs-wheels that are not coaligned: the ICR, defined by wheels one and two, coincides with the contact point of wheel three.

Let us again start at the root cause for restriction of motion, the sliding
constraints from Eq. (

We are looking for the “largest” connected submanifold

The submanifold

Since inv(

Let us now apply this theory to wheeled mobile robots. Consider a robot with
only one center-steerable or fixed standard wheel. In this case, there is
only one 1-form

For two sliding constraints, the involutive closure of the
coannihilator of the generalized codistribution

However, for two sliding constraints, the situation already gets more
complicated. Consider the robot with two center-steerable wheels from
Example 1. Without loss of generality, place the origin of the
robot fixed frame in the contact point of wheel 1 and let the

For three sliding constraints, the integrability condition
(Eq.

Constraint (Eq.

A

If

The foliation of

It can be concluded that WMR with three center-steerable wheels are (i) nonholonomic,
(ii) the state manifold

The case of three sliding constraints can be analogously extended to an arbitrary number of wheels: for every further wheel, either its axis must be co-aligned to at least one of the existing wheels or coordinated by a coordination function asserting the existence of a unique ICR.

For three sliding constraints, the involutive closure of
the coannihilator of the generalized codistribution

This article presented a detailed view on the kinematics of single-bodied wheeled mobile robots.

One suggestion is to drop the notion of degeneracy of wheeled mobile robots and replace it with controllability. The usage of this term is consistent with the standard definition in control theory. Furthermore, a simple condition for controllability was presented.

Based on this controllability study, a classification of wheeled mobile robots into six hardware types is introduced. This classification is solely based on the type, location and number of wheels. Each hardware type is able to operate in one or more modes of operation. A mode of operation corresponds to a set of configurations, in which a specific model is an accurate representation of the kinematic capabilities of the robot.

Moreover, we provide a detailed analysis of the geometry of wheeled mobile robots by which we are able to give a general view on state manifolds and singularities.

This work was supported by the Austrian Science Fund (FWF) under Grant PN 20041. Edited by: A. Müller Reviewed by: two anonymous referees