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Chapter

Mobile Robot Vehicles

helpful to consider some general, but important, concepts regarding mobility.

4.1

Mobility

We have already touched on the diversity of mobile robots and their modes of locomo-

tion. In this section we will discuss mobility which is concerned with how a vehicle

moves in space.

We first consider the simple example of a train. The train moves along rails and its posi-

tion is described by its distance along the rail from some datum. The configuration of the

train can be completely described by a scalar parameter q which is called its generalized

coordinate. The set of all possible configurations is the configuration space, or C-space, de-

noted by C and q∈C. In this case C ⊂R. We also say that the train has one degree of free-

dom since q is a scalar. The train also has one actuator (motor) that propels it forwards or

backwards along the rail. With one motor and one degree of freedom the train is fully

actuated and can achieve any desired configuration, that is, any position along the rail.

Another important concept is task space which is the set of all possible poses ξ of

the vehicle and ξ ∈ T. The task space depends on the application or task. If our task

was motion along the rail then T ⊂ R. If we cared only about the position of the train

in a plane then T ⊂ R

. If we considered a 3-dimensional world then T ⊂ SE(3), and its

height changes as it moves up and down hills and its orientation changes as it moves

around curves. Clearly for these last two cases the dimensions of the task space exceed

the dimensions of the configuration space and the train cannot attain an arbitrary

pose since it is constrained to move along fixed rails. In these cases we say that the

train moves along a manifold in the task space and there is a mapping from q  ξ.

Interestingly many vehicles share certain characteristics with trains – they are good

at moving forward but not so good at moving sideways. Cars, hovercrafts, ships and

aircraft all exhibit this characteristic and require complex manoeuvring in order to

move sideways. Nevertheless this is a very sensible design approach since it caters to

the motion we most commonly require of the vehicle. The less common motions such

as parking a car, docking a ship or landing an aircraft are more complex, but not im-

possible, and humans can learn this skill. The benefit of this type of design comes

from simplification and in particular reducing the number of actuators required.

This chapter discusses how a robot platform moves, that is, how its pose

changes with time as a function of its control inputs. There are many differ-

ent types of robot platform as shown on pages 61–63 but in this chapter we

will consider only two which are important exemplars. The first is a wheeled

vehicle like a car which operates in a 2-dimensional world. It can be pro-

pelled forwards or backwards and its heading direction controlled by chang-

ing the angle of its steered wheels. The second platform is a quadcopter, a

flying vehicle, which is an example of a robot that moves in 3-dimensional

space. Quadcopters are becoming increasing popular as a robot platform

since they can be quite easily modelled and controlled.

However before we start to discuss these two robot platforms it will be

Next consider a hovercraft which has two propellors whose axes are parallel but not

collinear. The sum of their thrusts provide a forward force and the difference in thrusts

generates a yawing torque for steering. The hovercraft moves over a planar surface

and its configuration is entirely described by three generalized coordinates

q = (x, y, θ) ∈ C and in this case C ⊂ R

× S. The configuration space has 3 dimen-

sions and the vehicle therefore has three degrees of freedom.

The hovercraft has only two actuators, one fewer than it has degrees of freedom,

and it is therefore an under-actuated system. This imposes limitations on the way in

which it can move. At any point in time we can control the forward (parallel to the

thrust vectors) acceleration and the rotational acceleration of the the hovercraft but

there is zero sideways (or lateral) acceleration since it does not generate any lateral

thrust. Nevertheless with some clever manoeuvring, like with a car, the hovercraft can

follow a path that will take it to a place to one side of where it started. The advantage of

under-actuation is the reduced number of actuators, in this case two instead of three.

The penalty is that the vehicle cannot move directly to an any point in its configura-

tion space, it must follow some path. If we added a third propellor to the hovercraft

with its axis normal to the first two then it would be possible to command an arbitrary

forward, sideways and rotational acceleration. The task space of the hovercraft is

T ⊂ SE(2) which is equivalent, in this case, to the configuration space.

A helicopter has four actuators. The main rotor generates a thrust vector whose

magnitude is controlled by the collective pitch, and the thrust vector’s direction is

controlled by the lateral and longitudinal cyclic pitch. The fourth actuator, the tail

rotor, provides a yawing moment. The helicopter’s configuration can be described by

six generalized coordinates q = (x, y, z, θ

, θ

) ∈ C which is its position and orienta-

tion in 3-dimensional space, with orientation expressed in roll-pitch-yaw angles. The

configuration space C ⊂ R

× S

has six dimensions and therefore the vehicle has six

degrees of freedom. The helicopter is under-actuated and it has no means to rotationally

accelerate in the pitch and roll directions but cleverly these unactuated degrees of

freedom are not required for helicopter operation – the helicopter naturally maintains

stable equilibrium values for roll and pitch angle. Gravity acts like an additional actua-

tor and provides a constant downward force. This allows the helicopter to accelerate

sideways using the horizontal component of its thrust vector, while the vertical com-

ponent of thrust is counteracted by gravity – without gravity a helicopter could not fly

sideways. The task space of the helicopter is T ⊂ SE(3).

A fixed-wing aircraft moves forward very efficiently and also has four actuators



(forward thrust, ailerons, elevator and rudder). The aircraft’s thrust provides accelera-

tion in the forward direction and the control surfaces exert various moments on the

aircraft: rudder (yaw torque), ailerons (roll torque), elevator (pitch torque). The aircraft’s

configuration space is the same as the helicopter and has six dimensions. The aircraft

is under-actuated and it has no way to accelerate in the lateral direction. The task

space of the aircraft is T ⊂ SE(3).

The DEPTHX underwater robot shown on page 62 also has a configuration space

C ⊂ R

× S

of six dimensions, but by contrast is fully actuated. Its six actuators can

exert an arbitrary force and torque on the vehicle, allowing it to accelerate in any di-

rection or about any axis. Its task space is T ⊂ SE(3).

Finally we come to the wheel – one of humanity’s greatest achievements. The wheel

was invented around 3000 bce and the two wheeled cart was invented around 2000 bce.

Today four wheeled vehicles are ubiquitous and the total automobile population of the

planet is approaching one billion.



The effectiveness of cars, and our familiarity with

them, makes them a natural choice for robot platforms that move across the ground.

The configuration of a car moving over a plane is described by its generalized coor-

dinates q = (x, y, θ) ∈ C and C ⊂ R

× S which has 3 dimensions. A car has only two

actuators, one to move forwards or backwards and one to change the heading direc-

tion. The car is therefore under-actuated.



As we have already remarked an under-

actuated vehicle cannot move directly to an any point in its configuration space, it

Some low-cost hobby aircraft have no

rudder and rely only on ailerons to bank

and turn the aircraft. Even cheaper

hobby aircraft have no elevator and rely

on engine speed to control height.

http://hypertextbook.com/facts/2001/

MarinaStasenko.shtml.

Unlike the aircraft and underwater ro-

bot the motion of a car is generally con-

sidered in terms of velocities rather

than forces and torques.

Chapter 4

Mobile Robot Vehicles

must follow generally nonlinear some path. We know from our everyday experience

with cars that it is not possible to drive sideways, but with some practice we can learn

to follow a path that results in the vehicle being to one side of its initial position – this

is parallel parking. Neither can a car rotate on spot, but we can follow a path that

results in the vehicle being at the same position but rotated by 180° – a three-point

turn. The challenges this introduces for control and path planning will be discussed in

the rest of this part of the book. Despite this limitation the car is the simplest and most

effective means of moving in a planar world that we have yet found.

The standard wheel is highly directional and prefers to roll in the direction normal

to the axis of rotation. We might often wish for an ability to roll sideways but the stan-

dard wheel provides significant benefit when cornering – lateral friction between the

wheels and the road counteracts, for free, the centripetal acceleration which would

otherwise require an extra actuator to provide that force. More radical types of wheels

have been developed that can roll sideways. An omni-directional wheel or Swedish

wheel is shown in Fig. 4.1. It is similar to a normal wheel but has a number of passive

rollers around its circumference and their rotational axes lie in the plane of the wheel.

It is driven like an ordinary wheel but has very low friction in the lateral direction. A

spherical wheel is similar to the roller ball in an old-fashioned computer mouse but

driven by two actuators so that it can achieve achieve a velocity in any direction.

In robotics a car is often described as a non-holonomic vehicle. The term non-

holonomic comes from mathematics and means that the motion of the car is subject

to one or more non-holonomic constraints. A holonomic constraint is an equation

that can be written in terms of the configuration variables x, y and θ. A non-holonomic

constraint can only be written in terms of the derivatives of the configuration vari-

ables and cannot be integrated to a constraint in terms of configuration variables. Such

systems are therefore also known as non-integrable systems. A key characteristic of

these systems, as we have already discussed, is that they cannot move directly from one

configuration to another – they must perform a manoeuvre or sequence of motions



A skid-steered vehicle, such as a tank, can turn on the spot but to move sideways it

would have to stop, turn, proceed, stop then turn – this is a manoeuvre or time-vary-

ing control strategy which is the hallmark of a non-holonomic system. The tank has

two actuators, one for each track, and just like a car is under-actuated.

Mobility parameters for the vehicles that we have discussed are tabulated in Table 4.1.

The second column is the number of degrees of freedom of the vehicle or the dimen-

sion of its configuration space. The third column is the number of actuators and the

fourth column indicates whether or not the vehicle is fully actuated.

4.2

Car-like Mobile Robots

Wheeled cars are a very effective class of vehicle and the archetype for most ground

robots such as those shown on page 62. In this section we will create a model for a car-

like vehicle and develop controllers that can drive the car to a point, along a line, follow

an arbitrary path, and finally, drive to a specific pose.

Fig. 4.1. Omni-directional (or

Swedish) wheel. Note the circum-

ferential rollers which make mo-

tion in the direction of the wheel’s

axis almost frictionless. (Courtesy

Vex Robotics)

Table 4.1.

Summary of parameters for

three different types of vehicle.

The +g notation indicates that

the gravity field can be consid-

ered as an extra actuator

We can also consider this in control theo-

retic terms. Brockett’s theorem (Brockett

1983) states that such systems are con-

trollable but they cannot be stabilized

to a desired state using a differentiable,

or even continuous, pure state feedback

controller. A time varying or non-linear

control strategy is required.

4.2 · Car-like Mobile Robots

For a fixed steering wheel angle the car moves along a circular arc. For this reason

curves on roads are circular arcs or clothoids



which makes life easier for the driver since

constant or smoothly varying steering wheel angle allow the car to follow the road. Note

that R

> R

which means the front wheel must follow a longer path and therefore rotate

more quickly than the back wheel. When a four-wheeled vehicle goes around a corner the

two steered wheels follow circular paths of different radius and therefore the angles of the

steered wheels γ

and γ

should be very slightly different. This is achieved by the com-

monly used Ackerman steering mechanism which results in lower wear and tear on the

tyres. The driven wheels must rotate at different speeds on corners which is why a differ-

ential gearbox is required between the motor and the driven wheels.

The velocity of the robot in the world frame is (v cosθ, vsinθ) and combined with

Eq. 4.1 we write the equations of motion as

(4.2)

This model is referred to as a kinematic model since it describes the velocities of the vehi-

cle but not the forces or torques that cause the velocity. The rate of change of heading Ë is

referred to as turn rate, heading rate or yaw rate and can be measured by a gyroscope. It can

also be deduced from the angular velocity of the wheels on the left- and right-hand sides

of the vehicle which follow arcs of different radius and therefore rotate at different speeds.

In the world coordinate frame we can write an expression for velocity in the vehicle’s

y-direction

which is the non-holonomic constraint. This equation cannot be integrated to form a

relationship between x, y and θ.

Equation 4.2 captures some other important characteristics of a wheeled vehicle.

When v = 0 then Ë = 0, that is, it is not possible to change the vehicle’s orientation

when it is not moving. As we know from driving we must be moving in order to turn.

If the steering angle is ü then the front wheel is orthogonal to the back wheel, the

vehicle cannot move forward and the model enters an undefined region.

Vehicle coordinate system. The coordinate system that we will use, and a common one for vehicles

of all sorts is that the x-axis is forward (longitudinal motion), the y-axis is to the left side (lateral

motion) which implies that the z-axis is upward. For aerospace and underwater applications the

z-axis is often downward and the x-axis is forward.

Paths that arcs with smoothly varying

radius.

Rudolph Ackerman (1764–1834) was a German inventor born at Schneeberg, in Saxony. For finan-

cial reasons he was unable to attend university and became a saddler like his father. For a time he

worked as a saddler and coach-builder and in 1795 established a print-shop and drawing-school in

London. He published a popular magazine “The Repository of Arts, Literature, Commerce, Manu-

factures, Fashion and Politics” that included an eclectic mix of articles on water pumps, gas-light-

ing, and lithographic presses, along with fashion plates and furniture designs. He manufactured

paper for landscape and miniature painters, patented a method for waterproofing cloth and paper

and built a factory in Chelsea to produce it. He is buried in Kensal Green Cemetery, London.

In 1818 Ackermann took out British patent 4212 on behalf of the German inventor George

Lankensperger for a steering mechanism which ensures that the steered wheels move on circles

with a common centre. The same scheme was proposed and tested by Erasmus Darwin (grandfa-

ther of Charles) in the 1760s. Subsequent refinement by the Frenchman Charles Jeantaud led to

the mechanism used in cars to this day which is known as Ackermann steering.

4.2 · Car-like Mobile Robots

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