INTRODUCTION R - site.iugaza.edu.pssite.iugaza.edu.ps/sbaraqouni/files/2011/09/first-chapters.pdf · INTRODUCTION R obotics is a relatively young ﬁeld of modern technology that

Chapter 1

INTRODUCTION

Robotics is a relatively young field of modern technology that crossestraditional engineering boundaries. Understanding the complexity of robotsand their application requires knowledge of electrical engineering, mechan-ical engineering, systems and industrial engineering, computer science, eco-nomics, and mathematics. New disciplines of engineering, such as manu-facturing engineering, applications engineering, and knowledge engineeringhave emerged to deal with the complexity of the field of robotics and factoryautomation.

This book is concerned with fundamentals of robotics, including kine-matics, dynamics, motion planning, computer vision, and control.Our goal is to provide an introduction to the most important concepts inthese subjects as applied to industrial robot manipulators and other me-chanical systems.

The term robot was first introduced by the Czech playwright KarelCapek in his 1920 play Rossum’s Universal Robots, the word robota beingthe Czech word for work. Since then the term has been applied to a greatvariety of mechanical devices, such as teleoperators, underwater vehicles,autonomous land rovers, etc. Virtually anything that operates with somedegree of autonomy, usually under computer control, has at some point beencalled a robot. In this text the term robot will mean a computer controlledindustrial manipulator of the type shown in Figure 1.1.

This type of robot is essentially a mechanical arm operating under com-puter control. Such devices, though far from the robots of science fiction, arenevertheless extremely complex electromechanical systems whose analyticaldescription requires advanced methods, presenting many challenging andinteresting research problems. An official definition of such a robot comesfrom the Robot Institute of America (RIA):

2 CHAPTER 1. INTRODUCTION

Figure 1.1: Examples of typical industrial manipulators, the AdeptSix 600robot (left) and the AdeptSix 300 robot (right). Both are six-axis, high per-formance robots designed for materials handling or assembly applications.(Photo courtesy of Adept Technology, Inc.)

Definition: A robot is a reprogrammable, multifunctional manipulator de-

signed to move material, parts, tools, or specialized devices through variable

programmed motions for the performance of a variety of tasks.

The key element in the above definition is the reprogrammability, whichgives a robot its utility and adaptability. The so-called robotics revolutionis, in fact, part of the larger computer revolution.

Even this restricted definition of a robot has several features that make itattractive in an industrial environment. Among the advantages often citedin favor of the introduction of robots are decreased labor costs, increasedprecision and productivity, increased flexibility compared with specializedmachines, and more humane working conditions as dull, repetitive, or haz-ardous jobs are performed by robots.

The robot, as we have defined it, was born out of the marriage of twoearlier technologies: teleoperators and numerically controlled millingmachines. Teleoperators, or master-slave devices, were developed during

1.1. MATHEMATICAL MODELING OF ROBOTS 3

the second world war to handle radioactive materials. Computer numericalcontrol (CNC) was developed because of the high precision required in themachining of certain items, such as components of high performance air-craft. The first robots essentially combined the mechanical linkages of theteleoperator with the autonomy and programmability of CNC machines.

The first successful applications of robot manipulators generally involvedsome sort of material transfer, such as injection molding or stamping, inwhich the robot merely attends a press to unload and either transfer orstack the finished parts. These first robots could be programmed to executea sequence of movements, such as moving to a location A, closing a gripper,moving to a location B, etc., but had no external sensor capability. Morecomplex applications, such as welding, grinding, deburring, and assemblyrequire not only more complex motion but also some form of external sensingsuch as vision, tactile, or force sensing, due to the increased interaction ofthe robot with its environment.

Worldwide there are currently over 800, 000 industrial robots in oper-ation, mostly in Japan, the European Union and North America (see Fig-ure 1.2). After a period of stagnation in the late 1980’s, the sale of industrialrobots began to rise in the 1990’s and sales growth is likely to remain strongfor the remainder of this decade.

It should be pointed out that the important applications of robots areby no means limited to those industrial jobs where the robot is directly re-placing a human worker. In fact,there are over 600, 000 household robotscurrently in use primarily as vacuum cleaning and lawn mowing robots.There are many other applications of robotics in areas where the use of hu-mans is impractical or undesirable. Among these are undersea and planetaryexploration, satellite retrieval and repair, the defusing of explosive devices,and work in radioactive environments. Finally, prostheses, such as artifi-cial limbs, are themselves robotic devices requiring methods of analysis anddesign similar to those of industrial manipulators.

1.1 MATHEMATICAL MODELING OF ROBOTS

In this text we will be primarily concerned with developing and analyzingmathematical models for robots. In particular, we will develop methodsto represent basic geometric aspects of robotic manipulation, dynamic as-pects of manipulation, and the various sensors available in modern roboticsystems. Equipped with these mathematical models, we will be developmethods for planning and controlling robot motions to perform specifiedtasks. We begin here by describing some of the basic notation and termi-


Number of Robots in Various Countries at the end of 2003

14,000

18,000

26,000

50,000

112,000

112,700

350,000

UK

Spain

France

Italy

North America

Germany

Japan

Figure 1.2: Number of robots in use at the end of 2003. Japan has thelargest number of industrial robots, followed by the European Union andNorth America. Source: UNECE - United Nations Economic Commissionfor Europe, October, 2004.

nology that we will use in later chapters to develop mathematical modelsfor robot manipulators.

1.1.1 Symbolic Representation of Robots

Robot manipulators are composed of links connected by joints to forma kinematic chain. Joints are typically rotary (revolute) or linear (pris-matic). A revolute joint is like a hinge and allows relative rotation betweentwo links. A prismatic joint allows a linear relative motion between twolinks. We denote revolute joints by R and prismatic joints by P, and drawthem as shown in Figure 1.3. For example, a three-link arm with threerevolute joints will be referred to as an RRR arm.

Each joint represents the interconnection between two links. We denotethe axis of rotation of a revolute joint, or the axis along which a prismaticjoint translates by zi if the joint is the interconnection of links i and i+1. Thejoint variables, denoted by θ for a revolute joint and d for the prismaticjoint, represent the relative displacement between adjacent links. We willmake this precise in Chapter 3.

1.1. MATHEMATICAL MODELING OF ROBOTS 5

Prismatic

2D

3D

Revolute

Figure 1.3: Symbolic representation of robot joints. Each joint allows asingle degree of freedom of motion between adjacent links of the manipulator.The revolute joint (shown in 2D and 3D on the left) produces a relativerotation between adjacent links. The prismatic joint (shown in 2D and 3Don the right) produces a linear or telescoping motion between adjacent links.

1.1.2 The Configuration Space

A configuration of a manipulator is a complete specification of the locationof every point on the manipulator. The set of all possible configurations iscalled the configuration space. In our case, if we know the values forthe joint variables (i.e., the joint angle for revolute joints, or the joint offsetfor prismatic joints), then it is straightforward to infer the position of anypoint on the manipulator, since the individual links of the manipulator areassumed to be rigid and the base of the manipulator is assumed to be fixed.Therefore, in this text, we will represent a configuration by a set of valuesfor the joint variables. We will denote this vector of values by q, and saythat the robot is in configuration q when the joint variables take on thevalues q1, . . . , qn, with qi = θi for a revolute joint and qi = di for a prismaticjoint.

An object is said to have n degrees of freedom (DOF) if its configura-tion can be minimally specified by n parameters. Thus, the number of DOFis equal to the dimension of the configuration space. For a robot manipula-tor, the number of joints determines the number of DOF. A rigid object inthree-dimensional space has six DOF: three for positioning and three fororientation. Therefore, a manipulator should typically possess at least sixindependent DOF. With fewer than six DOF the arm cannot reach everypoint in its work space with arbitrary orientation. Certain applications suchas reaching around or behind obstacles may require more than six DOF. Amanipulator having more than six DOF is referred to as a kinematically


redundant manipulator.

1.1.3 The State Space

A configuration provides an instantaneous description of the geometry ofa manipulator, but says nothing about its dynamic response. In contrast,the state of the manipulator is a set of variables that, together with adescription of the manipulator’s dynamics and future inputs, is sufficient todetermine the future time response of the manipulator. The state space isthe set of all possible states. In the case of a manipulator arm, the dynamicsare Newtonian, and can be specified by generalizing the familiar equationF = ma. Thus, a state of the manipulator can be specified by giving thevalues for the joint variables q and for joint velocities q (acceleration isrelated to the derivative of joint velocities).

1.1.4 The Workspace

The workspace of a manipulator is the total volume swept out by the endeffector as the manipulator executes all possible motions. The workspaceis constrained by the geometry of the manipulator as well as mechanicalconstraints on the joints. For example, a revolute joint may be limited toless than a full 360◦ of motion. The workspace is often broken down intoa reachable workspace and a dexterous workspace. The reachableworkspace is the entire set of points reachable by the manipulator, whereasthe dexterous workspace consists of those points that the manipulator canreach with an arbitrary orientation of the end effector. Obviously the dex-terous workspace is a subset of the reachable workspace. The workspaces ofseveral robots are shown later in this chapter.

1.2 ROBOTS AS MECHANICAL DEVICES

There are a number of physical aspects of robotic manipulators that we willnot necessarily consider when developing our mathematical models. Theseinclude mechanical aspects (e.g., how are the joints actually implemented),accuracy and repeatability, and the tooling attached at the end effector. Inthis section, we briefly describe some of these.

1.2.1 Classification of Robotic Manipulators

Robot manipulators can be classified by several criteria, such as their powersource, or the way in which the joints are actuated; their geometry, or

1.2. ROBOTS AS MECHANICAL DEVICES 7

kinematic structure; their method of control; and their intended applica-tion area. Such classification is useful primarily in order to determine whichrobot is right for a given task. For example, an hydraulic robot would notbe suitable for food handling or clean room applications whereas a SCARArobot would not be suitable for automobile spray painting. We explain thisin more detail below.

Power Source

Most robots are either electrically, hydraulically, or pneumatically powered.Hydraulic actuators are unrivaled in their speed of response and torque pro-ducing capability. Therefore hydraulic robots are used primarily for liftingheavy loads. The drawbacks of hydraulic robots are that they tend to leakhydraulic fluid, require much more peripheral equipment (such as pumps,which require more maintenance), and they are noisy. Robots driven byDC or AC motors are increasingly popular since they are cheaper, cleanerand quieter. Pneumatic robots are inexpensive and simple but cannot becontrolled precisely. As a result, pneumatic robots are limited in their rangeof applications and popularity.

Method of Control

Robots are classified by control method into servo and nonservo robots.The earliest robots were nonservo robots. These robots are essentially open-loop devices whose movements are limited to predetermined mechanicalstops, and they are useful primarily for materials transfer. In fact, accordingto the definition given above, fixed stop robots hardly qualify as robots.Servo robots use closed-loop computer control to determine their motionand are thus capable of being truly multifunctional, reprogrammable devices.

Servo controlled robots are further classified according to the methodthat the controller uses to guide the end effector. The simplest type ofrobot in this class is the point-to-point robot. A point-to-point robot canbe taught a discrete set of points but there is no control of the path ofthe end effector in between taught points. Such robots are usually taughta series of points with a teach pendant. The points are then stored andplayed back. Point-to-point robots are limited in their range of applications.With continuous path robots, on the other hand, the entire path of theend effector can be controlled. For example, the robot end effector canbe taught to follow a straight line between two points or even to follow acontour such as a welding seam. In addition, the velocity and/or accelerationof the end effector can often be controlled. These are the most advancedrobots and require the most sophisticated computer controllers and softwaredevelopment.


Application Area

Robots are often classified by application area into assembly and nonassem-bly robots. Assembly robots tend to be small, electrically driven and eitherrevolute or SCARA (described below) in design. Typical nonassembly appli-cation areas to date have been in welding, spray painting, material handling,and machine loading and unloading.

One of the primary differences between assembly and nonassembly appli-cations is the increased level of precision required in assembly due to signif-icant interaction with objects in the workspace. For example, an assemblytask may require part insertion (the so-called peg-in-hole problem) orgear meshing. A slight mismatch between the parts can result in wedgingand jamming, which can cause large interaction forces and failure of thetask. As a result assembly tasks are difficult to accomplish without specialfixtures and jigs, or without sensing and controlling the interaction forces.

Geometry

Most industrial manipulators at the present time have six or fewer DOF.These manipulators are usually classified kinematically on the basis of thefirst three joints of the arm, with the wrist being described separately. Themajority of these manipulators fall into one of five geometric types: articu-lated (RRR), spherical (RRP), SCARA (RRP), cylindrical (RPP),or Cartesian (PPP). We discuss each of these below in Section 1.3.

Each of these five manipulator arms is a serial link robot. A sixthdistinct class of manipulators consists of the so-called parallel robot. Ina parallel manipulator the links are arranged in a closed rather than openkinematic chain. Although we include a brief discussion of parallel robotsin this chapter, their kinematics and dynamics are more difficult to derivethan those of serial link robots and hence are usually treated only in moreadvanced texts.

1.2.2 Robotic Systems

A robot manipulator should be viewed as more than just a series of me-chanical linkages. The mechanical arm is just one component in an over-all robotic system, illustrated in Figure 1.4, which consists of the arm,external power source, end-of-arm tooling, external and internalsensors, computer interface, and control computer. Even the pro-grammed software should be considered as an integral part of the overallsystem, since the manner in which the robot is programmed and controlledcan have a major impact on its performance and subsequent range of appli-cations.


Sensors

��

Power supply

��Input device

orteach pendant

oo // Computercontroller

oo //

OO

��

Mechanicalarm

OO

��Programstorage

or network

End-of-armtooling

Figure 1.4: The integration of a mechanical arm, sensing, computation,user interface and tooling forms a complex robotic system. Many modernrobotic systems have integrated computer vision, force/torque sensing, andadvanced programming and user interface features.

1.2.3 Accuracy and Repeatability

The accuracy of a manipulator is a measure of how close the manipulatorcan come to a given point within its workspace. Repeatability is a measureof how close a manipulator can return to a previously taught point. The pri-mary method of sensing positioning errors is with position encoders locatedat the joints, either on the shaft of the motor that actuates the joint or onthe joint itself. There is typically no direct measurement of the end-effectorposition and orientation. One relies instead on the assumed geometry ofthe manipulator and its rigidity to calculate the end-effector position fromthe measured joint positions. Accuracy is affected therefore by computa-tional errors, machining accuracy in the construction of the manipulator,flexibility effects such as the bending of the links under gravitational andother loads, gear backlash, and a host of other static and dynamic effects.It is primarily for this reason that robots are designed with extremely highrigidity. Without high rigidity, accuracy can only be improved by some sortof direct sensing of the end-effector position, such as with computer vision.

Once a point is taught to the manipulator, however, say with a teachpendant, the above effects are taken into account and the proper encodervalues necessary to return to the given point are stored by the controllingcomputer. Repeatability therefore is affected primarily by the controllerresolution. Controller resolution means the smallest increment of mo-tion that the controller can sense. The resolution is computed as the totaldistance traveled divided by 2n, where n is the number of bits of encoderaccuracy. In this context, linear axes, that is, prismatic joints, typically


have higher resolution than revolute joints, since the straight line distancetraversed by the tip of a linear axis between two points is less than thecorresponding arc length traced by the tip of a rotational link.

In addition, as we will see in later chapters, rotational axes usually resultin a large amount of kinematic and dynamic coupling among the links, witha resultant accumulation of errors and a more difficult control problem. Onemay wonder then what the advantages of revolute joints are in manipulatordesign. The answer lies primarily in the increased dexterity and compactnessof revolute joint designs. For example, Figure 1.5 shows that for the samerange of motion, a rotational link can be made much smaller than a linkwith linear motion.

dd

Figure 1.5: Linear vs. rotational link motion showing that a smaller revolutejoint can cover the same distance d as a larger prismatic joint. The tip of aprismatic link can cover a distance equal to the length of the link. The tipof a rotational link of length a, by contrast, can cover a distance of 2a byrotating 180 degrees.

Thus, manipulators made from revolute joints occupy a smaller workingvolume than manipulators with linear axes. This increases the ability of themanipulator to work in the same space with other robots, machines, andpeople. At the same time revolute joint manipulators are better able tomaneuver around obstacles and have a wider range of possible applications.

1.2.4 Wrists and End Effectors

The joints in the kinematic chain between the arm and end effector arereferred to as the wrist. The wrist joints are nearly always all revolute.It is increasingly common to design manipulators with spherical wrists,by which we mean wrists whose three joint axes intersect at a commonpoint, known as the wrist center point. Such a spherical wrist is shownin Figure 1.6.

The spherical wrist greatly simplifies kinematic analysis, effectively al-lowing one to decouple the position and orientation of the end effector.Typically the manipulator will possess three DOF for position, which are


Yaw

Pitch

Roll

Wrist Center Point

Figure 1.6: The spherical wrist. The axes of rotation of the spherical wristare typically denoted roll, pitch, and yaw and intersect at a point called thewrist center point

produced by three or more joints in the arm. The number of DOF for ori-entation will then depend on the DOF of the wrist. It is common to findwrists having one, two, or three DOF depending on the application. Forexample, the SCARA robot shown in Figure 1.14 has four DOF: three forthe arm, and one for the wrist, which has only a rotation about the finalz-axis.

The arm and wrist assemblies of a robot are used primarily for position-ing the hand, end effector, and any tool it may carry. It is the end effectoror tool that actually performs the task. The simplest type of end effectoris a gripper, such as shown in Figure 1.7 which is usually capable of onlytwo actions, opening and closing. While this is adequate for materialstransfer, some parts handling, or gripping simple tools, it is not adequatefor other tasks such as welding, assembly, grinding, etc.

Figure 1.7: Examples of robot grippers. Shown here from left to right area two-fingered parallel jaw gripper, a scissor-type gripper, and a verticalgripper. (Photos courtesy of ASG-Jergen’s, Cleveland Ohio.)

A great deal of research is therefore devoted to the design of special pur-pose end effectors as well as of tools that can be rapidly changed as the task


dictates. There is also much research on the development of anthropomor-phic hands such as that shown in Figure 1.8. Since we are concerned withthe analysis and control of the manipulator itself and not in the particularapplication or end effector, we will not discuss the design of end effectors orthe study of grasping and manipulation.

Figure 1.8: A three-fingered anthropomorphic hand developed by BarrettTechnologies. Such grippers allow for more dexterity and the ability to ma-nipulate objects of various sizes and geometries. (Photo courtesy of BarrettTechnologies.)

1.3 COMMON KINEMATIC ARRANGEMENTS

There are many possible ways to construct kinematic chains using prismaticand revolute joints. However, in practice, only a few kinematic designs areused. Here we briefly describe the most typical arrangements.

1.3.1 Articulated Manipulator (RRR)

The articulated manipulator is also called a revolute, elbow, or anthro-pomorphic manipulator. The ABB IRB1400 articulated arm is shown inFigure 1.9. In the anthropomorphic design the three links are designatedas the body, upper arm, and forearm, respectively, as shown in Figure 1.9.The joint axes are designated as the waist (z0), shoulder (z1), and elbow(z2). Typically, the joint axis z2 is parallel to z1 and both z1 and z2 are

1.3. COMMON KINEMATIC ARRANGEMENTS 13

z1

z0

z2

θ3θ2

θ1

Shoulder

ForearmElbow

Base

Body

Figure 1.9: The ABB IRB1400 Robot, a six degree-of-freedom elbow manip-ulator (right). The symbolic representation of this manipulator (left) showswhy it is referred to as an anthropomorphic robot. The links and joints areanalogous to human joints and limbs. (Photo courtesy of ABB.)

perpendicular to z0. The workspace of the revolute manipulator is shown inFigure 1.10. The revolute manipulator provides for relatively large freedomof movement in a compact space.

Side Top

Figure 1.10: Workspace of the elbow manipulator. The elbow manipulatorprovides a larger workspace than other kinematic designs relative to its size.

An alternate revolute joint design is the parallelogram linkage suchas the ABB IRB6400, shown in Figure 1.11. The parallelogram linkage isless dexterous than the elbow manipulator but has several advantages thatmake it an attractive and popular design. The most notable feature of the


parallelogram linkage manipulator is that the actuator for joint 3 is locatedon link 1. Since the weight of the motor is born by link 1, links 2 and 3 canbe made more lightweight and the motors themselves can be less powerful.Also, the dynamics of the parallelogram manipulator are simpler than thoseof the elbow manipulator making it easier to control.

Figure 1.11: The ABB IRB6400 manipulator utilizes a parallelogram linkagedesign. The motor that actuates the elbow joint is located on the shoulder,which reduces the weight of the upper arm. A general principle in manip-ulator design is to locate as much of the mass of the robot away from thedistal links as possible. (Photo courtesy of ABB.)

1.3.2 Spherical Manipulator (RRP)

By replacing the third or elbow joint in the revolute manipulator by a pris-matic joint one obtains the spherical manipulator shown in Figure 1.12. Theterm spherical manipulator derives from the fact that the joint coordi-nates coincide with the spherical coordinates of the end effector relative toa coordinate frame located at the shoulder joint. Figure 1.12 shows theStanford Arm, one of the most well-known spherical robots.


z1

z0

θ2

θ1

d3

z2

Figure 1.12: The Stanford Arm is an example of a spherical manipulator.The earliest manipulator designs were spherical robots. (Photo courtesy ofthe Coordinated Science Lab, University of Illinois at Urbana-Champaign.)

1.3.3 SCARA Manipulator (RRP)

The SCARA arm (for Selective Compliant Articulated Robot for Assembly)shown in Figure 1.14 is a popular manipulator, which, as its name suggests,is tailored for assembly operations. Although the SCARA has an RRP struc-ture, it is quite different from the spherical manipulator in both appearanceand in its range of applications. Unlike the spherical design, which has z0perpendicular to z1, and z1 perpendicular to z2, the SCARA has z0, z1, andz2 mutually parallel. Figure 1.13 shows the symbolic representation of theSCARA arm and Figure 1.14 shows the Adept Cobra Smart600.

θ1

z0

z1 z2

θ2

d3

Figure 1.13: Symbolic representation of the SCARA arm


Figure 1.14: The Adept Cobra Smart600 SCARA Robot. The SCARAdesign is ideal for table top assembly, pick-and-place tasks, and certain typesof packaging applications. (Photo Courtesy of Adept Technology, Inc.)

1.3.4 Cylindrical Manipulator (RPP)

The cylindrical manipulator is shown in Figure 1.15. The first joint is rev-olute and produces a rotation about the base, while the second and thirdjoints are prismatic. As the name suggests, the joint variables are the cylin-drical coordinates of the end effector with respect to the base.

1.3.5 Cartesian Manipulator (PPP)

A manipulator whose first three joints are prismatic is known as a Carte-sian manipulator. The joint variables of the Cartesian manipulator are theCartesian coordinates of the end effector with respect to the base. As mightbe expected the kinematic description of this manipulator is the simplest ofall manipulators. Cartesian manipulators are useful for table-top assemblyapplications and, as gantry robots, for transfer of material or cargo. Anexample of a Cartesian robot, from Epson, is shown in Figure 1.16.

The workspaces of the spherical, SCARA, cylindrical, and Cartesiangeometries are shown in Figure 1.17


θ1

d3

z2

z0

z1

d2

Figure 1.15: The Seiko RT3300 Robot cylindrical robot. Cylindrical robotsare often used in materials transfer tasks. (Photo courtesy of Epson Robots.)

d2

z1

z0d1

d3

z2

Figure 1.16: The Epson Cartesian Robot. Cartesian robot designs allowincreased structural rigidity and hence higher precision. Cartesian robots areoften used in pick and place operations. (Photo courtesy of Epson Robots.)


(a) (b)

(c) (d)

Figure 1.17: Comparison of the workspaces of the a) spherical, b) SCARA,c) cylindrical, and d) Cartesian robots. The nature of the workspace dictatesthe types of application for which each design can be used.

1.4. OUTLINE OF THE TEXT 19

1.3.6 Parallel Manipulator

A parallel manipulator is one in which some subset of the links form aclosed chain. More specifically, a parallel manipulator has two or more kine-matic chains connecting the base to the end effector. Figure 1.18 shows theABB IRB940 Tricept robot, which is a parallel manipulator. The closed-chain kinematics of parallel robots can result in greater structural rigidity,and hence greater accuracy, than open chain robots. The kinematic de-scription of parallel robots is fundamentally different from that of serial linkrobots and therefore requires different methods of analysis.

Figure 1.18: The ABB IRB940 Tricept parallel robot. Parallel robots gen-erally have much higher structural rigidity than serial link robots. (Photocourtesy of ABB.)

1.4 OUTLINE OF THE TEXT

A typical application involving an industrial manipulator is shown in Fig-ure 1.19. The manipulator is shown with a grinding tool that it must useto remove a certain amount of metal from a surface. In the present text weare concerned with the following question: What are the basic issues to be

resolved and what must we learn in order to be able to program a robot

to perform such tasks? The ability to answer this question for a full sixdegree-of-freedom manipulator represents the goal of the present text. Theanswer is too complicated to be presented at this point. We can, however,


B

F

A

S

Home

Camera

Figure 1.19: Two-link planar robot example. Each chapter of the text dis-cusses a fundamental concept applicable to the task shown.

use the simple two-link planar mechanism to illustrate some of the majorissues involved and to preview the topics covered in this text.

Suppose we wish to move the manipulator from its home position toposition A, from which point the robot is to follow the contour of the surfaceS to the point B, at constant velocity, while maintaining a prescribed forceF normal to the surface. In so doing the robot will cut or grind the surfaceaccording to a predetermined specification. To accomplish this and evenmore general tasks, we must solve a number of problems. Below we giveexamples of these problems, all of which will be treated in more detail inthe remainder of the text.

Forward Kinematics

The first problem encountered is to describe both the position of the tooland the locations A and B (and most likely the entire surface S) with respectto a common coordinate system. In Chapter 2 we describe representations ofcoordinate systems and transformations among various coordinate systems.

Typically, the manipulator will be able to sense its own position in somemanner using internal sensors (position encoders located at joints 1 and 2)that can measure directly the joint angles θ1 and θ2. We also need thereforeto express the positions A and B in terms of these joint angles. This leadsto the forward kinematics problem studied in Chapter 3, which is todetermine the position and orientation of the end effector or tool in termsof the joint variables.


It is customary to establish a fixed coordinate system, called the worldor base frame to which all objects including the manipulator are referenced.In this case we establish the base coordinate frame o0x0y0 at the base of therobot, as shown in Figure 1.20.

y0

x0θ1

x1

x2

θ2

y1

y2

Figure 1.20: Coordinate frames attached to the links of a two-link planarrobot. Each coordinate frame moves as the corresponding link moves. Themathematical description of the robot motion is thus reduced to a mathe-matical description of moving coordinate frames.

The coordinates (x, y) of the tool are expressed in this coordinate frameas

x = a1 cos θ1 + a2 cos(θ1 + θ2) (1.1)

y = a1 sin θ1 + a2 sin(θ1 + θ2) (1.2)

in which a1 and a2 are the lengths of the two links, respectively. Also theorientation of the tool frame relative to the base frame is given by thedirection cosines of the x2 and y2 axes relative to the x0 and y0 axes, thatis,

x2 · x0 = cos(θ1 + θ2) ; y2 · x0 = − sin(θ1 + θ2)x2 · y0 = sin(θ1 + θ2) ; y2 · y0 = cos(θ1 + θ2)

(1.3)

which we may combine into a rotation matrix

[

x2 · x0 y2 · x0

x2 · y0 y2 · y0

]

=

[

cos(θ1 + θ2) − sin(θ1 + θ2)sin(θ1 + θ2) cos(θ1 + θ2)

]

(1.4)

Equations (1.1), (1.2) and (1.4) are called the forward kinematicequations for this arm. For a six degree-of-freedom robot these equations


are quite complex and cannot be written down as easily as for the two-linkmanipulator. The general procedure that we discuss in Chapter 3 estab-lishes coordinate frames at each joint and allows one to transform system-atically among these frames using matrix transformations. The procedurethat we use is referred to as the Denavit-Hartenberg convention. We thenuse homogeneous coordinates and homogeneous transformations tosimplify the transformation among coordinate frames.

Inverse Kinematics

Now, given the joint angles θ1, θ2 we can determine the end-effector co-ordinates x and y. In order to command the robot to move to location Awe need the inverse; that is, we need the joint variables θ1, θ2 in terms ofthe x and y coordinates of A. This is the problem of inverse kinematics.In other words, given x and y in Equations (1.1) and (1.2), we wish to solvefor the joint angles. Since the forward kinematic equations are nonlinear, asolution may not be easy to find, nor is there a unique solution in general.We can see in the case of a two-link planar mechanism that there may beno solution, for example if the given (x, y) coordinates are out of reach ofthe manipulator. If the given (x, y) coordinates are within the manipula-tor’s reach there may be two solutions as shown in Figure 1.21, the so-called

elbow up

elbow down

Figure 1.21: The two-link elbow robot has two solutions to the inversekinematics except at singular configurations, the elbow up solution and theelbow down solution.

elbow up and elbow down configurations, or there may be exactly onesolution if the manipulator must be fully extended to reach the point. Theremay even be an infinite number of solutions in some cases (Problem 1-20).

Consider the diagram of Figure 1.22. Using the law of cosines1 we see

1See Appendix A


x

a1

a2

θ1

θ2

y

Figure 1.22: Solving for the joint angles of a two-link planar arm.

that the angle θ2 is given by

cos θ2 =x2 + y2 − a2

1 − a22

2a1a2:= D (1.5)

We could now determine θ2 as θ2 = cos−1(D). However, a better way tofind θ2 is to notice that if cos(θ2) is given by Equation (1.5) then sin(θ2) isgiven as

sin(θ2) = ±√

1 −D2 (1.6)

and, hence, θ2 can be found by

θ2 = tan−1 ±√

1 −D2

D(1.7)

The advantage of this latter approach is that both the elbow-up andelbow-down solutions are recovered by choosing the negative and positivesigns in Equation (1.7), respectively.

It is left as an exercise (Problem 1-18) to show that θ1 is now given as

θ1 = tan−1(y/x) − tan−1

(

a2 sin θ2a1 + a2 cos θ2

)

(1.8)

Notice that the angle θ1 depends on θ2. This makes sense physicallysince we would expect to require a different value for θ1, depending on whichsolution is chosen for θ2.


Velocity Kinematics

In order to follow a contour at constant velocity, or at any prescribedvelocity, we must know the relationship between the velocity of the tool andthe joint velocities. In this case we can differentiate Equations (1.1) and(1.2) to obtain

x = −a1 sin θ1 · θ1 − a2 sin(θ1 + θ2)(θ1 + θ2)

y = a1 cos θ1 · θ1 + a2 cos(θ1 + θ2)(θ1 + θ2)(1.9)

Using the vector notation x =

[

xy

]

and θ =

[

θ1θ2

]

we may write these

equations as

x =

[

−a1 sin θ1 − a2 sin(θ1 + θ2) −a2 sin(θ1 + θ2)a1 cos θ1 + a2 cos(θ1 + θ2) a2 cos(θ1 + θ2)

]

θ (1.10)

= Jθ

The matrix J defined by Equation (1.10) is called the Jacobian of themanipulator and is a fundamental object to determine for any manipulator.In Chapter 4 we present a systematic procedure for deriving the manipulatorJacobian.

The determination of the joint velocities from the end-effector velocitiesis conceptually simple since the velocity relationship is linear. Thus, the jointvelocities are found from the end-effector velocities via the inverse Jacobian

θ = J−1x (1.11)

where J−1 is given by

J−1 =1

a1a2 sin θ2

[

a2 cos(θ1+θ2) a2 sin(θ1+θ2)

−a1 cos θ1−a2 cos(θ1+θ2) −a1 sin θ1−a2 sin(θ1+θ2)

]

The determinant of the Jacobian in Equation (1.10) is equal to a1a2 sin θ2.Therefore, this Jacobian does not have an inverse when θ2 = 0 or θ2 = π,in which case the manipulator is said to be in a singular configuration,such as shown in Figure 1.23 for θ2 = 0.

The determination of such singular configurations is important for sev-eral reasons. At singular configurations there are infinitesimal motions thatare unachievable; that is, the manipulator end effector cannot move in cer-tain directions. In the above example the end effector cannot move in thepositive x2 direction when θ2 = 0. Singular configurations are also relatedto the nonuniqueness of solutions of the inverse kinematics. For example,


θ1

y0

x0

θ2 = 0α1

α2

Figure 1.23: A singular configuration results when the elbow is straight. Inthis configuration the two-link robot has only one degree of freedom.

for a given end-effector position of the two-link planar manipulator, thereare in general two possible solutions to the inverse kinematics. Note thata singular configuration separates these two solutions in the sense that themanipulator cannot go from one to the other without passing through a sin-gularity. For many applications it is important to plan manipulator motionsin such a way that singular configurations are avoided.

Path Planning and Trajectory Generation

The robot control problem is typically decomposed hierarchically intothree tasks: path planning, trajectory generation, and trajectorytracking. The path planning problem, considered in Chapter 5, is to deter-mine a path in task space (or configuration space) to move the robot to agoal position while avoiding collisions with objects in its workspace. Thesepaths encode position and orientation information without timing considera-tions, i.e. without considering velocities and accelerations along the plannedpaths. The trajectory generation problem, also considered in Chapter 5, isto generate reference trajectories that determine the time history of themanipulator along a given path or between initial and final configurations.These are typically given in joint space as polynomial functions of time. Wediscuss the most common polynomial interpolation schemes used to generatethese trajectories.

Independent Joint Control

Once reference trajectories for the robot are specified, it is the task of thecontrol system to track them. In Chapter 6 we discuss the motion controlproblem. We treat the twin problems of tracking and disturbance re-jection, which are to determining the control inputs necessary to follow, or


track, a reference trajectory, while simultaneously rejecting disturbancesdue to unmodeled dynamic effects such as friction and noise. We first modelthe actuator and drive-train dynamics and discuss the design of independentjoint control algorithms. A block diagram of a single-input/single-output(SISO) feedback control system is shown in Figure 1.24.

Disturbance

+��Referencetrajectory +//

⊕

// Compensator // Poweramplifier

+//⊕

// PlantOutput//

ooSensor

−OO

Figure 1.24: Basic structure of a feedback control system. The Compensatormeasures the error between a reference and a measured output and producesa signal to the plant that is designed to drive the error to zero despite thepresences of disturbances.

We detail the standard approaches to robot control based on both fre-quency domain and state space techniques. We also introduce the notion offeedforward control for tracking time varying trajectories.

Dynamics

The simple control strategies considered in Chapter 6 are based on theactuator and drive-train dynamics but ignores the coupling effects due tothe motion of the links. In Chapter 7 we develop techniques based on La-grangian dynamics for systematically deriving the equations of motion ofrigid-link robots. Deriving the dynamic equations of motion for robots isnot a simple task due to the large number of degrees of freedom and thenonlinearities present in the system. We also discuss the so-called recur-sive Newton-Euler method for deriving the robot equations of motion.The Newton-Euler formulation is well-suited to real-time computation forboth simulation and control.

Multivariable Control

In Chapter 8 we discuss more advanced control techniques based on theLagrangian dynamic equations of motion derived in Chapter 7. We introducethe fundamental notions of computed torque and inverse dynamics as ameans for compensating the complex nonlinear interaction forces among thelinks of the manipulator. Robust and adaptive control are also introducedin using the second method of Lyapunov. Chapter 10 provides some

PROBLEMS 27

additional advanced techniques from geometric nonlinear control theory thatare useful for controlling high performance robots. We also discuss thecontrol of so-called nonholonomic systems such as mobile robots.

Force Control

In the example robot task above, once the manipulator has reached loca-tion A, it must follow the contour S maintaining a constant force normal tothe surface. Conceivably, knowing the location of the object and the shapeof the contour, one could carry out this task using position control alone.This would be quite difficult to accomplish in practice, however. Since themanipulator itself possesses high rigidity, any errors in position due to uncer-tainty in the exact location of the surface or tool would give rise to extremelylarge forces at the end effector that could damage the tool, the surface, orthe robot. A better approach is to measure the forces of interaction directlyand use a force control scheme to accomplish the task. In Chapter 9we discuss force control and compliance, along with common approaches toforce control, namely hybrid control and impedance control.

Computer Vision

Cameras have become reliable and relatively inexpensive sensors in manyrobotic applications. Unlike joint sensors, which give information aboutthe internal configuration of the robot, cameras can be used not only tomeasure the position of the robot but also to locate objects robot in therobot’s workspace. In Chapter 11 we discuss the use of computer vision todetermine position and orientation of objects.

Vision-Based Control

In some cases, we may wish to control the motion of the manipulatorrelative to some target as the end effector moves through free space. Here,force control cannot be used. Instead, we can use computer vision to closethe control loop around the vision sensor. This is the topic of Chapter 12.There are several approaches to vision-based control, but we will focus on themethod of Image-Based Visual Servo (IBVS). With IBVS, an error measuredin image coordinates is directly mapped to a control input that governs themotion of the camera. This method has become very popular in recentyears, and it relies on mathematical development analogous to that given inChapter 4.

PROBLEMS

1-1 What are the key features that distinguish robots from other forms ofautomation such as CNC milling machines?


1-2 Briefly define each of the following terms: forward kinematics, inversekinematics, trajectory planning, workspace, accuracy, repeatability,resolution, joint variable, spherical wrist, end effector.

1-3 What are the main ways to classify robots?

1-4 Make a list of 10 robot applications. For each application discuss whichtype of manipulator would be best suited; which least suited. Justifyyour choices in each case.

1-5 List several applications for nonservo robots; for point-to-point robots;for continuous path robots.

1-6 List five applications that a continuous path robot could do that apoint-to-point robot could not do.

1-7 List five applications for which computer vision would be useful inrobotics.

1-8 List five applications for which either tactile sensing or force feedbackcontrol would be useful in robotics.

1-9 Find out how many industrial robots are currently in operation inJapan. How many are in operation in the United States? What coun-try ranks third in the number of industrial robots in use?

1-10 Suppose we could close every factory today and reopen them tomorrowfully automated with robots. What would be some of the economicand social consequences of such a development?

1-11 Suppose a law were passed banning all future use of industrial robots.What would be some of the economic and social consequences of suchan act?

1-12 Discuss applications for which redundant manipulators would be use-ful.

1-13 Referring to Figure 1.25, suppose that the tip of a single link travels adistance d between two points. A linear axis would travel the distanced while a rotational link would travel through an arc length ℓθ asshown. Using the law of cosines show that the distance d is given by

d = ℓ√

2(1 − cos θ)

which is of course less than ℓθ. With 10-bit accuracy and ℓ = 1m,θ = 90◦ what is the resolution of the linear link? of the rotationallink?

PROBLEMS 29

ℓ

ℓθ

d

Figure 1.25: Diagram for Problem 1-15

1-14 For the single-link revolute arm shown in Figure 1.25, if the length ofthe link is 50 cm and the arm travels 180 degrees, what is the controlresolution obtained with an 8-bit encoder?

1-15 Repeat Problem 1.14 assuming that the 8-bit encoder is located on themotor shaft that is connected to the link through a 50:1 gear reduction.Assume perfect gears.

1-16 Why is accuracy generally less than repeatability?

1-17 How could manipulator accuracy be improved using endpoint sensing?What difficulties might endpoint sensing introduce into the controlproblem?

1-18 Derive Equation (1.8).

1-19 For the two-link manipulator of Figure 1.20 suppose a1 = a2 = 1.

1. Find the coordinates of the tool when θ1 = π6 and θ2 = π

2 .

2. If the joint velocities are constant at θ1 = 1, θ2 = 2, what isthe velocity of the tool? What is the instantaneous tool velocitywhen θ1 = θ2 = π

4 ?

3. Write a computer program to plot the joint angles as a functionof time given the tool locations and velocities as a function oftime in Cartesian coordinates.

4. Suppose we desire that the tool follow a straight line between thepoints (0,2) and (2,0) at constant speed s. Plot the time historyof joint angles.

1-20 For the two-link planar manipulator of Figure 1.20 is it possible forthere to be an infinite number of solutions to the inverse kinematicequations? If so, explain how this can occur.


1-21 Explain why it might be desirable to reduce the mass of distal links ina manipulator design. List some ways this can be done. Discuss anypossible disadvantages of such designs.

NOTES AND REFERENCES

We give below some of the important milestones in the history of modernrobotics.

1947 — The first servoed electric powered teleoperator is developed.

1948 — A teleoperator is developed incorporating force feedback.

1949 — Research on numerically controlled milling machine is initiated.

1954 — George Devol designs the first programmable robot

1956 — Joseph Engelberger, a Columbia University physics student, buysthe rights to Devol’s robot and founds the Unimation Company.

1961 — The first Unimate robot is installed in a Trenton, New Jersey plantof General Motors to tend a die casting machine.

1961 — The first robot incorporating force feedback is developed.

1963 — The first robot vision system is developed.

1971 — The Stanford Arm is developed at Stanford University.

1973 — The first robot programming language (WAVE) is developed atStanford.

1974 — Cincinnati Milacron introduced the T 3 robot with computer con-trol.

1975 — Unimation Inc. registers its first financial profit.

1976 — The Remote Center Compliance (RCC) device for part insertionin assembly is developed at Draper Labs in Boston.

1976 — Robot arms are used on the Viking I and II space probes and landon Mars.

1978 — Unimation introduces the PUMA robot, based on designs from aGeneral Motors study.

1979 — The SCARA robot design is introduced in Japan.

1981 — The first direct-drive robot is developed at Carnegie-Mellon Uni-versity.

NOTES AND REFERENCES 31

1982 — Fanuc of Japan and General Motors form GM Fanuc to marketrobots in North America.

1983 — Adept Technology is founded and successfully markets the direct-drive robot.

1986 — The underwater robot, Jason, of the Woods Hole OceanographicInstitute, explores the wreck of the Titanic, found a year earlier byDr. Robert Barnard.

1988 — Staubli Group purchases Unimation from Westinghouse.

1988 — The IEEE Robotics and Automation Society is formed.

1993 — The experimental robot, ROTEX, of the German Aerospace Agency(DLR) was flown aboard the space shuttle Columbia and performeda variety of tasks under both teleoperated and sensor-based offlineprogrammed modes.

1996 — Honda unveils its Humanoid robot; a project begun in secret in1986.

1997 — The first robot soccer competition, RoboCup-97, is held in Nagoya,Japan and draws 40 teams from around the world.

1997 — The Sojourner mobile robot travels to Mars aboard NASA’s MarsPathFinder mission.

2001 — Sony begins to mass produce the first household robot, a robotdog named Aibo.

2001 — The Space Station Remote Manipulation System (SSRMS) islaunched in space on board the space shuttle Endeavor to facilitatecontinued construction of the space station.

2001 — The first telesurgery is performed when surgeons in New Yorkperform a laparoscopic gall bladder removal on a woman in Strasbourg,France.

2001 — Robots are used to search for victims at the World Trade Centersite after the September 11th tragedy.

2002 — Honda’s Humanoid Robot ASIMO rings the opening bell at theNew York Stock Exchange on February 15th.

2005 — ROKVISS (Robotic Component Verification on board the Interna-tional Space Station), the experimental teleoperated arm built by theGerman Aerospace Center (DLR), undergoes its first tests in space.


Many books have been written about basic and advanced topics inrobotics. Below is an incomplete list of references.

• H. Asada and J-J. Slotine. Robot Analysis and Control. Wiley, NewYork, 1986.

• G. A. Bekey, Autonomous Robots. MIT Press, Cambridge, MA, 2005.

• M. Brady et al., editors. Robot Motion: Planning and Control. MITPress, Cambridge, MA, 1983.

• H. Choset, K. M. Lynch, S. Hutchinson, G. Kantor, W. Burgard, L. E.Kavraki, and S. Thrun. Principles of Robot Motion: Theory, Algorithms,and Implementations. MIT Press, Cambridge, MA, 2005.

• J. Craig. Introduction to Robotics: Mechanics and Control. AddisonWesley, Reading, MA, 1986.

• R. Dorf. Robotics and Automated Manufacturing. Reston, VA, 1983.

• J. Engleberger. Robotics in Practice. Kogan Page, London, 1980.

• K.S. Fu, R. C. Gonzalez, and C.S.G. Lee. Robotics: Control Sensing,Vision, and Intelligence. McGraw-Hill, St Louis, 1987.

• B. K. Ghosh, N. Xi and T. J. Tarn. Control in Robotics and Automation:Sensor-Based Integration, Academic Press, San Diego, CA, 1999.

• T. R. Kurfess. Robotics and Automation Handbook, CRC Press, BocaRaton, FL, 2005.

• J. C. Latombe. Robot Motion Planning. Kluwer Academic Publishers,Boston, 1991.

• M. T. Mason. Mechanics of Robotic Manipulation, MIT Press, Cam-bridge, MA, 2001.

• R.M. Murray, Z. Li, and S.S. Sastry. A Mathematical Introduction toRobotics. CRC Press, Boca Raton, FL, 1994.

• S. B. Niku. Introduction to Robotics: Analysis, Systems, Applications.Prentice Hall, Upper Saddle River, NJ, 2001.

• R. Paul. Robot Manipulators: Mathematics, Programming and Control.MIT Press, Cambridge, MA, 1982.

• L. Sciavicco and B. Siciliano. Modelling and Control of Robot Manipula-tors, 2nd Edition, Springer-Verlag, London, 2000.

• M. Shahinpoor. Robot Engineering Textbook. Harper and Row, NewYork, 1987.

• W. Snyder. Industrial Robots: Computer Interfacing and Control. Prentice-Hall, Englewood Cliffs, NJ, 1985.

NOTES AND REFERENCES 33

• M.W. Spong, F.L. Lewis and C.T. Abdallah. Robot Control: Dynamics,Motion Planning, and Analysis, IEEE Press, Boca Raton, FL, 1992.

• M. Spong and M. Vidyasagar. Robot Dynamics and Control. John Wileyand Sons, NY, NY, 1989.

• W. Wolovich. Robotics: Basic Analysis and Design. Holt, Rinehart, andWinston, New York, 1985.

There is a great deal of ongoing research in robotics. Current researchresults can be found in journals such as• IEEE Transactions on Robotics (previously IEEE Transactions on Robotics

and Automation)

• IEEE Robotics and Automation Magazine

• International Journal of Robotics Research

• Robotics and Autonomous Systems

• Journal of Robotic Systems

• Robotica

• Journal of Intelligent and Robotic Systems

• Autonomous Robots

• Advanced Robotics


Chapter 2

RIGID MOTIONS ANDHOMOGENEOUSTRANSFORMATIONS

A large part of robot kinematics is concerned with establishing variouscoordinate frames to represent the positions and orientations of rigid ob-jects, and with transformations among these coordinate frames. Indeed, thegeometry of three-dimensional space and of rigid motions plays a central rolein all aspects of robotic manipulation. In this chapter we study the opera-tions of rotation and translation, and introduce the notion of homogeneoustransformations.1 Homogeneous transformations combine the operations ofrotation and translation into a single matrix multiplication, and are usedin Chapter 3 to derive the so-called forward kinematic equations of rigidmanipulators.

We begin by examining representations of points and vectors in a Eu-clidean space equipped with multiple coordinate frames. Following this, weintroduce the concept of a rotation matrix to represent relative orientationsamong coordinate frames. Then we combine these two concepts to buildhomogeneous transformation matrices, which can be used to simultaneouslyrepresent the position and orientation of one coordinate frame relative toanother. Furthermore, homogeneous transformation matrices can be usedto perform coordinate transformations. Such transformations allow us torepresent various quantities in different coordinate frames, a facility that wewill often exploit in subsequent chapters.

1Since we make extensive use of elementary matrix theory, the reader may wish toreview Appendix B before beginning this chapter.

36 CHAPTER 2. RIGID MOTIONS

Figure 2.1: Two coordinate frames, a point p, and two vectors v1 and v2.

2.1 REPRESENTING POSITIONS

Before developing representation schemes for points and vectors, it is in-structive to distinguish between the two fundamental approaches to geo-metric reasoning: the synthetic approach and the analytic approach. Inthe former, one reasons directly about geometric entities (e.g., points orlines), while in the latter, one represents these entities using coordinatesor equations, and reasoning is performed via algebraic manipulations. Thelatter approach requires the choice of a reference coordinate frame. A co-ordinate frame consists of an origin (a single point in space), and two orthree orthogonal coordinate axes, for two- and three-dimensional spaces,respectively.

Consider Figure 2.1, which shows two coordinate frames that differ inorientation by an angle of 45◦. Using the synthetic approach, without everassigning coordinates to points or vectors, one can say that x0 is perpendic-ular to y0, or that v1 × v2 defines a vector that is perpendicular to the planecontaining v1 and v2, in this case pointing out of the page.

In robotics, one typically uses analytic reasoning, since robot tasks areoften defined using Cartesian coordinates. Of course, in order to assigncoordinates it is necessary to specify a reference coordinate frame. Consideragain Figure 2.1. We could specify the coordinates of the point p with respectto either frame o0x0y0 or frame o1x1y1. In the former case, we might assignto p the coordinate vector [5, 6]T , and in the latter case [−2.8, 4.2]T . So thatthe reference frame will always be clear, we will adopt a notation in whicha superscript is used to denote the reference frame. Thus, we would write

p0 =

[

56

]

, p1 =

[

−2.84.2

]

2.1. REPRESENTING POSITIONS 37

Geometrically, a point corresponds to a specific location in space. Westress here that p is a geometric entity, a point in space, while both p0 andp1 are coordinate vectors that represent the location of this point in spacewith respect to coordinate frames o0x0y0 and o1x1y1, respectively.

Since the origin of a coordinate frame is just a point in space, we canassign coordinates that represent the position of the origin of one coordinateframe with respect to another. In Figure 2.1, for example, we have

o01 =

[

105

]

, o10 =

[

−10.63.5

]

In cases where there is only a single coordinate frame, or in which thereference frame is obvious, we will often omit the superscript. This is a slightabuse of notation, and the reader is advised to bear in mind the differencebetween the geometric entity called p and any particular coordinate vectorthat is assigned to represent p. The former is independent of the choiceof coordinate frames, while the latter obviously depends on the choice ofcoordinate frames.

While a point corresponds to a specific location in space, a vector specifiesa direction and a magnitude. Vectors can be used, for example, to representdisplacements or forces. Therefore, while the point p is not equivalent tothe vector v1, the displacement from the origin o0 to the point p is givenby the vector v1. In this text, we will use the term vector to refer to whatare sometimes called free vectors, i.e., vectors that are not constrained to belocated at a particular point in space. Under this convention, it is clear thatpoints and vectors are not equivalent, since points refer to specific locationsin space, but a vector can be moved to any location in space. Under thisconvention, two vectors are equal if they have the same direction and thesame magnitude.

When assigning coordinates to vectors, we use the same notational con-vention that we used when assigning coordinates to points. Thus, v1 and v2are geometric entities that are invariant with respect to the choice of coordi-nate frames, but the representation by coordinates of these vectors dependsdirectly on the choice of reference coordinate frame. In the example of Fig-ure 2.1, we would obtain

v01 =

[

56

]

, v11 =

[

7.770.8

]

, v02 =

[

−5.11

]

, v12 =

[

−2.894.2

]

In order to perform algebraic manipulations using coordinates, it is es-sential that all coordinate vectors be defined with respect to the same coor-dinate frame. In the case of free vectors, it is enough that they be defined


with respect to “parallel” coordinate frames, i.e. frames whose respectivecoordinate axes are parallel, since only their magnitude and direction arespecified and not their absolute locations in space.

Using this convention, an expression of the form v11 +v2

2, where v11 and v2

2

are as in Figure 2.1, is not defined since the frames o0x0y0 and o1x1y1 are notparallel. Thus, we see a clear need not only for a representation system thatallows points to be expressed with respect to various coordinate frames, butalso for a mechanism that allows us to transform the coordinates of pointsfrom one coordinate frame to another. Such coordinate transformations arethe topic for much of the remainder of this chapter.

2.2 REPRESENTING ROTATIONS

In order to represent the relative position and orientation of one rigid bodywith respect to another, we will attach coordinate frames to each body, andthen specify the geometric relationships between these coordinate frames.In Section 2.1 we saw how one can represent the position of the origin ofone frame with respect to another frame. In this section, we address theproblem of describing the orientation of one coordinate frame relative toanother frame. We begin with the case of rotations in the plane, and thengeneralize our results to the case of orientations in a three-dimensional space.

2.2.1 Rotation in the Plane

Figure 2.2 shows two coordinate frames, with frame o1x1y1 being obtainedby rotating frame o0x0y0 by an angle θ. Perhaps the most obvious way torepresent the relative orientation of these two frames is to merely specifythe angle of rotation θ. There are two immediate disadvantages to such arepresentation. First, there is a discontinuity in the mapping from relativeorientation to the value of θ in a neighborhood of θ = 0. In particular, forθ = 2π − ǫ, small changes in orientation can produce large changes in thevalue of θ (i.e., a rotation by ǫ causes θ to “wrap around” to zero). Second,this choice of representation does not scale well to the three-dimensionalcase.

A slightly less obvious way to specify the orientation is to specify thecoordinate vectors for the axes of frame o1x1y1 with respect to coordinateframe o0x0y0:

R01 =

[

x01 | y0

1

]

in which x01 and y0

1 are the coordinates in frame o0x0y0 of unit vectors x1

2.2. REPRESENTING ROTATIONS 39

o0, o1

y0

y1

θ

x1

sin θ

cos θ

x0

Figure 2.2: Coordinate frame o1x1y1 is oriented at an angle θ with respectto o0x0y0.

and y1, respectively2. A matrix in this form is called a rotation matrix.Rotation matrices have a number of special properties that we will discussbelow.

In the two-dimensional case, it is straightforward to compute the entriesof this matrix. As illustrated in Figure 2.2,

x01 =

[

cos θsin θ

]

, y01 =

[

− sin θcos θ

]

which gives

R01 =

[

cos θ − sin θsin θ cos θ

]

(2.1)

Note that we have continued to use the notational convention of allowingthe superscript to denote the reference frame. Thus, R0

1 is a matrix whosecolumn vectors are the coordinates of the unit vectors along the axes offrame o1x1y1 expressed relative to frame o0x0y0.

Although we have derived the entries for R01 in terms of the angle θ,

it is not necessary that we do so. An alternative approach, and one thatscales nicely to the three-dimensional case, is to build the rotation matrix byprojecting the axes of frame o1x1y1 onto the coordinate axes of frame o0x0y0.Recalling that the dot product of two unit vectors gives the projection of

2We will use xi, yi to denote both coordinate axes and unit vectors along the coordinateaxes depending on the context.


one onto the other, we obtain

x01 =

[

x1 · x0

x1 · y0

]

, y01 =

[

y1 · x0

y1 · y0

]

which can be combined to obtain the rotation matrix

R01 =

[

x1 · x0 y1 · x0

x1 · y0 y1 · y0

]

Thus, the columns of R01 specify the direction cosines of the coordinate axes

of o1x1y1 relative to the coordinate axes of o0x0y0. For example, the firstcolumn [x1 · x0, x1 · y0]

T of R01 specifies the direction of x1 relative to the

frame o0x0y0. Note that the right hand sides of these equations are defined interms of geometric entities, and not in terms of their coordinates. ExaminingFigure 2.2 it can be seen that this method of defining the rotation matrixby projection gives the same result as was obtained in Equation (2.1).

If we desired instead to describe the orientation of frame o0x0y0 withrespect to the frame o1x1y1 (i.e., if we desired to use the frame o1x1y1 asthe reference frame), we would construct a rotation matrix of the form

R10 =

[

x0 · x1 y0 · x1

x0 · y1 y0 · y1

]

Since the dot product is commutative, (i.e. xi · yj = yj · xi), we see that

R10 = (R0

1)T

In a geometric sense, the orientation of o0x0y0 with respect to the frameo1x1y1 is the inverse of the orientation of o1x1y1 with respect to the frameo0x0y0. Algebraically, using the fact that coordinate axes are mutually or-thogonal, it can readily be seen that

(R01)T = (R0

1)−1

The column vectors of R01 are of unit length and mutually orthogonal

(Problem 2-4). Such a matrix is said to be orthogonal. It can also be shown(Problem 2-5) that detR0

1 = ±1. If we restrict ourselves to right-handedcoordinate frames, as defined in Appendix B, then detR0

1 = +1 (Problem2-5). It is customary to refer to the set of all such n × n matrices by thesymbol SO(n), which denotes the Special Orthogonal group of ordern. The properties of such matrices are summarized in Figure 2.3.

To provide further geometric intuition for the notion of the inverse of arotation matrix, note that in the two-dimensional case, the inverse of the


For any R ∈ SO(n) the following hold.

• RT = R−1 ∈ SO(n)

• The columns (and therefore the rows) of R are mutually orthogonal

• Each column (and therefore each row) of R is a unit vector

• detR = 1

Figure 2.3: Properties of the Matrix group SO(n)

rotation matrix corresponding to a rotation by angle θ can also be easilycomputed simply by constructing the rotation matrix for a rotation by theangle −θ:

[

cos(−θ) − sin(−θ)sin(−θ) cos(−θ)

]

=

[

cos θ sin θ− sin θ cos θ

]

=

[


]T

2.2.2 Rotations in Three Dimensions

The projection technique described above scales nicely to the three-dimensionalcase. In three dimensions, each axis of the frame o1x1y1z1 is projected ontocoordinate frame o0x0y0z0. The resulting rotation matrix is given by

R01 =

x1 · x0 y1 · x0 z1 · x0

x1 · y0 y1 · y0 z1 · y0

x1 · z0 y1 · z0 z1 · z0

As was the case for rotation matrices in two dimensions, matrices in thisform are orthogonal, with determinant equal to 1. In this case, 3×3 rotationmatrices belong to the group SO(3).

Example 2.1Suppose the frame o1x1y1z1 is rotated through an angle θ about the z0-

axis, and we wish to find the resulting transformation matrix R01. By con-

vention, the right hand rule (see Appendix B) defines the positive sense forthe angle θ to be such that rotation by θ about the z-axis would advance aright-hand threaded screw along the positive z-axis. From Figure 2.4 we seethat

x1 · x0 = cos θ, y1 · x0 = − sin θ,

x1 · y0 = sin θ, y1 · y0 = cos θ


sinθx1

x0

y0

y1

z0, z1

θcos

θ

Figure 2.4: Rotation about z0 by an angle θ.

and

z0 · z1 = 1

while all other dot products are zero. Thus, the rotation matrix R01 has a

particularly simple form in this case, namely

R01 =

cos θ − sin θ 0sin θ cos θ 0

0 0 1

(2.2)

⋄The rotation matrix given in Equation (2.2) is called a basic rotation

matrix (about the z-axis). In this case we find it useful to use the moredescriptive notation Rz,θ instead of R0

1 to denote the matrix. It is easy toverify that the basic rotation matrix Rz,θ has the properties

Rz,0 = I (2.3)

Rz,θRz,φ = Rz,θ+φ (2.4)

which together imply

(

Rz,θ)−1

= Rz,−θ (2.5)

Similarly, the basic rotation matrices representing rotations about the x


and y-axes are given as (Problem 2-8)

Rx,θ =

1 0 00 cos θ − sin θ0 sin θ cos θ

(2.6)

Ry,θ =

cos θ 0 sin θ0 1 0

− sin θ 0 cos θ

(2.7)

which also satisfy properties analogous to Equations (2.3)-(2.5).

Example 2.2

x1

z0

y0, z1

y1

x045

o

Figure 2.5: Defining the relative orientation of two frames.

Consider the frames o0x0y0z0 and o1x1y1z1 shown in Figure 2.5. Project-ing the unit vectors x1, y1, z1 onto x0, y0, z0 gives the coordinates of x1, y1, z1in the o0x0y0z0 frame. We see that the coordinates of x1, y1 and z1 aregiven by

x1 =

1√2

01√2

, y1 =

1√2

0− 1√

2

, z1 =

010

The rotation matrix R01 specifying the orientation of o1x1y1z1 relative to


o0x0y0z0 has these as its column vectors, that is,

R01 =

1√2

1√2

0

0 0 11√2

− 1√2

0

⋄

2.3 ROTATIONAL TRANSFORMATIONS

y1

z1

z0

x0

x1

0

S

p

yo

Figure 2.6: Coordinate frame attached to a rigid body.

Figure 2.6 shows a rigid object S to which a coordinate frame o1x1y1z1is attached. Given the coordinates p1 of the point p (i.e., given the coor-dinates of p with respect to the frame o1x1y1z1), we wish to determine thecoordinates of p relative to a fixed reference frame o0x0y0z0. The coordinatesp1 = [u, v, w]T satisfy the equation

p = ux1 + vy1 + wz1

In a similar way, we can obtain an expression for the coordinates p0 byprojecting the point p onto the coordinate axes of the frame o0x0y0z0, giving

p0 =

p · x0

p · y0

p · z0

Combining these two equations we obtain

2.3. ROTATIONAL TRANSFORMATIONS 45

z0

x0 x0

z0

pby0 y0

(a) (b)

pa

Figure 2.7: The block in (b) is obtained by rotating the block in (a) by πabout z0.

p0 =

(ux1 + vy1 + wz1) · x0

(ux1 + vy1 + wz1) · y0(ux1 + vy1 + wz1) · z0

=

ux1 · x0 + vy1 · x0 + wz1 · x0

ux1 · y0 + vy1 · y0 + wz1 · y0

ux1 · z0 + vy1 · z0 + wz1 · z0

=

x1 · x0 y1 · x0 z1 · x0

x1 · y0 y1 · y0 z1 · y0

x1 · z0 y1 · z0 z1 · z0

uvw

But the matrix in this final equation is merely the rotation matrix R01, which

leads top0 = R0

1p1 (2.8)

Thus, the rotation matrix R01 can be used not only to represent the

orientation of coordinate frame o1x1y1z1 with respect to frame o0x0y0z0,but also to transform the coordinates of a point from one frame to another.If a given point is expressed relative to o1x1y1z1 by coordinates p1, thenR0

1p1 represents the same point expressed relative to the frame o0x0y0z0.We can also use rotation matrices to represent rigid motions that cor-

respond to pure rotation. Consider Figure 2.7. One corner of the block inFigure 2.7(a) is located at the point pa in space. Figure 2.7(b) shows thesame block after it has been rotated about z0 by the angle π. In Figure2.7(b), the same corner of the block is now located at point pb in space. Itis possible to derive the coordinates for pb given only the coordinates for pa


and the rotation matrix that corresponds to the rotation about z0. To seehow this can be accomplished, imagine that a coordinate frame is rigidly at-tached to the block in Figure 2.7(a), such that it is coincident with the frameo0x0y0z0. After the rotation by π, the block’s coordinate frame, which isrigidly attached to the block, is also rotated by π. If we denote this rotatedframe by o1x1y1z1, we obtain

R01 = Rz,π =

−1 0 00 −1 00 0 1

In the local coordinate frame o1x1y1z1, the point pb has the coordinaterepresentation p1

b . To obtain its coordinates with respect to frame o0x0y0z0,we merely apply the coordinate transformation Equation (2.8), giving

p0b = Rz,πp

1b

It is important to notice that the local coordinates p1b of the corner of the

block do not change as the block rotates, since they are defined in termsof the block’s own coordinate frame. Therefore, when the block’s frame isaligned with the reference frame o0x0y0z0 (i.e., before the rotation is per-formed), the coordinates p1

b equals p0a, since before the rotation is performed,

the point pa is coincident with the corner of the block. Therefore, we cansubstitute p0

a into the previous equation to obtain

p0b = Rz,πp

0a

This equation shows how to use a rotation matrix to represent a rotationalmotion. In particular, if the point pb is obtained by rotating the point pa asdefined by the rotation matrix R, then the coordinates of pb with respect tothe reference frame are given by

p0b = Rp0

a

This same approach can be used to rotate vectors with respect to a coordi-nate frame, as the following example illustrates.

Example 2.3The vector v with coordinates v0 = [0, 1, 1]T is rotated about y0 by π

2 asshown in Figure 2.8. The resulting vector v1 has coordinates given by

v01 = Ry,π

2

v0 (2.9)

=

0 0 10 1 0

−1 0 0

011

=

110

(2.10)

2.3. ROTATIONAL TRANSFORMATIONS 47

y0

z0

x0

v1

v

θ

Figure 2.8: Rotating a vector about axis y0.

Thus, a third interpretation of a rotation matrix R is as an operatoracting on vectors in a fixed frame. In other words, instead of relating thecoordinates of a fixed vector with respect to two different coordinate frames,Equation (2.9) can represent the coordinates in o0x0y0z0 of a vector v1 thatis obtained from a vector v by a given rotation.

⋄As we have seen, rotation matrices can serve several roles. A rotation

matrix, either R ∈ SO(3) or R ∈ SO(2), can be interpreted in three distinctways:

1. It represents a coordinate transformation relating the coordinates of apoint p in two different frames.

2. It gives the orientation of a transformed coordinate frame with respectto a fixed coordinate frame.

3. It is an operator taking a vector and rotating it to give a new vectorin the same coordinate frame.

The particular interpretation of a given rotation matrix R will be made clearby the context.

2.3.1 Similarity Transformations

A coordinate frame is defined by a set of basis vectors, for example, unitvectors along the three coordinate axes. This means that a rotation matrix,


as a coordinate transformation, can also be viewed as defining a change ofbasis from one frame to another. The matrix representation of a generallinear transformation is transformed from one frame to another using aso-called similarity transformation3. For example, if A is the matrixrepresentation of a given linear transformation in o0x0y0z0 and B is therepresentation of the same linear transformation in o1x1y1z1 then A and Bare related as

B = (R01)

−1AR01 (2.11)

where R01 is the coordinate transformation between frames o1x1y1z1 and

o0x0y0z0. In particular, if A itself is a rotation, then so is B, and thusthe use of similarity transformations allows us to express the same rotationeasily with respect to different frames.

Example 2.4

Henceforth, whenever convenient we use the shorthand notation cθ =cos θ, sθ = sin θ for trigonometric functions. Suppose frames o0x0y0z0 ando1x1y1z1 are related by the rotation

R01 =

0 0 10 1 0

−1 0 0

If A = Rz,θ relative to the frame o0x0y0z0, then, relative to frame o1x1y1z1we have

B = (R01)

−1AR01 =

1 0 00 cθ sθ0 −sθ cθ

In other words, B is a rotation about the z0-axis but expressed relative tothe frame o1x1y1z1. This notion will be useful below and in later sections.

⋄

2.4 COMPOSITION OF ROTATIONS

In this section we discuss the composition of rotations. It is important forsubsequent chapters that the reader understand the material in this sectionthoroughly before moving on.

3See Appendix B.

2.4. COMPOSITION OF ROTATIONS 49

2.4.1 Rotation with Respect to the Current Frame

Recall that the matrix R01 in Equation (2.8) represents a rotational transfor-

mation between the frames o0x0y0z0 and o1x1y1z1. Suppose we now add athird coordinate frame o2x2y2z2 related to the frames o0x0y0z0 and o1x1y1z1by rotational transformations. A given point p can then be represented bycoordinates specified with respect to any of these three frames: p0, p1 andp2. The relationship among these representations of p is

p0 = R01p

1 (2.12)

p1 = R12p

2 (2.13)

p0 = R02p

2 (2.14)

where each Rij is a rotation matrix. Substituting Equation (2.13) into Equa-tion (2.12) gives

p0 = R01R

12p

2 (2.15)

Note that R01 and R0

2 represent rotations relative to the frame o0x0y0z0while R1

2 represents a rotation relative to the frame o1x1y1z1. ComparingEquations (2.14) and (2.15) we can immediately infer

R02 = R0

1R12 (2.16)

Equation (2.16) is the composition law for rotational transformations. Itstates that, in order to transform the coordinates of a point p from itsrepresentation p2 in the frame o2x2y2z2 to its representation p0 in the frameo0x0y0z0, we may first transform to its coordinates p1 in the frame o1x1y1z1using R1

2 and then transform p1 to p0 using R01.

We may also interpret Equation (2.16) as follows. Suppose that ini-tially all three of the coordinate frames coincide. We first rotate the frameo2x2y2z2 relative to o0x0y0z0 according to the transformation R0

1. Then,with the frames o1x1y1z1 and o2x2y2z2 coincident, we rotate o2x2y2z2 rela-tive to o1x1y1z1 according to the transformation R1

2. The resulting frame,o2x2y2z2 has orientation with respect to o0x0y0z0 given by R0

1R12. In each

case we call the frame relative to which the rotation occurs the currentframe.

Example 2.5

Suppose a rotation matrix R represents a rotation of angle φ about thecurrent y-axis followed by a rotation of angle θ about the current z-axis as


z0

x 0

y0

y2

+ =

z0

x 0

z1

x 1 φ y1x 2

y2x 1

y0, y1

z2z1, z2

x 2

θ

Figure 2.9: Composition of rotations about current axes.

shown in Figure 2.9. Then the matrix R is given by

R = Ry,φRz,θ (2.17)

=

cφ 0 sφ0 1 0

−sφ 0 cφ

cθ −sθ 0sθ cθ 00 0 1

=

cφcθ −cφsθ sφsθ cθ 0

−sφcθ sφsθ cφ

⋄It is important to remember that the order in which a sequence of ro-

tations is performed, and consequently the order in which the rotation ma-trices are multiplied together, is crucial. The reason is that rotation, unlikeposition, is not a vector quantity and so rotational transformations do notcommute in general.

Example 2.6Suppose that the above rotations are performed in the reverse order, that

is, first a rotation about the current z-axis followed by a rotation about thecurrent y-axis. Then the resulting rotation matrix is given by

R′ = Rz,θRy,φ (2.18)

=

cθ −sφ 0sθ cθ 00 0 1

cφ 0 sφ0 1 0

−sφ 0 cφ

=

cθcφ −sθ cθsφsθcφ cθ sθsφ−sφ 0 cφ

2.4. COMPOSITION OF ROTATIONS 51

Comparing Equations (2.17) and (2.18) we see that R 6= R′.⋄

2.4.2 Rotation with Respect to the Fixed Frame

Many times it is desired to perform a sequence of rotations, each about agiven fixed coordinate frame, rather than about successive current frames.For example we may wish to perform a rotation about x0 followed by arotation about y0 (and not y1!). We will refer to o0x0y0z0 as the fixedframe. In this case the composition law given by Equation (2.16) is notvalid. It turns out that the correct composition law in this case is simply tomultiply the successive rotation matrices in the reverse order from that givenby Equation (2.16). Note that the rotations themselves are not performedin reverse order. Rather they are performed about the fixed frame insteadof about the current frame.

To see this, suppose we have two frames o0x0y0z0 and o1x1y1z1 relatedby the rotational transformation R0

1. If R ∈ SO(3) represents a rotationrelative to o0x0y0z0 we know from Section 2.3.1 that the representationfor R in the current frame o1x1y1z1 is given by (R0

1)−1RR0

1. Therefore,applying the composition law for rotations about the current axis yields

R02 = R0

1

[

(R01)

−1RR01

]

= RR01 (2.19)

Thus, when a rotation R is performed with respect to the world coordi-nate frame, the current rotation matrix is premultiplied by R to obtain thedesired rotation matrix.

Example 2.7 Rotations about Fixed Axes

y0

y1

θ

z0

x 0

x 1

z0

x 0

z0

x 0

y0

z1 z2

x 2

y2

x 1 φ

+ =

y0

Figure 2.10: Composition of rotations about fixed axes.

Referring to Figure 2.10, suppose that a rotation matrix R representsa rotation of angle φ about y0 followed by a rotation of angle θ about the


fixed z0. The second rotation about the fixed axis is given by Ry,−φRz,θRy,φ,which is the basic rotation about the z-axis expressed relative to the frameo1x1y1z1 using a similarity transformation. Therefore, the composition rulefor rotational transformations gives us

R = Ry,φ[

Ry,−φRz,θRy,φ]

= Rz,θRy,φ (2.20)

It is not necessary to remember the above derivation, only to note by com-paring Equation (2.20) with Equation (2.17) that we obtain the same basicrotation matrices, but in the reverse order.⋄

2.4.3 Rules for Composition of Rotational Transformations

We can summarize the rule of composition of rotational transformations bythe following recipe. Given a fixed frame o0x0y0z0 a current frame o1x1y1z1,together with rotation matrix R0

1 relating them, if a third frame o2x2y2z2is obtained by a rotation R performed relative to the current frame thenpostmultiply R0

1 by R = R12 to obtain

R02 = R0

1R12 (2.21)

If the second rotation is to be performed relative to the fixed frame thenit is both confusing and inappropriate to use the notation R1

2 to representthis rotation. Therefore, if we represent the rotation by R, we premultiplyR0

1 by R to obtain

R02 = RR0

1 (2.22)

In each case R02 represents the transformation between the frames o0x0y0z0

and o2x2y2z2. The frame o2x2y2z2 that results from Equation (2.21) will bedifferent from that resulting from Equation (2.22).

Using the above rule for composition of rotations, it is an easy matterto determine the result of multiple sequential rotational transformations.

Example 2.8Suppose R is defined by the following sequence of basic rotations in the

order specified:

1. A rotation of θ about the current x-axis

2. A rotation of φ about the current z-axis

3. A rotation of α about the fixed z-axis

2.5. PARAMETERIZATIONS OF ROTATIONS 53

4. A rotation of β about the current y-axis

5. A rotation of δ about the fixed x-axis

In order to determine the cumulative effect of these rotations we simply beginwith the first rotation Rx,θ and pre- or postmultiply as the case may be toobtain

R = Rx,δRz,αRx,θRz,φRy,β (2.23)

⋄

2.5 PARAMETERIZATIONS OF ROTATIONS

The nine elements rij in a general rotational transformation R ∈ SO(3) arenot independent quantities. Indeed a rigid body possesses at most three ro-tational degrees of freedom, and thus at most three quantities are required tospecify its orientation. This can be easily seen by examining the constraintsthat govern the matrices in SO(3):

∑

i

r2ij = 1, j ∈ {1, 2, 3} (2.24)

r1ir1j + r2ir2j + r3ir3j = 0, i 6= j (2.25)

Equation (2.24) follows from the fact that the columns of a rotation matrixare unit vectors, and Equation (2.25) follows from the fact that columns of arotation matrix are mutually orthogonal. Together, these constraints definesix independent equations with nine unknowns, which implies that there arethree free variables.

In this section we derive three ways in which an arbitrary rotation canbe represented using only three independent quantities: the Euler-anglerepresentation, the roll-pitch-yaw representation, and the axis/angle rep-resentation.

2.5.1 Euler Angles

A common method of specifying a rotation matrix in terms of three in-dependent quantities is to use the so-called Euler angles. Consider thefixed coordinate frame o0x0y0z0 and the rotated frame o1x1y1z1 shown inFigure 2.11. We can specify the orientation of the frame o1x1y1z1 relativeto the frame o0x0y0z0 by three angles (φ, θ, ψ), known as Euler angles, andobtained by three successive rotations as follows. First rotate about thez-axis by the angle φ. Next rotate about the current y-axis by the angle


ya ya , yb

x 0

φ

zb

θy0

za

yb

x 1

y1

ψzb, z1

x a

z0, za

x a

x b

(b )(a) (c)

x b

Figure 2.11: Euler angle representation.

θ. Finally rotate about the current z-axis by the angle ψ. In Figure 2.11,frame oaxayaza represents the new coordinate frame after the rotation byφ, frame obxbybzb represents the new coordinate frame after the rotation byθ, and frame o1x1y1z1 represents the final frame, after the rotation by ψ.Frames oaxayaza and obxbybzb are shown in the figure only to help visualizethe rotations.

In terms of the basic rotation matrices the resulting rotational transfor-mation can be generated as the product

RZY Z = Rz,φRy,θRz,ψ

=

cφ −sφ 0sφ cφ 00 0 1

cθ 0 sθ0 1 0

−sθ 0 cθ

cψ −sψ 0sψ cψ 00 0 1

=

cφcθcψ − sφsψ −cφcθsψ − sφcψ cφsθsφcθcψ + cφsψ −sφcθsψ + cφcψ sφsθ

−sθcψ sθsψ cθ

(2.26)

The matrix RZY Z in Equation (2.26) is called the ZY Z-Euler angle trans-formation.

The more important and more difficult problem is to determine for aparticular R = (rij) the set of Euler angles φ, θ, and ψ, that satisfy

R =

cφcθcψ − sφsψ −cφcθsψ − sφcψ cφsθsφcθcψ + cφsψ −sφcθsψ + cφcψ sφsθ

−sθcψ sθsψ cθ

(2.27)

for a matrix R ∈ SO(3). This problem will be important later when weaddress the inverse kinematics problem for manipulators in Section 3.3.


To find a solution for this problem we break it down into two cases. First,suppose that not both of r13, r23 are zero. Then from Equation (2.26) wededuce that sθ 6= 0, and hence that not both of r31, r32 are zero. If not bothr13 and r23 are zero, then r33 6= ±1, and we have cθ = r33, sθ = ±

√

1 − r233so

θ = Atan2

(

r33,√

1 − r233

)

(2.28)

or

θ = Atan2

(

r33,−√

1 − r233

)

(2.29)

where the function Atan2 is the two-argument arctangent function de-fined in Appendix A.

If we choose the value for θ given by Equation (2.28), then sθ > 0, and

φ = Atan2(r13, r23) (2.30)

ψ = Atan2(−r31, r32) (2.31)

If we choose the value for θ given by Equation (2.29), then sθ < 0, and

φ = Atan2(−r13,−r23) (2.32)

ψ = Atan2(r31,−r32) (2.33)

Thus, there are two solutions depending on the sign chosen for θ.If r13 = r23 = 0, then the fact that R is orthogonal implies that r33 = ±1,

and that r31 = r32 = 0. Thus, R has the form

R =

r11 r12 0r21 r22 00 0 ±1

(2.34)

If r33 = 1, then cθ = 1 and sθ = 0, so that θ = 0. In this case Equation(2.26) becomes

cφcψ − sφsψ −cφsψ − sφcψ 0sφcψ + cφsψ −sφsψ + cφcψ 0

0 0 1

=

cφ+ψ −sφ+ψ 0sφ+ψ cφ+ψ 0

0 0 1

Thus, the sum φ+ ψ can be determined as

φ+ ψ = Atan2(r11, r21) = Atan2(r11,−r12) (2.35)


Since only the sum φ+ψ can be determined in this case, there are infinitelymany solutions. In this case, we may take φ = 0 by convention. If r33 = −1,then cθ = −1 and sθ = 0, so that θ = π. In this case Equation (2.26)becomes

−cφ−ψ −sφ−ψ 0sφ−ψ cφ−ψ 0

0 0 −1

=

r11 r12 0r21 r22 0

0 0 −1

(2.36)

The solution is thus

φ− ψ = Atan2(−r11,−r12) (2.37)

As before there are infinitely many solutions.

2.5.2 Roll, Pitch, Yaw Angles

A rotation matrix R can also be described as a product of successive rota-tions about the principal coordinate axes x0, y0, and z0 taken in a specificorder. These rotations define the roll, pitch, and yaw angles, which weshall also denote φ, θ, ψ, and which are shown in Figure 2.12.

x0

Yaw

Roll

y0

z0

Pitch

Figure 2.12: Roll, pitch, and yaw angles.

We specify the order of rotation as x− y− z, in other words, first a yawabout x0 through an angle ψ, then pitch about the y0 by an angle θ, andfinally roll about the z0 by an angle φ4. Since the successive rotations are

4It should be noted that other conventions exist for naming the roll, pitch and yawangles.


relative to the fixed frame, the resulting transformation matrix is given by

R = Rz,φRy,θRx,ψ

=

cφ −sφ 0sφ cφ 00 0 1

cθ 0 sθ0 1 0

−sθ 0 cθ

1 0 00 cψ −sψ0 sψ cψ

=

cφcθ −sφcψ + cφsθsψ sφsψ + cφsθcψsφcθ cφcψ + sφsθsψ −cφsψ + sφsθcψ−sθ cθsψ cθcψ

(2.38)

Of course, instead of yaw-pitch-roll relative to the fixed frames we couldalso interpret the above transformation as roll-pitch-yaw, in that order, eachtaken with respect to the current frame. The end result is the same matrixas in Equation (2.38).

The three angles φ, θ, and ψ can be obtained for a given rotation matrixusing a method that is similar to that used to derive the Euler angles above.

2.5.3 Axis/Angle Representation

Rotations are not always performed about the principal coordinate axes.We are often interested in a rotation about an arbitrary axis in space. Thisprovides both a convenient way to describe rotations, and an alternativeparameterization for rotation matrices. Let k = [kx, ky, kz]

T , expressed inthe frame o0x0y0z0, be a unit vector defining an axis. We wish to derive therotation matrix Rk,θ representing a rotation of θ about this axis.

There are several ways in which the matrix Rk,θ can be derived. Oneapproach is to note that the rotational transformation R = Rz,αRy,β willbring the world z-axis into alignment with the vector k. Therefore, a rotationabout the axis k can be computed using a similarity transformation as

Rk,θ = RRz,θR−1 (2.39)

= Rz,αRy,βRz,θRy,−βRz,−α (2.40)

From Figure 2.13 we see that

sinα =ky

√

k2x + k2

y

cosα =kx

√

k2x + k2

y

(2.41)

sinβ =√

k2x + k2

y cosβ = kz (2.42)


βθ

x0

y0

z0

kx

ky

kz

k

α

Figure 2.13: Rotation about an arbitrary axis.

Note that the final two equations follow from the fact that k is a unit vector.Substituting Equations (2.41)-(2.42) into Equation (2.40) we obtain aftersome lengthy calculation (Problem 2-17)

Rk,θ =

k2xvθ + cθ kxkyvθ − kzsθ kxkzvθ + kysθ

kxkyvθ + kzsθ k2yvθ + cθ kykzvθ − kxsθ

kxkzvθ − kysθ kykzvθ + kxsθ k2zvθ + cθ

(2.43)

where vθ = vers θ = 1 − cθ.In fact, any rotation matrix R ∈ S0(3) can be represented by a single

rotation about a suitable axis in space by a suitable angle,

R = Rk,θ (2.44)

where k is a unit vector defining the axis of rotation, and θ is the angleof rotation about k. The pair (k, θ) is called the axis/angle representa-tion of R. Given an arbitrary rotation matrix R with components rij , theequivalent angle θ and equivalent axis k are given by the expressions

θ = cos−1

(

r11 + r22 + r33 − 1

2

)

and

k =1

2 sin θ

r32 − r23r13 − r31r21 − r12

(2.45)


These equations can be obtained by direct manipulation of the entries ofthe matrix given in Equation (2.43). The axis/angle representation is notunique since a rotation of −θ about −k is the same as a rotation of θ aboutk, that is,

Rk,θ = R−k,−θ (2.46)

If θ = 0 then R is the identity matrix and the axis of rotation is undefined.

Example 2.9Suppose R is generated by a rotation of 90◦ about z0 followed by a rota-

tion of 30◦ about y0 followed by a rotation of 60◦ about x0. Then

R = Rx,60Ry,30Rz,90 (2.47)

=

0 −√

32

12

12 −

√3

4 −34√

32

14

√3

4

We see that Tr(R) = 0 and hence the equivalent angle is given by Equation(2.45) as

θ = cos−1

(

−1

2

)

= 120◦ (2.48)

The equivalent axis is given from Equation (2.45) as

k =

[

1√3,

1

2√

3− 1

2,

1

2√

3+

1

2

]T

(2.49)

⋄The above axis/angle representation characterizes a given rotation by

four quantities, namely the three components of the equivalent axis k andthe equivalent angle θ. However, since the equivalent axis k is given asa unit vector only two of its components are independent. The third isconstrained by the condition that k is of unit length. Therefore, only threeindependent quantities are required in this representation of a rotation R.We can represent the equivalent axis/angle by a single vector r as

r = [rx, ry, rz]T = [θkx, θky, θkz]

T (2.50)

Note, since k is a unit vector, that the length of the vector r is the equivalentangle θ and the direction of r is the equivalent axis k.

One should be careful not to interpret the representation in Equation (2.50)to mean that two axis/angle representations may be combined using stan-dard rules of vector algebra, as doing so would imply that rotations commutewhich, as we have seen, is not true in general.


2.6 RIGID MOTIONS

We have now seen how to represent both positions and orientations. Wecombine these two concepts in this section to define a rigid motion and,in the next section, we derive an efficient matrix representation for rigidmotions using the notion of homogeneous transformation.

Definition 2.1 A rigid motion is an ordered pair (d,R) where d ∈ R3

and R ∈ SO(3). The group of all rigid motions is known as the SpecialEuclidean Group and is denoted by SE(3). We see then that SE(3) =R

3 × SO(3).

A rigid motion is a pure translation together with a pure rotation.5 LetR0

1 be the rotation matrix that specifies the orientation of frame o1x1y1z1with respect to o0x0y0z0, and d be the vector from the origin of frameo0x0y0z0 to the origin of frame o1x1y1z1. Suppose the point p is rigidlyattached to coordinate frame o1x1y1z1, with local coordinates p1. We canexpress the coordinates of p with respect to frame o0x0y0z0 using

p0 = R01p

1 + d0 (2.51)

Now consider the three coordinate frames o0x0y0z0, o1x1y1z1 and o2x2y2z2.Let d1 be the vector from the origin of o0x0y0z0 to the origin of o1x1y1z1and d2 be the vector from the origin of o1x1y1z1 to the origin of o2x2y2z2. Ifthe point p is attached to frame o2x2y2z2 with local coordinates p2, we cancompute its coordinates relative to frame o0x0y0z0 using

p1 = R12p

2 + d12 (2.52)

and

p0 = R01p

1 + d01 (2.53)

The composition of these two equations defines a third rigid motion, whichwe can describe by substituting the expression for p1 from Equation (2.52)into Equation (2.53)

p0 = R01R

12p

2 +R01d

12 + d0

1 (2.54)

5The definition of rigid motion is sometimes broadened to include reflections, whichcorrespond to detR = −1. We will always assume in this text that detR = +1, i.e. thatR ∈ SO(3).

2.7. HOMOGENEOUS TRANSFORMATIONS 61

Since the relationship between p0 and p2 is also a rigid motion, we canequally describe it as

p0 = R02p

2 + d02 (2.55)

Comparing Equations (2.54) and (2.55) we have the relationships

R02 = R0

1R12 (2.56)

d02 = d0

1 +R01d

12 (2.57)

Equation (2.56) shows that the orientation transformations can simply bemultiplied together and Equation (2.57) shows that the vector from theorigin o0 to the origin o2 has coordinates given by the sum of d0

1 (the vectorfrom o0 to o1 expressed with respect to o0x0y0z0) and R0

1d12 (the vector from

o1 to o2, expressed in the orientation of the coordinate frame o0x0y0z0).

2.7 HOMOGENEOUS TRANSFORMATIONS

One can easily see that the calculation leading to Equation (2.54) wouldquickly become intractable if a long sequence of rigid motions were con-sidered. In this section we show how rigid motions can be represented inmatrix form so that composition of rigid motions can be reduced to matrixmultiplication as was the case for composition of rotations.

In fact, a comparison of Equations (2.56) and (2.57) with the matrixidentity

[

R01 d0

1

0 1

] [

R12 d2

1

0 1

]

=

[

R01R

12 R0

1d21 + d0

1

0 1

]

(2.58)

where 0 denotes the row vector (0, 0, 0), shows that the rigid motions canbe represented by the set of matrices of the form

H =

[

R d0 1

]

, R ∈ SO(3), d ∈ R3 (2.59)

Transformation matrices of the form given in Equation (2.59) are calledhomogeneous transformations. A homogeneous transformation is there-fore nothing more than a matrix representation of a rigid motion and wewill use SE(3) interchangeably to represent both the set of rigid motionsand the set of all 4 × 4 matrices H of the form given in Equation (2.59).

Using the fact that R is orthogonal it is an easy exercise to show thatthe inverse transformation H−1 is given by

H−1 =

[

RT −RTd0 1

]

(2.60)


In order to represent the transformation given in Equation (2.51) bya matrix multiplication, we must augment the vectors p0 and p1 by theaddition of a fourth component of 1 as follows,

P 0 =

[

p0

1

]

(2.61)

P 1 =

[

p1

1

]

(2.62)

The vectors P 0 and P 1 are known as homogeneous representations ofthe vectors p0 and p1, respectively. It can now be seen directly that thetransformation given in Equation (2.51) is equivalent to the (homogeneous)matrix equation

P 0 = H01P

1 (2.63)

A set of basic homogeneous transformations generating SE(3) isgiven by

Transx,a =

1 0 0 a0 1 0 00 0 1 00 0 0 1

, Rotx,α =

1 0 0 00 cα −sα 00 sα cα 00 0 0 1

(2.64)

Transy,b =

1 0 0 00 1 0 b0 0 1 00 0 0 1

, Roty,β =

cβ 0 sβ 00 1 0 0

−sβ 0 cβ 00 0 0 1

(2.65)

Transz,c =

1 0 0 00 1 0 00 0 1 c0 0 0 1

, Rotx,γ =

cγ −sγ 0 0sγ cγ 0 00 0 1 00 0 0 1

(2.66)

for translation and rotation about the x, y, z-axes, respectively.The most general homogeneous transformation that we will consider may

be written now as

H01 =

nx sx ax dxny sy ay dynz sx az dz0 0 0 1

=

[

n s a d0 0 0 1

]

(2.67)

2.8. SUMMARY 63

In the above equation n = [nx, ny, nz]T is a vector representing the direction

of x1 in the o0x0y0z0 frame, s = [sx, sy, sz]T represents the direction of

y1, and a = [ax, ay, az]T represents the direction of z1. The vector d =

[dx, dy, dz]T represents the vector from the origin o0 to the origin o1 expressed

in the frame o0x0y0z0. The rationale behind the choice of letters n, s and ais explained in Chapter 3.

The same interpretation regarding composition and ordering of transfor-mations holds for 4× 4 homogeneous transformations as for 3× 3 rotations.Given a homogeneous transformation H0

1 relating two frames, if a secondrigid motion, represented by H ∈ SE(3) is performed relative to the currentframe, then

H02 = H0

1H

whereas if the second rigid motion is performed relative to the fixed frame,then

H02 = HH0

1

Example 2.10The homogeneous transformation matrix H that represents a rotation by

angle α about the current x-axis followed by a translation of b units along thecurrent x-axis, followed by a translation of d units along the current z-axis,followed by a rotation by angle θ about the current z-axis, is given by

H = Rotx,αTransx,bTransz,dRotz,θ

=

cθ −sθ 0 bcαsθ cαcθ −sα −dsαsαsθ sαcθ cα dcα

0 0 0 1

⋄

2.8 SUMMARY

In this chapter, we have seen how matrices in SE(n) can be used to representthe relative position and orientation of two coordinate frames for n = 2, 3.We have adopted a notional convention in which a superscript is used toindicate a reference frame. Thus, the notation p0 represents the coordinatesof the point p relative to frame 0.


The relative orientation of two coordinate frames can be specified bya rotation matrix, R ∈ SO(n), with n = 2, 3. In two dimensions, theorientation of frame 1 with respect to frame 0 is given by

R01 =

[

x1 · x0 y1 · x0

x1 · y0 y1 · y0

]

=

[


]

in which θ is the angle between the two coordinate frames. In the three-dimensional case, the rotation matrix is given by

R01 =

x1 · x0 y1 · x0 z1 · x0

x1 · y0 y1 · y0 z1 · y0

x1 · z0 y1 · z0 z1 · z0

In each case, the columns of the rotation matrix are obtained by projectingan axis of the target frame (in this case, frame 1) onto the coordinate axesof the reference frame (in this case, frame 0).

The set of n × n rotation matrices is known as the special orthogonalgroup of order n, and is denoted by SO(n). An important property of thesematrices is that R−1 = RT for any R ∈ SO(n).

Rotation matrices can be used to perform coordinate transformationsbetween frames that differ only in orientation. We derived rules for thecomposition of rotational transformations as

R02 = R0

1R

for the case where the second transformation, R, is performed relative tothe current frame and

R02 = RR0

1

for the case where the second transformation, R, is performed relative tothe fixed frame.

In the three-dimensional case, a rotation matrix can be parameterizedusing three angles. A common convention is to use the Euler angles (φ, θ, ψ),which correspond to successive rotations about the z, y and z axes. Thecorresponding rotation matrix is given by

R(φ, θ, ψ) = Rz,φRy,θRz,ψ

Roll, pitch and yaw angles are similar, except that the successive rotationsare performed with respect to the fixed, world frame instead of being per-formed with respect to the current frame.

PROBLEMS 65

Homogeneous transformations combine rotation and translation. In thethree-dimensional case, a homogeneous transformation has the form

H =

[

R d0 1

]

, R ∈ SO(3), d ∈ R3

The set of all such matrices comprises the set SE(3), and these matricescan be used to perform coordinate transformations, analogous to rotationaltransformations using rotation matrices.

Homogeneous transformation matrices can be used to perform coordinatetransformations between frames that differ in orientation and translation.We derived rules for the composition of rotational transformations as

H02 = H0

1H

for the case where the second transformation, H, is performed relative tothe current frame and

H02 = HH0

1

for the case where the second transformation, H, is performed relative tothe fixed frame.

PROBLEMS

2-1 Using the fact that v1 · v2 = vT1 v2, show that the dot product of twofree vectors does not depend on the choice of frames in which theircoordinates are defined.

2-2 Show that the length of a free vector is not changed by rotation, i.e.,that ‖v‖ = ‖Rv‖.

2-3 Show that the distance between points is not changed by rotation i.e.,that ‖p1 − p2‖ = ‖Rp1 −Rp2‖.

2-4 If a matrix R satisfies RTR = I, show that the column vectors of Rare of unit length and mutually perpendicular.

2-5 If a matrix R satisfies RTR = I, thena) Show that detR = ±1b) Show that detR = +1 if we restrict ourselves to right-handed co-ordinate frames.

2-6 Verify Equations (2.3)-(2.5).


2-7 A group is a set X together with an operation ∗ defined on that setsuch that

• x1 ∗ x2 ∈ X for all x1, x2 ∈ X

• (x1 ∗ x2) ∗ x3 = x1 ∗ (x2 ∗ x3)

• There exists an element I ∈ X such that I ∗ x = x ∗ I = x for allx ∈ X.

• For every x ∈ X, there exists some element y ∈ X such thatx ∗ y = y ∗ x = I.

Show that SO(n) with the operation of matrix multiplication is agroup.

2-8 Derive Equations (2.6) and (2.7).

2-9 Suppose A is a 2 × 2 rotation matrix. In other words ATA = I anddetA = 1. Show that there exists a unique θ such that A is of theform

A =

[


]

2-10 Consider the following sequence of rotations:

1. Rotate by φ about the world x-axis.

2. Rotate by θ about the current z-axis.

3. Rotate by ψ about the world y-axis.

Write the matrix product that will give the resulting rotation matrix(do not perform the matrix multiplication).



2. Rotate by θ about the world z-axis.

3. Rotate by ψ about the current x-axis.




PROBLEMS 67

2. Rotate by θ about the current z-axis.


4. Rotate by α about the world z-axis.




2. Rotate by θ about the world z-axis.


4. Rotate by α about the world z-axis.


2-14 If the coordinate frame o1x1y1z1 is obtained from the coordinate frameo0x0y0z0 by a rotation of π

2 about the x-axis followed by a rotation ofπ2 about the fixed y-axis, find the rotation matrix R representing thecomposite transformation. Sketch the initial and final frames.

2-15 Suppose that three coordinate frames o1x1y1z1, o2x2y2z2 and o3x3y3z3are given, and suppose

R12 =

1 0 0

0 12 −

√3

2

0√

32

12

, R1

3 =

0 0 −10 1 01 0 0

Find the matrix R23.

2-16 Derive equations for the roll, pitch and yaw angles corresponding tothe rotation matrix R = (rij).

2-17 Verify Equation (2.43).


2-19 If R is a rotation matrix show that +1 is an eigenvalue of R. Let k bea unit eigenvector corresponding to the eigenvalue +1. Give a physicalinterpretation of k.

2-20 Let k = 1√3[1, 1, 1]T , θ = 90◦. Find Rk,θ.


2-21 Show by direct calculation that Rk,θ given by Equation (2.43) is equalto R given by Equation (2.47) if θ and k are given by Equations (2.48)and (2.49), respectively.

2-22 Compute the rotation matrix given by the product

Rx,θRy,φRz,πRy,−φRx,−θ

2-23 Suppose R represents a rotation of 90◦ about y0 followed by a rotationof 45◦ about z1. Find the equivalent axis/angle to represent R. Sketchthe initial and final frames and the equivalent axis vector k.

2-24 Find the rotation matrix corresponding to the Euler angles φ = π2 ,

θ = 0 and ψ = π4 . What is the direction of the x1 axis relative to the

base frame?

2-25 Section 2.5.1 described only the Z-Y-Z Euler angles. List all possiblesets of Euler angles. Is it possible to have Z-Z-Y Euler angles? Whyor why not?

2-26 Unit magnitude complex numbers (i.e., a + ib such that a2 + b2 = 1)can be used to represent orientation in the plane. In particular, for thecomplex number a+ ib, we can define the angle θ = Atan2(a, b). Showthat multiplication of two complex numbers corresponds to additionof the corresponding angles.

2-27 Show that complex numbers together with the operation of complexmultiplication define a group. What is the identity for the group?What is the inverse for a+ ib?

2-28 Complex numbers can be generalized by defining three independentsquare roots for −1 that obey the multiplication rules

−1 = i2 = j2 = k2,

i = jk = −kj,j = ki = −ik,k = ij = −ji

Using these, we define a quaternion by Q = q0+iq1+jq2+kq3, whichis typically represented by the 4-tuple (q0, q1, q2, q3). A rotation by θabout the unit vector n = [nx, ny, nz]

T can be represented by the unitquaternion Q =

(

cos θ2 , nx sin θ2 , ny sin θ

2 , nz sin θ2

)

. Show that such aquaternion has unit norm, i.e., that q20 + q21 + q22 + q23 = 1.

PROBLEMS 69

2-29 Using Q =(

cos θ2 , nx sin θ2 , ny sin θ

2 , nz sin θ2

)

, and the results from Sec-tion 2.5.3, determine the rotation matrix R that corresponds to therotation represented by the quaternion (q0, q1, q2, q3).

2-30 Determine the quaternion Q that represents the same rotation as givenby the rotation matrix R.

2-31 The quaternion Q = (q0, q1, q2, q3) can be thought of as having a scalarcomponent q0 and a vector component = [q1, q2, q3]

T . Show that theproduct of two quaternions, Z = XY is given by

z0 = x0y0 − xT y

z = x0y + y0x+ x× y,

Hint: Perform the multiplication (x0 +ix1+jx2 +kx3)(y0 +iy1 +jy2 +ky3) and simplify the result.

2-32 Show that QI = (1, 0, 0, 0) is the identity element for unit quaternionmultiplication, i.e., that QQI = QIQ = Q for any unit quaternion Q.

2-33 The conjugate Q∗ of the quaternion Q is defined as

Q∗ = (q0,−q1,−q2,−q3)

Show that Q∗ is the inverse of Q, i.e., that Q∗Q = QQ∗ = (1, 0, 0, 0).

2-34 Let v be a vector whose coordinates are given by [vx, vy, vz]T . If the

quaternion Q represents a rotation, show that the new, rotated coor-dinates of v are given by Q(0, vx, vy, vz)Q

∗, in which (0, vx, vy, vz) is aquaternion with zero as its real component.

2-35 Let the point p be rigidly attached to the end effector coordinate framewith local coordinates (x, y, z). If Q specifies the orientation of the endeffector frame with respect to the base frame, and T is the vector fromthe base frame to the origin of the end effector frame, show that thecoordinates of p with respect to the base frame are given by

Q(0, x, y, z)Q∗ + T (2.68)

in which (0, x, y, z) is a quaternion with zero as its real component.



2-37 Compute the homogeneous transformation representing a translationof 3 units along the x-axis followed by a rotation of π2 about the currentz-axis followed by a translation of 1 unit along the fixed y-axis. Sketchthe frame. What are the coordinates of the origin o1 with respect tothe original frame in each case?

2-38 Consider the diagram of Figure 2.14. Find the homogeneous transfor-

Figure 2.14: Diagram for Problem 2-38.

mations H01 ,H

02 ,H

12 representing the transformations among the three

frames shown. Show that H02 = H0

1 ,H12 .

2-39 Consider the diagram of Figure 2.15. A robot is set up 1 meter froma table. The table top is 1 meter high and 1 meter square. A frameo1x1y1z1 is fixed to the edge of the table as shown. A cube measur-ing 20 cm on a side is placed in the center of the table with frameo2x2y2z2 established at the center of the cube as shown. A camera issituated directly above the center of the block 2m above the table topwith frame o3x3y3z3 attached as shown. Find the homogeneous trans-formations relating each of these frames to the base frame o0x0y0z0.Find the homogeneous transformation relating the frame o2x2y2z2 tothe camera frame o3x3y3z3.

2-40 In Problem 2-39, suppose that, after the camera is calibrated, it is ro-tated 90◦ about z3. Recompute the above coordinate transformations.

2-41 If the block on the table is rotated 90◦ about z2 and moved so thatits center has coordinates [0, .8, .1]T relative to the frame o1x1y1z1,

PROBLEMS 71

Figure 2.15: Diagram for Problem 2-39.

compute the homogeneous transformation relating the block frame tothe camera frame; the block frame to the base frame.

2-42 Consult an astronomy book to learn the basic details of the Earth’srotation about the sun and about its own axis. Define for the Eartha local coordinate frame whose z-axis is the Earth’s axis of rotation.Define t = 0 to be the exact moment of the summer solstice, and theglobal reference frame to be coincident with the Earth’s frame at timet = 0. Give an expression R(t) for the rotation matrix that representsthe instantaneous orientation of the earth at time t. Determine asa function of time the homogeneous transformation that specifies theEarth’s frame with respect to the global reference frame.

2-43 In general, multiplication of homogeneous transformation matrices is


not commutative. Consider the matrix product

H = Rotx,αTransx,bTransz,dRotz,θ

Determine which pairs of the four matrices on the right hand sidecommute. Explain why these pairs commute. Find all permutationsof these four matrices that yield the same homogeneous transformationmatrix, H .

NOTES AND REFERENCES

Rigid body motions and the groups SO(n) and SE(n) are often addressedin mathematics books on the topic of linear algebra. Standard texts for thismaterial include [8] [23] [40]. These topics are also often covered in appliedmathematics texts for physics and engineering, such as [108] [119] [139]. Inaddition to these, a detailed treatment of rigid body motion developed withthe aid of exponential coordinates and Lie groups is given in [93].

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