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PDF Instructions. Scientific Method. Three joints allow the position of the hand to attain any position in three-dimensional space, with a fourth joint added to allow the hand to rotate the grasped component about a vertical axis. In the case of a universal robot, it is interesting that fundamental properties of the physical world we live in dictate the "correct" minimum number of joints—that minimum number is six.
Integral to the design of the manipulator are issues involving the choice and location of actuators, transmission systems, and internal-position and sometimes force sensors. These and other design issues will be discussed in Chapter 8. Linear position control Some manipulators are equipped with stepper motors or other actuators that can execute a desired trajectory directly.
However, the vast majority of manipulators are driven by actuators that supply a force or a torque to cause motion of the links. In this case, an algorithm is needed to compute torques that will cause the desired motion. The problem of dynamics is central to the design of such algorithms, but does not in itself constitute a solution. A primary concern of a position control system is to compensate automatically for errors in knowledge of the parameters of a system and to suppress disturbances that tend to perturb the system from the desired trajectory.
To accomplish this, position and velocity sensors are monitored by the control algorithm, which computes torque commands for the actuators. Such a system uses feedback from joint sensors to keep the manipulator on course. In Chapter 9, we wifi consider control algorithms whose synthesis is based on linear approximations to the dynamics of a manipulator.
These linear methods are prevalent in current industrial practice. Nonlinear position control Although control systems based on approximate linear models are popular in current industrial robots, it is important to consider the complete nonlinear dynamics of the manipulator when synthesizing control algorithms.
These nonlinear techniques of controlling a manipulator promise better performance than do simpler linear schemes. Chapter 10 will introduce nonlinear control systems for mechanical manipulators. Force control The ability of a manipulator to control forces of contact when it touches parts, tools, or work surfaces seems to be of great importance in applying manipulators to many real-world tasks.
Force control is complementary to position control, in that we usually think of only one or the other as applicable in a certain situation. When a manipulator is moving in free space, only position control makes sense, because there is no surface to react against.
When a manipulator is touching a rigid surface, however, position-control schemes can cause excessive forces to build up at the contact or cause contact to be lost with the surface when it was desired for some application. Manipulators are rarely constrained by reaction surfaces in all directions simultaneously, so a mixed or hybrid control is required, with some directions controlled by a position-control law and remaining directions controlled by a force-control law.
Chapter 11 introduces a methodology for implementing such a force-control scheme. A robot should be instructed to wash a window by maintaining a certain force in the direction perpendicular to the plane of the glass, while following a motion trajectory in directions tangent to the plane. Such split or hybrid control specifications are natural for such tasks. Programming robots A robot progranuning language serves as the interface between the human user and the industrial robot.
Central questions arise: How are motions through space described easily by the programmer? How are sensor-based actions described in a language? Robot manipulators differentiate themselves from fixed automation by being "flexible," which means programmable. Not only are the movements of manipulators programmable, but, through the use of sensors and communications with other factory automation, manipulators can adapt to variations as the task proceeds.
In typical robot systems, there is a shorthand way for a human user to instruct the robot which path it is to follow. First of all, a special point on the hand or perhaps on a grasped tool is specified by the user as the operational point, sometimes also called the TCP for Tool Center Point. Motions of the robot wifi be described by the user in terms of desired locations of the operational point relative to a user-specified coordinate system.
Generally, the user wifi define this reference coordinate system relative to the robot's base coordinate system in some task-relevant location.
Most often, paths are constructed by specifying a sequence of via points. Via points are specified relative to the reference coordinate system and denote locations along the path through which the TCP should pass. Along with specifying the via points, the user may also indicate that certain speeds of the TCP be used over various portions of the path. Sometimes, other modifiers can also be specified to affect the motion of the robot e.
From these inputs, the trajectory-generation algorithm must plan all the details of the motion: velocity profiles for the joints, time duration of the move, and so on.
The sophistication of the user interface is becoming extremely important as manipulators and other programmable automation are applied to more and more demanding industrial applications. The problem of programming manipu- lators encompasses all the issues of "traditional" computer programming and so is an extensive subject in itself.
Additionally, some particular attributes of the manipulator-programming problem cause additional issues to arise.
Some of these topics will be discussed in Chapter Off-line programming and simulation An off-line programming system is a robot programming environment that has been sufficiently extended, generally by means of computer graphics, that the development of robot programs can take place without access to the robot itself. A common argument raised in their favor is that an off-line programming system wifi not cause production equipment i.
They also serve as a natural vehicle to tie computer-aided design CAD data bases used in the design phase of a product to the actual manufacturing of the product. In some cases, this direct use of CAD data can dramatically reduce the programming time required for the manufacturing process.
Chapter 13 discusses the elements of industrial robot off-line programming systems. In this book, we use the following conventions: 1. Usually, variables written in uppercase represent vectors or matrices. Lower- case variables are scalars. Leading subscripts and superscripts identify which coordinate system a quantity is written in.
Trailing superscripts are used as widely accepted for indicating the inverse or transpose of a matrix e. Trailing subscripts are not subject to any strict convention but may indicate a vector component e. We will use many trigonometric fi. Vectors are taken to be column vectors; hence, row vectors wifi have the transpose indicated explicitly. A note on vector notation in general: Many mechanics texts treat vector quantities at a very abstract level and routinely use vectors defined relative to different coordinate systems in expressions.
The clearest example is that of addition of vectors which are given or known relative to differing reference systems. This is often very convenient and leads to compact and somewhat elegant formulas. For example, consider the angular velocity, 0w4 of the last body in a series connection of four rigid bodies as in the links of a manipulator relative to the fixed base of the chain. For the particular case of the study of mechanical manipulators, statements like that of 1.
Therefore, in this book, we carry frame-of-reference information in the nota- tion for vectors, and we do not sum vectors unless they are in the same coordinate system. In this way, we derive expressions that solve the "bookkeeping" problem and can be applied directly to actual numerical computation.
Exercises 17 [2] R. General-reference books [4] R. Brady et al. Asada and J. Fu, R. Gonzalez, and C. Sciavicco and B. Schmierer and R. Schraft, Service Robots, A. Peters, Natick, MA, General-reference journals and magazines [17] Robotics World. See Bibliography and general references. Base your chart on the most recent data you can find. Find data on the cost of human labor in various specific industries e. From this, derive approximate dates when robotics first became cost effective for use in various industries.
See reference section. Make sure you can create and edit files and can compile and execute programs. Learn how to create arrays matrices and vectors , and explore the built-in MATLAB linear-algebra functions for matrix and vector multiplication, dot and cross products, transposes, determinants, and inverses, and for the solution of linear equations.
Learn how to use subprograms and functions. Check out www. This product can be downloaded for free from www. The source code is readable and changeable, and there is an international community of users, at robot-toolbox lists.
Find the robot. Don't worry if you can't understand the purpose of these functions yet; they deal with robotics mathematics concepts covered in Chapters 2 through 7 of this book.
This naturally leads to a need for representing positions and orientations of parts, of tools, and of the mechanism itself. To define and manipulate mathematical quantities that represent position and orientation, we must define coordinate systems and develop conventions for representation. Many of the ideas developed here in the context of position and orientation will form a basis for our later consideration of linear and rotational velocities, forces, and torques.
We adopt the philosophy that somewhere there is a universe coordinate system to which everything we discuss can be referenced. We wifi describe all positions and orientations with respect to the universe coordinate system or with respect to other Cartesian coordinate systems that are or could be defined relative to the universe system.
These objects are parts, tools, and the manipulator itself. In this section, we discuss the description of positions, of orientations, and of an entity that contains both of these descriptions: the frame. Because we wifi often define many coordinate systems in addition to the universe coordinate system, vectors must be tagged with information identifying which coordinate system they are defined within.
Each of these distances along an axis can be thought of as the result of projecting the vector onto the corresponding axis. Figure 2. A point A P is represented as a vector and can equivalently be thought of as a position in space, or simply as an ordered set of three numbers.
Individual elements of a vector are given the subscripts x, y, and z: r 1. Other 3-tuple descriptions of the position of points, such as spherical or cylindrical coordinate representations, are discussed in the exercises at the end of the chapter. Description of an orientation Often, we wifi find it necessary not only to represent a point in space but also to describe the orientation of a body in space. For example, if vector Ap in Fig. Section 2. In order to describe the orientation of a body, we wifi attach a coordinate system to the body and then give a description of this coordinate system relative to the reference system.
In Fig. Thus, positions of points are described with vectors and orientations of bodies are described with an attached coordinate system. For convenience, we wifi construct a 3 x 3 matrix that has these three vectors as its colunms. Hence, whereas the position of a point is represented with a vector, the is often convenient to use three, although any two would suffice.
The third can always be recovered by taking the cross product of the two given. In Section 2. We can give expressions for the scalars in 2. Hence, each component of in 2. ZA YB. ZA ZB. ZAJ For brevity, we have omitted the leading superscripts in the rightmost matrix of 2. In fact, the choice of frame in which to describe the unit vectors is arbitrary as long as it is the same for each pair being dotted.
The dot product of two unit vectors yields the cosine of the angle between them, so it is clear why the components of rotation matrices are often referred to as direcfion cosines. Further inspection of 2. We have just shown this geometrically. Description of a frame The information needed to completely specify the whereabouts of the manipulator hand in Fig.
The point on the body whose position we describe could be chosen arbitrarily, however. The situation of a position and an orientation pair arises so often in robotics that we define an entity called a frame, which is a set of four vectors giving position and orientation information. For example, in Fig. Equivalently, the description of a frame can be thought of as a position vector and a rotation matrix. Note that a frame is a coordinate system where, in addition to the orientation, we give a position vector which locates its origin relative to some other embedding frame.
A frame is depicted by three arrows representing unit vectors defining the principal axes of the frame. An arrow representing a vector is drawn from one origin to another. This vector represents the position of the origin at the head of the arrow in tenns of the frame at the tail of the arrow. The direction of this locating arrow tells us, for example, in Fig.
In summary, a frame can be used as a description of one coordinate system relative to another. A frame encompasses two ideas by representing both position and orientation and so may be thought of as a generalization of those two ideas. Positions could be represented by a frame whose rotation-matrix part is the identity matrix and whose position-vector part locates the point being described. Likewise, an orientation could be represented by a frame whose position-vector part was the zero vector.
The previous section introduced descriptions of positions, orientations, and frames; we now consider the mathematics of mapping in order to change descriptions from frame to frame. Mappings involving translated frames In Fig. In this simple example, we have illustrated mapping a vector from one frame to another.
This idea of mapping, or changing the description from one frame to another, is an extremely important concept. The quantity itself here, a point in space is not changed; only its description is changed. This is illustrated in Fig. Mappings involving rotated frames Section 2. For convenience, we stack these three unit vectors together as the columns of a 3 x 3 matrix.
So a rotation matrix can be interpreted as a set of three column vectors or as a set of three row vectors, as follows: Bkr 2. In order to calculate A P, we note that the components of any vector are simply the projections of that vector onto the unit directions of its frame. The projection is calculated as the vector dot product. Bp In order to express 2. We now see that our notation is of great help in keeping track of mappings and frames of reference.
A helpful way of viewing the notation we have introduced is to imagine that leading subscripts cancel the leading superscripts of the following entity, for example the Bs in 2. Here, Z is pointing out of the page. Rather, we compute a new description of the vector relative to another frame.
We now consider the general case of mapping. This is done by premultiplying by as in the last section. Note the following interpretation of our notation as exemplified in 2. The form of 2. This aids in writing compact equations and is conceptually clearer than 2. In order that we may write the mathematics given in 2. We adopt the convention that a position vector is 3 x 1 or 4 x 1, depending on whether it appears multiplied by a 3 x 3 matrix or by a 4 x 4 matrix.
It is readily seen that 2. For our purposes, it can be regarded purely as a construction used to cast the rotation and translation of the general transform into a single matrix form. In other fields of study, it can be used to compute perspective and scaling operations when the last row is other than "[0 0 0 1]" or the rotation matrix is not orthonormal.
The interested reader should see [2]. Often, we wifi write an equation like 2. Note that, although homogeneous transforms are useful in writing compact equations, a computer program to transform vectors would generally not use them, because of time wasted multiplying ones and zeros. Thus, this representation is mainly for our convenience when thinking and writing equations down on paper.
Observe that, although we have introduced homogeneous transforms in the context of mappings, they also serve as descriptions of frames. This section illustrates this interpretation of the mathematics we have already developed. Translational operators A translation moves a point in space a finite distance along a given vector direc- tion.
With this interpretation of actually translating the point in space, only one coordinate system need be involved.
It turns out that translating the point in space is accomplished with the same mathematics as mapping the point to a second frame. Almost always, it is very important to understand which interpretation of the mathematics is being used. The distinction is as simple as this: When a vector is moved "forward" relative to a frame, we may consider either that the vector moved "forward" or that the frame moved "backward. Equations 2. This sign change would indicate the difference between moving the vector "forward" and moving the coordinate system "backward.
Now that the "DQ" notation has been introduced, we may also use it to describe frames and as a mapping. Rotational operators Another interpretation of a rotation matrix is as a rotational operator that operates on a vector A P1 and changes that vector to a new vector, A P2, by means of a rotation, R.
Usually, when a rotation matrix is shown as an operator, no sub- or superscripts appear, because it is not viewed as relating two frames. This fact also allows us to see how to obtain rotational matrices that are to be used as operators: The rotation matrix that rotates vectors through some rotation, R, is the same as the rotation matrix that describes a frame rotated by R relative to the reference frame.
This operator can be written as a homogeneous transform whose position-vector part is zero. For example, substitution into 2. The "RK" notation, therefore, may be considered to represent a 3 x 3 or a 4 x 4 matrix. Later in this chapter, we will see how to write the rotation matrix for a rotation about a general axis K.
We wish to compute the vector obtained by rotating this vector about 2 by 30 degrees. Call the new vector The rotation matrix that rotates vectors by 30 degrees about 2 is the same as the rotation matrix that describes a frame rotated 30 degrees about Z relative to the reference frame. Thus, the correct rotational operator is [0. Note that, if we had defined R instead of R in 2. This change would indicate the difference between rotating the vector "forward" versus rotating the coordinate system "backward.
In this interpretation, only one coordinate system is involved, and so the symbol T is used without sub- or superscripts. This fact also allows us to see how to obtain homogeneous transforms that are to be used as operators: The transform that rotates by R and translates by Q is the same as the transform that describes afraine rotated by Rand translated by Q relative to the reference frame. A transform is usually thought of as being in the form of a homogeneous transform with general rotation-matrix and position-vector parts.
Having understood the general case of rotation and translation, we wifi not need to explicitly consider the two simpler cases since they are contained within the general framework. As a general tool to represent frames, we have introduced the homogeneous transform, a 4 x 4 matrix containing orientation and position information. We have introduced three interpretations of this homogeneous transform: 1. It is a description of a frame. It is a transform mapping.
It is a transform operator. T operates on Ap1 to create Ap2 From this point on, the terms frame and transform wifi both be used to refer to a position vector plus an orientation. Frame is the term favored in speaking of a description, and transform is used most frequently when function as a mapping or operator is implied. Note that transformations are generalizations of and subsume translations and rotations; we wifi often use the term transform when speaking of a pure rotation or translation.
These two elementary operations form a functionally complete set of transform operators. Compound transformations In Fig. A straightforward way of calculating the inverse is to compute the inverse of the 4 x 4 homogeneous transform.
However, if we do so, we are not taking full advantage of the structure inherent in the transform. It is easy to find a computationally simpler method of computing the inverse, one that does take advantage of this structure.
Consider 2. Note that, in all figures, we have introduced a graphical representation of frames as an arrow pointing from one origin to another origin. The arrow's direction indicates which way the frames are defined: In Fig. In order to compound frames when the arrows line up, we simply compute the product of the transforms. If an arrow points the opposite way in a chain of transforms, we simply compute its inverse first. Again, we might equate 2.
As shown, rotation matrices are special in that all columns are mutually orthogonal and have unit magnitude. It is natural to ask whether it is possible to describe an orientation with fewer than nine numbers. Now a skew-symmetric matrix i. Clearly, the nine elements of a rotation matrix are not all independent. In fact, given a rotation matrix, R, it is easy to write down the six dependencies between the elements.
It is natural then to ask whether representations of orientation can be devised such that the representation is conveniently specified with three parameters.
This section will present several such representations. Whereas translations along three mutually perpendicular axes are quite easy to visualize, rotations seem less intuitive. Unfortunately, people have a hard time describing and specifying orientations in three-dimensional space. One difficulty is that rotations don't generally commute. Because rotations can be thought of either as operators or as descriptions of orientation, it is not surprising that different representations are favored for each of these uses.
Rotation matrices are useful as operators. Their matrix form is such that, when multiplied by a vector, they perform the rotation operation. However, rotation matrices are somewhat unwieldy when used to specify an orientation. A human operator at a computer terminal who wishes to type in the specification of the desired orientation of a robot's hand would have a hard time inputting a nine-element matrix with orthonormal colunms.
A representation that requires only three numbers would be simpler. The following sections introduce several such representations.
We will call this convention for specifying an orientation X—Y—Z fixed angles. The word "fixed" refers to the fact that the rotations are specified about the fixed i. Sometimes this convention is referred to as roll, pitch, yaw angles, but care must be used, as this name is often given to other related but different conventions. Rotations are performed in the order Rz a. It is extremely important to understand the order of rotations used in 2. Thinking in terms of rotations as operators, we have applied the rotations from the right of y , then p , and then Multiplying 2.
Equation 2. The inverse problem, that of extracting equivalent X—Y—Z fixed angles from a rotation matrix, is often of interest. The solution depends on solving a set of transcendental equations: there are nine equations and three unknowns if 2. Among the nine equations are six dependencies, so, essentially, we have three equations and three unknowns.
Let r r11 r12 r13 1 a r21 ifl 2. This is usually a good practice, because we can then define one-to-one mapping functions between various representations of orientation. However, in some cases, calculating all solutions is important more on this in Chapter 4. In those cases, only the sum or the difference of a and y can be computed. Such sets of three rotations 3Atan2 y, x computes tan1 but uses the signs of both x and y to identify the quadrant in which the resulting angle lies.
For example, Atan 2 —2. Note that Atan2 becomes undefflied when both arguments are zero. It is sometimes called a "4-quadrant arc tangent," and some programming-language libraries have it predeSned. Note that each rotation takes place about an axis whose location depends upon the preceding rotations.
An additional "prime" gets added to each axis with each rotation. A rotation matrix which is parameterized by Z—Y—X Euler angles wifi be indicated by the notation y. Note that we have added "primes" to the subscripts to indicate that this rotation is described by Euler angles. With reference to Fig. Because 2. That is, 2. In those cases, only the sum or the difference of a and y may be computed.
Each of these conventions requires performing three rotations about principal axes in a certain order. These conventions are examples of a set of 24 conventions that we will call angle-set conventions.
Of these, 12 conventions are for fixed-angle sets, and 12 are for Euler-angle sets. Note that, because of the duality of fixed-angle sets with Euler-angle sets, there are really only 12 unique parameterizations of a rotation matrix by using successive rotations about principal axes.
There is often no particular reason to favor one convention over another, but various authors adopt different ones, so it is useful to list the equivalent rotation matrices for all 24 conventions. Appendix B in the back of the book gives the equivalent rotation matrices for all 24 conventions. Equivalent angle—axis representation With the notation Rx This is an example of an equivalent angle—axis representation. If the axis is a general direction rather than one of the unit directions any orientation may be obtained through proper axis and angle selection.
Vector K is sometimes called the equivalent axis of a finite rotation. The angle specifies a third parameter.
Often, we wifi multiply the unit direction, K, with the amount of rotation, 9, to form a compact 3 x 1 vector description of orientation, denoted by K no "hat". Note that, given any axis of rotation and any angular amount, we can easily construct an equivalent rotation matrix.
The inverse problem, namely, that of computing K and 0 from a given rotation matrix, is mostly left for the exercises Exercises 2. Therefore, in converting from a rotation-matrix into an angle—axis representation, we are faced with choosing between solutions. A more serious problem is that, for small angular rotations, the axis becomes ill-defined. Clearly, if the amount of rotation goes to zero, the axis of rotation becomes completely undefined.
The solution given by 2. Substituting into 2. There was no translation of the origin, so the position vector is [0, 0, Hence, 0. If we encounter a problem for which this is not true, we can reduce the problem to the "axis through the origin" case by defining additional frames whose origins lie on the axis and then solving a transform equation. As is shown in Fig. We wifi choose 1. This is a rotation about an axis that passes through the origin, so we can use 2.
There was no translation of the origin, so the position vector is [0, 0, OjT. Thus, we have 0. Our particular choice of orientation was arbitrary, and our choice of the position of the origin was one of an infinity of possible choices lying along the axis of rotation.
See also Exercise 2. Euler parameters Another representation of orientation is by means of four numbers called the Euler parameters.
Although complete discussion is beyond the scope of the book, we state the convention here for reference. Hence, an orientation might be visualized as a point on a unit hypersphere in four-dimensional space.
Sometimes, the Euler parameters are viewed as a 3 x 1 vector plus a scalar. However, as a 4 x 1 vector, the Euler parameters are known as a unit quaternion. However, it can be shown that, in the limit, all the expressions in 2.
In fact, from the definitions in 2. Taught and predefined orientations In many robot systems, it wifi be possible to "teach" positions and orientations by using the robot itself. The manipulator is moved to a desired location, and this position is recorded. A frame taught in this manner need not necessarily be one to which the robot wifi be commanded to return; it could be a part location or a fixture location. In other words, the robot is used as a measuring tool having six degrees of freedom.
Teaching an orientation like this completely obviates the need for the human programmer to deal with orientation representation at all. In the computer, the taught point is stored as a rotation matrix or however , but the user never has to see or understand it.
Robot systems that allow teaching of frames by using the robot are thus highly recommended. Besides teaching frames, some systems have a set of predefined orientations, such as "pointing down" or "pointing left.
However, if this were the only means of describing and specifying orientation, the system would be very limited. In later chapters, we wifi discuss velocity and force vectors as well. These vectors will transform differently because they are a different type of vector.
In mechanics, one makes a distinction between the equality and the equivalence of vectors. Two vectors are equal if they have the same dimensions, magnitude, and direction.
Two vectors that are considered equal could have different lines of action—for example, the three equal vectors in Fig 2. These velocity vectors have the same dimensions, magnitude, and direction and so are equal according to our definition. Two vectors are equivalent in a certain capacity if each produces the very same effect in this capacity. Thus, if the criterion in Fig. If the criterion is height above the xy plane, then the vectors are not equivalent despite their equality.
Thus, relationships between vectors and notions of equivalence depend entirely on the situation at hand. Furthermore, vectors that are not equal might cause equivalent effects in certain cases.
We wifi define two basic classes of vector quantities that might be helpful. The term line vector refers to a vector that is dependent on its line of action, along with direction and magnitude, for causing its effects.
Often, the effects of a force vector depend upon its line of action or point of application , so it would then be considered a line vector. A free vector refers to a vector that may be positioned anywhere in space with- out loss or change of meaning, provided that magnitude and direction are preserved. The relative locations of the origins do not enter into the calculation.
The operation of rotation as in 2. Note that A which would appear in a position-vector transformation, does not appear in a velocity transform. The homogeneous representation is useful as a conceptual entity, but trans- formation software typically used in industrial manipulation systems does not make use of it directly, because the time spent multiplying by zeros and ones is wasteful. Usually, the computations shown in 2.
The order in which transformations are applied can make a large difference in the amount of computation required to compute the same quantity. Performing the final matrix-vector multiplication of 2. Of course, in some cases, the relationships and are constant, while there are many Dp.
In such a case, it is more efficient to calculate once, and then use it for all future mappings. Where L. Ballard and C. Bottema and B. Gorla and M. Give the rotation matrix that accomplishes these rotations in the given order. Give the rotation matrix that will change the descriptions of vectors from Bp to Ap 2. Give the rotation matrix that will change the description of vectors from B p to A p.
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