Planar Transformations and Displacements

Transcription

1 Chater Planar Transformations and Dislacements Kinematics is concerned with the roerties of the motion of oints. These oints are on objects in the environment or on a robot maniulator. Two features that make kinematics difficult are () rotations do not commute, and () the maing between joint angles and maniulator endoint osition is nonlinear. To begin, one needs to establish a formal method for describing oints and their dislacements; this is done b defining coordinate sstems and frames. A basic distinction is between oints and vectors, treated net. Another distinction is between coordinate transformations and dislacements: the former describe the relationshi between fied oints, the latter describe the motion of a oint. In this chater, we will develo the concets of coordinate transformations and dislacements in the lane; subsequent chaters will treat the 3D case. The basic concets are most easil introduced for the lane. First we consider rotations alone, then rotations lus translations. A dislacement will be distinguished from a coordinate transformation. Finall, we will discuss the ole of a lanar dislacement.. Points and Vectors Points and vectors are not the same, although the are intimatel related. A oint is a geometrical object, i.e., some definite location in the environment or on a maniulator, that eists indeendentl of how we describe it. Points belong to an affine sace, and are oerated on b affine transformations such as translation and rotation. Points are denoted b uer-case italic letters with subscrits, e.g.,, identifies a articular instance., etc. The subscrit just A vector reresents a dislacement between two oints; e.g., given oints and, the vector defines the distance and direction from to. Figure. reresents a table to or lane, where distinctive oints and have been identified. The vector lies on the lane of the table. c Vectors belong to a vector sace, and are described b coordinates relative to articular coordinate aes. Vectors are denoted b lower-case bold letters with subscrits and suerscrits, e.g.,. The suerscrit denotes the articular coordinate aes with resect to which the coordinates of are eressed. John M. Hollerbach, August

2 CHAPTER. PLANAR TRANSFRMATINS AND DISPLACEMENTS P. v. P v. v P. 3 P. P. P 4 Figure.: Points and on a table to and the dislacement vector vector describes the dislacement from different oints to. between them. The same Net we rovide more detail on affine saces and vector saces... Affine Saces As mentioned earlier, oints are geometric objects that are elements of an affine sace A. Points are not vectors, because there is no referred origin. For eamle, addition of oints makes no sense. In Figure. a different result would be obtained in adding oints and if eressed relative to different origins and. However, subtraction of oints makes sense, and gives rise to an associated vector sace. For eamle, from Figure.: (.) where vector is the dislacement from to. We ma also state that is obtained from b adding a dislacement : (.) The formal definition of an affine sace can now be stated: it consists of a set of oints, an associated vector sace, and the subtraction and addition oerators above. There are infinitel man oint airs whose relative dislacement is the same vector. In Figure., the dislacement from to is the same as from to, i.e., the vector. Another wa of thinking about a vector is that it encodes a length or distance and a direction. Eamle.: Suose ou are giving a erson directions from an originating oint, smbolicall reresented as, to a destination oint. An eamle might be go meters to the northeast. If a erson were starting out at a different originating oint, giving the same directions would cause the erson to end u at a different oint. But in both cases, the same distance and direction were secified. Thus a vector is not a geometrical object b itself, i.e., it is not tied to an oint. nl when added to a reeisting oint is there an association with a unique oint. More formall, the length of a vector is denoted as, the Euclidean norm of (discussed later). The unit vector reresents the direction. ften we will secif oints in terms of dislacements from a fied oint called origin i. For eamle, in Figure. oint is dislaced from the origin b, and is dislaced from b. Then is dislaced from b. More generall,, and if and onl if (iff).

3 .. PINTS AND VECTRS 3 P + P? P P P + P? v = v +v P v v P Figure.: Adding oints is not defined. Dislacement of oints and relative to... Vector Saces A vector sace is comosed of vectors, whose linear combinations are also vectors. Vectors in a two-dimensional (D) sace, i.e., the lane, have two comonents, while vectors in a 3D sace have three comonents. We will write resectivel that or deending on the contet. Given vectors and in an -dimensional sace, then is also a vector in, where and are arbitrar scalars. Scalars are reesented b subscrited lower-case italic letters, e.g., Figure. shows the familiar triangle which results from the addition of two vectors:, and. Vectors are usuall eressed in terms of other vectors, which form a basis for the vector sace. A basis is a set of vectors that sans the vector sace, that is to sa, an vector can be eressed as a linear combination of the basis vectors. There are as man vectors in a basis as the dimension of the sace. For the lane, i.e., the vector sace, an two vectors and that are not arallel can form a basis. Thus an vector in the lane can be reresented as: for some unique scalars and...3 rthonormal Bases,, etc. (.3) f all ossible bases, the most useful ones are orthonormal bases, i.e., bases in which the vectors are orthogonal and are unit vectors. rthogonal vectors in a lane are at right angles; the defining condition is that their dot roduct is zero. We will denote the two vectors in an orthonormal basis as and, where the right-hand rule determines that the angle from to is radians. A vector in terms of the orthonormal basis will then be written as: (.4) The unique scalars and are the coordinates of with resect to orthonormal basis. The suerscrit on the scalar multiliers indicates which orthonormal basis is the reference. For a different orthonormal basis, the same vector will have different coordinates: (.5)

4 4 CHAPTER. PLANAR TRANSFRMATINS AND DISPLACEMENTS v v - Figure.3: The vector eressed in basis. The same vector eressed in basis. The basis coordinates and will be different from the basis coordinates because the basis vectors are different. A ke oint is that the vector is the same in both cases, onl its descrition in terms of a articular basis differs. Eamle.: In Figure.3, a articular vector has been constructed such that with resect to orthonormal basis. Hence its coordinates are and. In Figure.3, there is a different orthonormal basis that is rotated 9 degrees from basis. Then, and its associated coordinates are and. Just like an other vectors, the basis vectors can be reresented in terms of some other basis, sa. Then (.6) where the double subscrits and refer to the first and second elements of the coordinates of the resective basis vectors. Thus and are the coordinates of, and and are the coordinates of, with resect to basis. Eamle.3: For the simle case of relative rotation b 9 as in Figure.3, we can see b insection that and. Hence the coordinates of are and, and the coordinates of are and, with resect to basis. What are the coordinates of the basis vectors with resect to their own basis? Clearl (.7) Hence and are the coordinates of, and and are the coordinates of, with resect to basis. When actuall evaluating an vector equation, for eamle (.4), all vectors have to be eressed with resect to the same basis. We did not state what basis the vectors and in (.4) were being referred to. We elicitl indicate the reference basis, sa, with resect to which all vectors are being eressed b a left suerscrit on the vectors: (.8) Note that the coordinates did not change, onl the eression of the three vectors.

5 $.. PINTS AND VECTRS 5 The simlest case to evaluate is when. For this case, the coordinates of a vector will be written as a column vector. First, consider the coordinates of basis vectors with resect to themselves, rewritten from (.7): (.9) This familiar result states that the coordinates of the and aes are and resectivel, also convenientl abbreviated as and. This is true of an set of basis vectors eressed with resect to themselves. Thus for a different basis, then and. Note that when a set of orthonormal basis vectors is eressed in terms of a different basis, for eamle and, the are no longer and. Net, the coordinates of an arbitrar vector eressed in basis will be written as: (.) where is the -b- identit matri. Thus the vector referred to basis is reresented as a -b- column vector of its coordinates. (Higher-dimensional vectors will also be reresented as column vectors.) Eamle.4: From Figure.3, we would write that We can also write that and, where means the transose of vector. In this contet, the transose oerator converts a -b- row vector into a -b- column vector. We are now in a osition to define the length of a vector and the dot roduct. The dot roduct of two vectors and is:!!!! (.) This definition of the dot roduct generalizes to vector saces of an dimensions. Although the dot roduct is evaluated with resect to a articular basis, the value of the dot roduct is the same irresective of the basis. This is because the dot roduct reresents the rojection of onto, and this rojection alwas has the same value. Two vectors and are orthogonal if their dot roduct is zero:. The length of a vector is defined in terms of the dot roduct: # (.) The length of a vector does not deend on the basis; it s alwas the same. Hence for different bases and. A unit vector has a length of :. An vector can be converted to a unit vector b dividing b its length:. Eamle.5: For the vector of Figure.3, we have that. The length is the same. and

6 6 CHAPTER. PLANAR TRANSFRMATINS AND DISPLACEMENTS j j v v jv j Figure.4: The rojections of a vector onto aes are its coordinates and. An orthonormal basis consists of orthogonal ( ) and unit ( ) vectors. Let s reeamine the meaning of the coordinates of a vector, in view of the dot roduct and its definition as a rojection (Figure.4). Take the dot roduct of (.8) first with, and then with : (.3) since and. Hence the coordinate reresents the rojection of onto, and is indeendent of the basis with resect to which the vectors in (.8) are eressed. Similarl, (.4) since and. Hence the coordinate is the rojection of onto...4 Coordinate Sstems The combination of an origin with a set of orthonormal basis vectors is called coordinate sstem j or frame j. The basis vectors are called the aes of coordinate sstem. The aes are ictured as dislacements from the origin in Figure.5, to imlied oints and. Thus a coordinate sstem reall consists of 3 oints, and the aes are deduced as and. While these imlied oints are to be understood, we will usuall draw a coordinate sstem with just the origin and aes indicated (Figure.5). Y j j j j j X j j j Figure.5: A coordinate sstem with imlied oints coordinate sstem is usuall deicted without the imlied oints. and. A To describe where objects in the environment are, we will attach coordinate sstems to them and relate the various coordinate sstems. We will also be attaching coordinate sstems to each link of a maniulator, in order to deduce joint angles and to relate the osition of a maniulator relative to objects.

7 .. PINTS AND VECTRS 7 How coordinate sstems are defined and attached to objects is a matter of convenience. For eamle, consider the two traezoids in arbitrar lacement in Figure.6, where the origins have been laced in corresonding corners and the aes on corresonding sides. In this general case, the origins and are different, as are the aes and. The osition and orientation of each traezoid is considered to be snonmous with the osition and orientation of its coordinate sstem.,, (C) Figure.6: Two traezoids with attached coordinate sstems in general osition relative to each other. The two traezoids are oriented the same. (C) The two traezoids are in the same osition and abutted. In Figure.6, traezoid has been rotated such that and. When the aes are aligned like this, we sa that the two traezoids are in the same orientation; however, the are not in the same osition becasue their origins differ. Furthermore, when we sa something like, we are not saing that these vectors are ointing to the same oint. What we are saing is that the two dislacements are the same. In terms of the imlied oints and (not shown, but in the manner of Figure.5), we are stating that, which are the definitions of and resectivel. In Figure.6(C), traezoid s origin has been made coincident with traezoid s origin, and traezoid has been rotated so that. Hence we have abutted the two traezoids. Their osition is now the same because. Their orientations differ b 9, which is the angle from to..5 o 45 d.5 Y / Y X / X Figure.7: Traezoid is dislaced b dislaced to origin. from traezoid, and rotated b 45. Aes are Eamle.6: Suose in Figure.6 that aes are rotated b 45 with reect to aes, and that the origin is dislaced from origin b (see Figure.7). The same linear dislacement also alies to Figure.6. What are the coordinates of with resect to aes in the three figures?

8 8 CHAPTER. PLANAR TRANSFRMATINS AND DISPLACEMENTS. The coordinate descrition for Figure.6(C) is the same as for Figure.3.. For Figure.6, and since aes are arallel to aes. Note that the dislacement not oints. between origins makes no difference, because we are comaring dislacements, 3. The same is true for Figure.6, where (.5) We are not describing where the oints reresented b and are located relative to. We are describing the dislacements and relative to the dislacements and. Another wa to view this roblem is to define oints and such that the dislacements of each oint relative to are (Figure.7): Again, we can do this because there are infinitel man airs of oints whose dislacement is the same vector. From the construction of Figure.7 where the various vectors in question all start from, it is clear that and. Hence when comaring vectors in coordinate sstems with different origins, we can just retend the origins coincide in the derivation. P Figure.8: An arbitrar oint on traezoid is located relative to b and relative to b...5 Locating Arbitrar Points Suose that there is an arbitrar oint on traezoid that we wish to locate (Figure.8). The vector locates oint relative to origin, while the vector locates oint relative to origin. We can relate either vector to bases or : (.6) (.7)

9 .. PINTS AND VECTRS 9 P P Figure.9: The coordinates of relative to aes and. Aes are referred to origin to find the coordinates of. P P Y P X Figure.: The coordinates of and. (.8) (.9) The coordinates of each vector are in general different from each other. Figure.9 illustrates the coordinates for vector. Relative to basis, the coordinates are siml the rojections of onto the aes as indicated. Relative to basis, again we imagine that a oint has been created such that describes the same dislacement. Then the coordinates are obtained b rojection onto the aes as before. A different construction for the coordinates of is to imagine oints and created such that and (Figure.9). This time we are dealing with the original vector, but coies of referred to origin. The coordinates are obtained b rojection onto the aes referred to origin. With regard to vector, the coordinates are obtained b rojecting onto the aes in Figure.. For the coordinates, we have chosen in Figure. to refer aes to origin before finding the coordinates b rojection. The rocess of actuall evaluating the coordinates of and with resect to different aes is the subject of subsequent sections.

10 CHAPTER. PLANAR TRANSFRMATINS AND DISPLACEMENTS cos sin θ θ θ θ, cos θ -sin θ (C) θ Figure.: Coordinate origins and are coincident, while aes are rotated b from aes. Derivation of and (C).. Planar Rotational Transformations Consider two coordinate sstems and, whose origins coincide and where aes are rotated b from aes (Figure.). The general eression of aes relative to aes is obtained from (.6): (.) What is the relationshi of the coordinates and to the rotation angle? Since these coordinates reresent the rojection of aes onto aes, from Figure.-(C) we have that (.) Substituting, (.) where again, the -b- identit matri, and and just equal their comonents relative to coordinate aes. Combining the aes equations (.) into one matri equation, (.3) where is the lanar rotation matri corresonding to angle from coordinate aes to coordinate aes. While this rotation matri in terms of ma be ver familiar, it is imortant to recognize that its columns are the coordinate aes referred to aes. We are able to erform a rotation because we know how aes look relative to aes. Eamle.7: Suose that, or 45, between aes and. Refer to Figure.7 for a similar relationshi between aes and. Then (.4)

11 .. PLANAR RTATINAL TRANSFRMATINS v v v θ v v v Figure.: Reresentation of a vector in the lane. Reresentation of with resect to coordinate aes, which are at an angle with resect to coordinate aes. Comare this result with (.5). Conversel, we ma eress coordinate aes relative to coordinate aes. The result is: (.5) The derivation is left as an eercise. Note that the rotation matri (.5) from aes to aes is the transose of the rotation matri (.3) from aes to, which is true in general. Net consider a vector, with comonents and relative to aes, and comonents and relative to aes (Figure.). Then (.6) From (.3) and the definition of the comonents of, The rotation matri converts into. (.7) Eamle.8: Let us return to the roblem of determining in Figure.9, when. Then and A similar develoment utilizing (.5) will show that (.8) This result can be gotten immediatel b inverting inverse is the transose: in (.7). For this and all rotation matrices, the (.9)

12 CHAPTER. PLANAR TRANSFRMATINS AND DISPLACEMENTS In the lanar case, another wa of looking at the inverse or transose of a rotation matri is to note that (.3) If we rotate forward b, then the inverse oeration is just to rotate backwards b. This works, because in the lane the rotation ais is imlicitl alwas the same. θ θ v θ + θ Figure.3: Comosition of rotations and.. Arbitrar vector reresented in frame... Comosition of Planar Rotations Suose that another set of coordinate aes, are introduced, that make an angle with resect to aes (Figure.3). Then the rotation matri relating aes to aes has the same form as (.3): (.3) Consequentl, the rotation matri relating aes to aes is obtained b using (.7): (.3) This result is clear, because is the angle between coordinate aes and. The comosition of rotations in the lane, in this eamle of with, is ver simle, as the result is the rotation matri which sums the angles, e.g.,. In general, for aes related to aes with and a number of intermediate aes, where (.33) Similar to the revious develoment, an arbitrar vector eressed with resect to coordinate aes, i.e., (Figure.3), can be related to, and then to, b: (.34) In keeing with the sirit of later chaters, we introduce an additional notation for rotation matrices derived from (.33): is the rotation matri from aes to aes.

13 .. PLANAR RTATINAL TRANSFRMATINS 3 P θ P d θ θ d d Figure.4: Coordinate transformation between frame and frame, and also including frame. The atomic relation is. More generall, (.35) For 3D rotations, we cannot eress a rotation matri as a function of a single arameter, such as for for the lanar case. Instead, we will see that 3 arameters are required to reresent 3D rotations. Nevertheless, will have the same meaning, as the rotation from aes to aes, whether for D or 3D. The reason for the notation is aarent from (.33): Thus has the same form as its column constituents. Another view is to reeress (.34) as The right subscrit in the left suerscrit in (.36) (.37) matches the left suerscrit in, while the left suerscrit in. More generall, matches (.38) Finall, the inverse relation (.9) is (.39) Eactl the same relation will hold for 3D rotations.

14 4 CHAPTER. PLANAR TRANSFRMATINS AND DISPLACEMENTS.3 Planar Coordinate Transformations For the rotational lanar transformations considered so far, it was assumed that the origins corresonding to different coordinate aes were the same. In general, the origins corresonding to coordinate aes will be located in different laces, for eamle the two traezoids in Figure.6. Let us relabel the coordinate sstems as and ; in general, the environmentall fied reference frame will be labeled as frame. Figure.4 shows origin dislaced from origin b the vector. is defined as the vector from to. The vector identifies some oint relative to, while identifies relative to. Clearl (.4) ften a oint will onl be known relative to the frame of the bod in which it is located. For eamle, oint in Figure.6 could be at the oosite corner of traezoid, and its location would be known relative to aes. In the resent eamle, we assume we know the coordinates. From the revious section, the rotation matri alied to ields (see equation (.7)): (.4) We assume that and are given, i.e., we know where frame is relative to frame. To find given, substitute (.4) into (.4): (.4) Equation (.4) is said to reresent a coordinate transformation: a oint originall reresented in frame is transformed to its reresentation in frame. Note the order in (.4): first is rotated b so that it is eressed relative to aes, i.e.,, then it is added to the dislacement between origins, which is also eressed relative to aes. Eamle.9: Let us return to the roblem of determining, and. Then in Figure. for, Given a coordinate transformation from to, we can reverse the transformation to go from to. Given (.4), b rearrangement we find: where again. (.43) Eamle.: For the roblem of determining in Figure. for, and,,

15 .4. TW-LINK PLANAR MANIPULATR KINEMATICS 5.3. Comosition of Coordinate Transformations Net introduce frame, which is dislaced from frame b and whose aes are at an angle with resect to aes (Figure.4). Suose oint is located relative to b, relative to b, and relative to b. Then Again, suose (.44) (.45) is known relative to frame, i.e., as. The location of relative to frame is: (.46) Utilizing (.46), the location of relative to frame is: (.47) Equation (.47) demonstrates the comosition of two coordinate transformations. Note again the comosition of rotations:..4 Two-Link Planar Maniulator Kinematics As an eamle of the alication of coordinate transformations, we will solve for the forward kinematics of a lanar two-link maniulator (Figure.5). a θ a θ Figure.5: Two-link lanar maniulator. Frame, the stationar or ground reference, is laced at the roimal end of link, such that origin is located at joint.

16 6 CHAPTER. PLANAR TRANSFRMATINS AND DISPLACEMENTS Frame is embedded in link at the distal end, such that origin is the location of joint. Ais lies along the long ais of link, which connects to. Frame is the intrinsic coordinate sstem of link. Frame is embedded in link in the distal end, where is considered the maniulator endoint. The endoint is the working oint of the manulator, which we wish to interact with the environment. Ais is along the long ais of link, which connects to. Frame is the intrinsic coordinate sstem of link. The length of link is, and that of link is. Joint angle is the angle from to, while joint angle is the angle from to. This method of setting u coordinate sstems in the links is consistent with the Denavit-Hartenberg convention, discussed in a later chater. Note that the coordinate sstem for link is laced in link s distal end. ther conventions, not emloed in this course, lace it in the roimal end. The roblem of forward kinematics is to find the endoint osition, i.e., oint relative to oint, given the joint angles and. From Figure.5, (.48) rdinaril, we wouldn t go through all these transformations for the lanar case, because this relation can be directl found from insection. Let the coordinates of the endoint be in the simlified diagram, Figure.6. Then where throughout this course, the shorthand notation is used for and for. c (.49) s (.5).5 Homogeneous Transformations in a Plane A coordinate transformation such as (.4) searatel reresents rotations and translations. These transformations can be combined into a single matri oeration b increasing the dimension of a lanar vector to 3, where the third dimension is. Let be a -D column vector with its comonents indicated. Then the homogeneous coordinate created from is indicated b the uer case italic letter : (.5)

17 .5. HMGENEUS TRANSFRMATINS IN A PLANE 7.(,) a a θ + θ Figure.6: Endoint osition θ of two-link lanar maniulator found b insection. A coordinate transformation can now be reresented as: (.5) or emloing the homogeneous coordinate notation, (.53) where and is the 3-b-3 homogeneous transformation, from frame to frame, for the lane. Note that a suerscrit can be emloed for the homogeneous coordinate descrition of a oint, to indicate the aes with resect to which the coordinates are eressed:.5. Comosition of Homogeneous Transformations (.54) Coordinate transformations can easil be chained between airs of coordinate sstems. Let reresent the homogeneous transformation from frame to frame and from frame to frame. Then the homogeneous transformation from frame to is (.55) In this notation, note how the subscrit of lines u with the suerscrit of. Alied to homogeneous coordinates, (.56) where. Note how much more comact the homogeneous transformation notation is than (.47). The inverse coordinate transformation is again easier to eress using homogeneous transformations. (.57) (.58)

18 8 CHAPTER. PLANAR TRANSFRMATINS AND DISPLACEMENTS where (.58) is derived from (.43). With our notation, : the inverse of the transformation from frame to is the transformation from frame to. The inverse of the comosition of transformations ma be siml written: (.59) Eamle.: Consider the following relative location of frames,, and (Figure.7): Frame is dislaced from frame b a rotation of and a translation of. Then and P 6 o / d sqrt(3)/ 3 o d Figure.7: Relative locations of frames,, and, and a oint. Frame is dislaced from frame b a rotation of and a translation of and. Then

19 .5. HMGENEUS TRANSFRMATINS IN A PLANE 9 Then Given a vector, then Also,.5. Homogeneous Coordinates and the Affine Plane In grahics, homogeneous coordinates and transformations are used etensivel to reresent scaling and shear, as well as translation and rotation []. In addition, homogeneous coordinates are a wa of concretel denoting a oint b a set of coordinates, which is distinct from denoting a vector b a set of coordinates. We urosel used the same notation for a homogeneous coordinate as was introduced for a oint, namel italic uer case letters such as, because we now consider that a homogeneous coordinate reresents the coordinates of a oint. The set of oints in whose last coordinate is is referred to as the standard affine lane in. It is the subsace of oints that lies in the lane (Figure.8). A vector is then considered to be a 3-coordinate vector whose last comonent is and that lies in the affine lane. In Figure.8, oint has coordinates, while the vector from to has coordinates V (.6) Even though a 3-tule eresses both oints and vectors, the third comonent as or distinguishes them.

20 CHAPTER. PLANAR TRANSFRMATINS AND DISPLACEMENTS z. P. v. P z= Figure.8: Points and on the affine lane, reresented as homogeneous coordinates, and the dislacement that lies on the lane. The notation to distinguish the 3-b- vector V from the -b- vector is uer case slanted bold font. Because a homogeneous transformation can be considered to oerate on oints, the are also referred to as affine transformations. Two of these affine transformations are secial. A translational affine transformation Trans is the secial homogeneous transformation with the form: (.6) There is no rotation, reflected b the identit transformation, and there is a dislacement between origins. A rotational affine transformation Rot imlements a ure rotation without translation: (.6) Thus a general homogeneous transformation is the comosition of a translational with a rotational affine transformation: (.63) Note the order: first a translation, then a rotation. The reverse comosition would not give the correct result. For, the rotational art contains as its two columns and ; refer to (.3). Thus (.64)

21 .6. PERATRS Hence the homogeneous transformation contains the aes of frame relative to frame, lus the dislacement of origin from origin. Because contains all the information about the coordinate sstem relative to coordinate sstem, it itself is often called frame..6 erators A coordinate transformation is a static descrition of the relative location of two coordinate sstems, erhas reresenting two different objects. An oerator or dislacement is an active descrition of the rocess of moving a oint relative to a fied coordinate sstem, such as moving an object somelace. The same mathematics for coordinate transformations can also be interreted as oerators. The difference from the coordinate transformation viewoint is: oerations or dislacements are with resect to the fied frame; coordinate transformations are with resect to the current frame. Another viewoint is that dislacing a oint forward is the same as moving the coordinate sstem backward. The translational oerator in Figure.9 translates a oint, reresented relative to the origin b, b the dislacement to the oint, reresented relative to the origin b : or (.65) A homogeneous transformation and a vector equation have both been resented to eress the results of the translational oerator. The homogeneous transformation emhasizes that oint is being moved to oint. All vectors in the vector equation are reresented in the same coordinate frame, which demonstrates that the dislacement is with resect to a fied frame. The coordinate transformation interretation of the translation oerator assumes coordinate frames and are initiall coincident. Then move coordinate frame backwards b (Figure.9) to describe the location of oint. The rotational oerator in Figure. rotates oint to b the angle. Then the new vector resulting after the dislacement b in the same coordinate frame is: or (.66) Again, the homogeneous transformation reresentation of a rotational oerator demonstrates that oint is being rotated about oint to oint. The vector equation again shows that all vectors are eressed with resect to the fied frame. The coordinate transformation interretation assumes coordinate frames and are initiall coincident (Figure.). Rotate frame backwards b to describe the location of oint. The general transformational oerator first rotates a oint, then translates it. The homogeneous transformation and vector forms are: or (.67) In Figure., oint is rotated to become the oint, then this intermediate oint is translated b to become. Again, all vectors in the vector form are eressed in the same, or fied, frame. We have adoted the coordinate transformation form to reresent the transformational

22 CHAPTER. PLANAR TRANSFRMATINS AND DISPLACEMENTS P. d. P. P,,, -d. P Figure.9: The translation oerator moves oint to oint via the dislacement equivalent coordinate transformation moves coordinate sstem backwards b. P P. θ.,.,, P. P, - θ. The Figure.: The rotation oerator moves oint to oint b rotating around b the angle. The equivalent coordinate transformation rotates coordinate sstem backwards b. R(θ ) θ P d P,,, P P -d θ Figure.: The transformational oerator moves oint to oint b rotating to, then translating b. The equivalent coordinate transformation rotates frame backwards b, then translates frame backwards b relative to frame s current or intermediate location.

23 .6. PERATRS 3 oerator for convenience, and in the future we will be elicit about whether this form is being interreted as an oerator or as a coordinate transformation. The same holds for and. The coordinate transformation interretation, starting from overlaing frames and, involves rotating frame backwards b, then translating backwards b relative to the current frame, i.e., the intermediate location of frame indicated b dashed lines..6. Comosition of erators As another viewoint to elain the order of translations and rotations in an oerator, recast the transformational oerator as a homogeneous transformation in terms of a ure translation and a ure rotation: (.68) That is to sa, we have a comosition of two elementar oerators or coordinate transformations. difference between oerators and coordinate transformations can be viewed as follows: The A coordinate transformation interretation involves a left-to-right evaluation of a comosition about the current frame. An oerator interretation involves a right-to-left evaluation of a comosition about the fied frame. As an eamle to illustrate these two different interretations, consider the homogeneous transformation (.69). Viewed as a coordinate transform, first the aes b.. Viewed as an oerator, first to frame. translates the origin of frame to, then rotates rotates frame b, then translates it b, all still relative There doesn t seem to be such a big difference between these two interretations for this simle eamle, but the comosition of two general oerators illustrates the difference more clearl. Consider the comosition of the homogeneous transformations, where is the same as above and Decomose these transformations in terms of the elementar transformations Trans and Rot: where we have written and are all initiall overlaing. The interretation of differs as follows. (.7) (.7) (.7) (.73). Consider that coordinate sstems,, and as a coordinate transformation versus as an oerator

24 4 CHAPTER. PLANAR TRANSFRMATINS AND DISPLACEMENTS d 3 o d 45 o Figure.: The comosition viewed as a coordinate transformation (left to right evaluation about the current frame), or as an oerator (right to left evaluation about the fied frame).. Viewed as a coordinate transform, the transformations in are interreted from left to right about the current frame. First translates the origin of frames and to, then rotates the aes b. Then translates frame relative to frame b, then rotates frame b relative to frame (Figure.).. Viewed as an oerator, the transformations in are interreted from right to left about the fied frame. First rotates frames and b, then translates frames and b, all relative to frame. Then rotates frame b, then translates it b, all still relative to frame. The result of viewed as an oerator or a coordinate transformation is the same, but the rocess of reaching the final lacement of frame is different. What is the relation to the interretation of the revious section? Figure. shows these oerations from an observer fied in frame. If the observer had been fied in frame as in the revious Figures.9-., then frame would have aeared to be undergoing the negative of the dislacements just discussed. o 3 d o 45 d Figure.3: erator frame moves to. P D P T P D P moves to relative to frame. The corresonding oerator in

25 .7. PLES F PLANAR DISPLACEMENTS 5.6. eration about a Different Frame Let homogeneous transformation oerate on homogeneous coordinate in frame to ield homogeneous coordinate (Figure.3): Consider another reference frame, located b homogeneous transformation.3). The corresonding homogeneous coordinates in frame are: (.74) relative to frame (Figure (.75) (.76) Then (.77) Hence the oerator moves to in the same wa that moves to. Thus frame rather than in frame. is the same oerator as (.78), but in.7 Poles of Planar Dislacements For an arbitrar lanar oerator, there is one oint called the ole that doesn t move []. Consider an arbitrar oerator reresented b rotation and translation, relative to frame. Figure.4 shows the effect of this oerator on a frame that is initiall coincident with frame. The ole, reresented b the vector, is that oint which is invariant under this oerator: (.79) There is one ecetion for which there is no solution: for a ure translation. Let be the oerator, reresented b the homogeneous transformation with elements and. Then (.8) where reresents the homogeneous coordinates fashioned from. Suose we change reference frames to frame b the homogeneous transformation ole in frame? The ole identified in frame is eressed in frame as: The corresonding oerator Hence is the ole of. in frame is given b (.78), and its effect on. What is the (.8) is: (.8)

26 6 CHAPTER. PLANAR TRANSFRMATINS AND DISPLACEMENTS d θ ole θ d Figure.4: The effect of oerator on frame, initiall coincident with frame. A ure rotation about frame b catures the oerator about frame. where Suose frame is located at the ole (Figure.4), with the same aes orientation as frame. Then, i.e., the ole is at the origin of frame. From this relation, it is eas to show that since aes and are arallel. Also. Hence (.83) since from (.79). Hence is a ure rotation about the ole. Figure.4 shows that frame is derived from frame with a ure rotation relative to frame at the ole. We reiterate this imortant result. An dislacement in the lane is equivalent to a ure rotation about the ole. This center of rotation is a useful concet in comliance control.

27 .7. PLES F PLANAR DISPLACEMENTS 7 Eamle.: Consider the oerator (.7). Its ole is found from (.79): bisector of - ole c h ole θ θ θ d d θ Figure.5: A circle constructed so that its center is the ole. A different circle results from a different rotation angle. A more intuitive viewoint is rovided b Figure.5, which shows some construction lines added to Figure.4. Because the dislacement referred to the ole is a ure rotation, origins and have to lie on the arc of a circle, and the ole is the center of the circle. To find where the center is, the ole has to lie on the erendicular bisector of the segment. Form an isosceles triangle b drawing lines and and including the line. The location of the ole on the bisector is that oint where the angle between lines and is the same as the rotation angle between aes and.

28 8 CHAPTER. PLANAR TRANSFRMATINS AND DISPLACEMENTS Let be the vector along the bisector from the midoint of to, given b: Then the location of, given b vector, is (.84) (.85) B moving the ole u and down on the bisector, different rotation angles are obtained. Figure.5 shows another eamle.

29 .8. SUMMARY 9.8 Summar. Points are distinct geometrical objects that eist indeendent of an coordinate descrition. The belong to an affine sace, whose onl meaningful oeration is the difference between oints.. Vectors reresent a dislacement between two oints, and are not themselves geometric objects. Vectors belong to a vector sace, defined as one in which the linear combination of two vectors is still a vector. 3. A basis is a minimal set of vectors that sans a vector sace, i.e., an vector can be eressed as a linear combination of the basis vectors. f the infinitel man ossible bases, the most imortant are the orthonormal bases, whose vectors are orthogonal and of unit length. 4. A coordinate sstem or frame is an origin lus an orthonormal basis basis vectors are referred to as the aes of the coordinate sstem.. The orthonormal 5. The coordinates of a vector with resect to aes are the coefficients on the linear combinations of the orthonormal basis vectors in which is eressed. The are formed into a -b- column vector. 6. A rotation matri describing the angle from aes to aes has as its columns the aes eressed in terms of aes. The inverse of a rotation matri is its transose. 7. A coordinate transformation reresents the relation between two arbitrar coordinate sstems b a dislacement between the origins and a rotation between the aes. 8. The homogeneous coordinates of a oint relative to origin are formed as the 3-b- column vector, b adding a 3rd coordinate to the vector. The homogeneous coordinates of a oint can be interreted as ling on the affine lane in three-dimensional sace. A vector reresented b homogeneous coordinates is then interreted as ling in the affine lane with 3rd coordinate. 9. A homogeneous transformation reresents a coordinate transformation as a 3-b-3 matri, alied to the homogeneous coordinates of a oint. It is a comact wa to combine rotations and translations into a single matri-vector oeration. The secial transformations and erform rotation and translation, resectivel, and can be combined to define an arbitrar homogeneous transformation.. An oerator activel moves oints with resect to the fied frame. B contrast, a coordinate transformation describes the instantaneous relationshi between two coordinate sstems. However, the same mathematics reresents oerators and coordinate transformations, but interreted differentl. Consider a comosition of several homogeneous transformations. A coordinate transformation is interreted as a left-to-right evaluation of the comosition about the current frame. An oerator is interreted as a right-to-left evaluation about the fied frame. Another interretation is that moving a oint forwards is the same as moving the reference coordinate sstem backwards. This viewoint corresonds to an observer in the last versus the first frame, resectivel.

30 3 CHAPTER. PLANAR TRANSFRMATINS AND DISPLACEMENTS. The ole of a lanar dislacement is that oint which does not move under an arbitrar lanar oerator. It is unique, unless the lanar oerator is a ure translation. An oerator can be related to a different frame b re- and ost-multiling b the coordinate transformation to the different frame and its inverse resectivel. When related to the ole, the equivalent oerator is a ure rotation.

31 .9. NTATIN SYNPSIS 3.9 Notation Snosis. Points are denoted b uer-case italic letters with subscrits, e.g.,, identif different instances., etc. The subscrits just (a) is the origin of coordinate sstem.. Scalars are reesented b subscrited lower-case italic letters, e.g., 3. Vectors are denoted b lower-case bold letters with subscrits and suerscrits, e.g.,. The suerscrit denotes the articular coordinate aes with resect to which the coordinates of are eressed. (a) The aes for coordinate sstem are (b) The coordinates of vector with resect to aes are. For a subscrited vector such as, the two coordinates are indicated b double subscrits:. (c) The coordinates of aes with resect to themselves are (d) All vectors are considered as column vectors.,, etc. (e) The transose of the column vector with resect to basis is the row vector:!! (f) The dot roduct of two vectors and is!! (g) The length or Euclidean norm of a vector is # (h) is the dislacement from origin to origin. 4. Matrices are reresented b uer-case bold letters, e.g.,,,, etc., which ma have suerscrits or subscrits. (a) is the rotation matri relative to rotation angle. (b) is the more general notation for rotation matri describing the orientation of aes to aes. An two arbitrar aes and are related b. (c) is the -b- identit matri.

32 3 CHAPTER. PLANAR TRANSFRMATINS AND DISPLACEMENTS (d) is the rotational homogeneous transformation (e) where. is the translational homogeneous transformation (f) The homogeneous transformation from coordinate sstem to coordinate sstem is has the form where. 5. The homogeneous coordinate reresentation of a oint is. It where the suerscrit indicates the reference coordinate sstem for the affine lane. (a) The homogeneous coordinate reresentation of a vector from origin to oint is P and is reresented as an uer-case letter in a slanted bold font. corresonding to the dislacement

33 Bibliograh [] Fole, J.D., van Dam, A., Feiner, S.K., and Hughes, J.F., Comuter Grahics: Princiles and Practice. Reading, MA: Addison-Wesle, 99. [] McCarth, J.M., Introduction to Theoretical Kinematics. Cambridge, MA: The MIT Press,