Theory of two-dimensional transformations

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1 Calhoun: The NPS Institutional Archive DSpace Repositor Facult and Researchers Facult and Researchers Collection Theor of two-dimensional transformations Kanaama, Yutaka J.; Krahn, Gar W. Downloaded from NPS Archive: Calhoun

2 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 14, NO. 5, OCTOBER [11] F. L. Lewis, K. Liu, and A. Yesildirek, Neural net robot controller with guaranteed tracking performance, IEEE Trans. Neural Networks, vol., pp , Ma [1] S. S. Ge, T. H. Lee, and C. J. Harris, Adaptive Neural Network Control of Robotic Manipulators. Singapore: World Scientific, [13] S. S. Ge and I. Postlethwaite, Adaptive neural network controller design for fleible joint robots using singular perturbation technique, Trans. Inst. Meas. Contr., vol. 17, no. 3, pp , Theor of Two-Dimensional Transformations Yutaka J. Kanaama and Gar W. Krahn Abstract This paper proposes a new heterogeneous two-dimensional (-D) transformation group ht ; i to solve motion analsis/planning problems in robotics. In this theor, we use a 3 1 matri to represent a transformation as opposed to a 3 3 matri in the homogeneous formulation. First, this theor is as capable as the homogeneous theor. Because of the minimal size, its implementation requires less memor space and less computation time, and it does not have the rotational matri inconsistenc problem. Furthermore, the raw rotation angle is more useful than the trigonometric values, cos and sin, in the homogeneous transformations. This paper also discusses how to appl the group ht ; i to solve problems related to motion analsis/planning, trajector generation, and others. This heterogeneous formulation has been successfull implemented in the MML software sstem for the autonomous mobile robot Yamabico-11 developed at the Naval Postgraduate School. Inde Terms Group theor, heterogeneous transformations, trajector generation, transformation, two-dimensional transformation. I. INTRODUCTION The three-dimensional (3-D) homogeneous transformation theor has been etensivel used in the robotics field [1], []. Therefore, when we need a two-dimensional (-D) transformation theor to deal with the problems in mobile robot motion control or computer graphics, a natural consequence is to appl the -D version of the 3-D homogeneous transformation formalism [3]. The general format of -D homogeneous transformations is T R 11 R 1 R 1 R 1 cos sin sin cos 1 where a transformation is represented b a 3 3 matri. As opposed to this classical method, in this paper, we propose a new -D transformation group ht ; i [4]. Each transformation in this theor is the 3 1 matri q (; ; ) T, where ; ; <. Since a transformation in a plane has three degrees of freedom (two for translation and one for rotation), this 3 1 form is the minimal mathematical structure we need. Therefore, if this new transformation Manuscript received Jul 19, 1994; revised June 1, This paper was recommended for publication b Associate Editors Y. Nakamura, V. Kumar, and A. De Luca and Editor A. Goldenberg upon evaluation of the reviewers comments. Y. Kanaama is with the Computer Science Department, Naval Postgraduate School, Montere, CA USA ( kanaama@cs.nps. nav.mil). G. Krahn is with the Department of Mathematical Sciences, United States Militar Academ, West Point, NY 199 USA ( ag9@usma. usma.edu). Publisher Item Identifier S 14-9X(98) (1) theor is as capable as the homogeneous theor, we can epect this theor to be an optimal one. The rotational information is contained in a matri in the homogeneous theor, however, in the proposed theor it is contained in one real number in the proposed theor. Since we do not use the homogeneous matri operations in this new theor, we call this new formulation heterogeneous. This paper reports four major results on the heterogeneous -D transformation theor: 1) a formulation that is as capable as the homogeneous transformation; ) this formulation requires less memor space and less computation time than the homogeneous formulation and it does not have the rotation matri consistenc problem; 3) the eplicit angle representation carries more information; 4) there are several applications of this theor in autonomous vehicle motion control. More detailed discussions are given in Section III. In the motion planning research, the concept of configuration, (; ; ) has been widel used [5]. It should be noted that this concept is distinct from that of the heterogeneous transformations. A configuration describes the static positioning of a rigid bod where (; ; ) and (; ; ) are equivalent. For a transformation of a rigid bod, however, there is a clear distinction between a no-rotation and a -rotation. In the 3-D transformations, to avoid its singular point, the quaternian algebra was introduced [], [7]. In the -D transformations, however, there is no singular point in both the homogeneous and heterogeneous formulations and we do not need an special consideration in this respect. The authors have implemented this heterogeneous transformation formulation as a part of the MML software sstem for the autonomous robot Yamabico-11 at the Naval Postgraduate School [8], [9]. The simple function set (the inverse and composition functions) solves numerous motion planning/analzing problems including the eamples described in Section IV. II. HETEROGENEOUS TRANSFORMATION THEORY Let < denote the set of all real numbers. A transformation, q, is defined b q where ; ; <. The set of all transformations is denoted b T. For eample, (; 1; ) T, (; 4; 4) T T (M T denotes the transposition of the matri M). A transformation q is interpreted as a -D coordinate transformation from one Cartesian coordinate sstem to another. Furthermore, q is interpreted as a composition of a translational transformation (; ) and a rotational transformation. Definition: The transformation group ht ; i consists of the set T of transformations, where T f(; ; ) T j; ; <g and the binar operator (composition function),, is defined as follows: Let q 1 ( 1 ; 1 ; 1 ) T, q ( ; ; ) T ht; i, then q 1 q () 1 + cos 1 sin sin 1 + cos 1 : (3) X/98$ IEEE

3 88 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 14, NO. 5, OCTOBER 1998 Fig. 1. Composition. Furthermore, we write q1 q if and onl if 1, 1, and 1. Notice that this relation is the most general equivalence relation in ht ; i. In this section we show that the transformation sstem ht ; i is an algebraic nonabelian group. The interpretation of q1 q in the domain of -D coordinate transformations is the composition of the coordinate transformations q1 and q (Fig. 1). For instance, if q1 8 5 then their compositions are q1 q p 7+3 p and q 4 and q q1 + p 3 + p : Thus, the composite function is not commutative. The following is an immediate result from the definition of ht ; i. Corollar 1: For an q (; ; ) T ht ; i q : (4) Therefore, a transformation (; ; ) T can alwas be decomposed into a translation (; ; ) T and a rotation (; ; ) T. Notice, however, that in general q (; ; ) T (; ; ) T. The order of two arguments of the composite function is important. The composition function satisfies the laws of closure, associativit, identit, and inverse as shown below. Lemma 1 Closure: For an q1; q ht; i, q1 q ht; i. Proof: Follows directl from the definition of the composition function given in (3). Lemma Associativit: For an q1; q; q3 ht ; i (q1 q) q3 q1 (q q3): (5) Proof: See () at the bottom of this page. (q1 q) q3 1 + cos 1 sin sin 1 + cos cos 1 sin cos(1 + ) 3 sin(1 + ) 1 + sin 1 + cos sin(1 + ) +3 cos(1 + ) ( + 3 cos 3 sin ) cos 1 ( + 3 sin + 3 cos ) sin 1 1 +( + 3 cos 3 sin ) sin 1 +( + 3 sin + 3 cos ) cos 1 1 +( + 3) cos 3 sin + 3 sin + 3 cos q1 (q q3) () 1 + 3

4 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 14, NO. 5, OCTOBER Let a transformation e ht; i be defined as e Lemma 3 Identit: For an q ht; i : (7) q e e q q: (8) Therefore, e (; ; ) T is the identit element in ht ; i. For an q (; ; ) T ht; i, q 1 is defined as q 1 cos sin sin cos : (9) Lemma 4 Inverse: For an q (; ; ) T ht; i, q 1 is the inverse element of q. Proof: Clearl, q 1 ht ; i. In addition, as shown in (9a) at the bottom of this page. Hence, q 1 is the inverse for each q ht; i. For eample, the inverse of a transformation (8; 5; ) T is p 1 8 : :48 p 5 4 :5 3 :33 : Proposition 1: The set, T, of transformations with the binar operation (composition function),, is a group denoted b ht ; i. Proof: The algebraic structure ht ; i satisfies the closure law b Lemma 1, the associative law b Lemma, the identit b Lemma 3, and the eistence of inverses b Lemma 4. It is worth summarizing some useful consequences of the four laws required in a group with the following well-known propositions [], [1]. Proposition : In an group G hg; 3i, where q; r; s G, we have the following: 1) Left cancellation law: If q 3 r q 3 s, then r s. ) Right cancellation law: If r 3 q s 3 q, then r s. 3) The identit element is unique. 4) The inverse of an element q is unique. 5) (q 3 r) 1 r 1 3 q 1. ) (q 1 ) 1 q. III. COMPARISONS BETWEEN TWO FORMULATIONS A. Homogeneous Two-Dimensional Transformations Before we compare the two transformation theories, let us briefl present the -D homogeneous transformation theor. The general form of a 3-D homogeneous transformation [1] is R 11 R 1 R 13 R 1 R R 3 R 31 R 3 R 33 z 1 (1) where the left-top 3 3 submatri represents rotation and the righttop 3 1 submatri a translation. Then its -D version becomes R 11 R 1 cos sin T R 1 R sin cos : (11) 1 1 Let H be the set of transformations of in this form, T, for some ; ; <. Notice that and +n represents the same homogeneous transformation T for an integer n, because each R ij is given b sin or cos. The composition of two transformations T; T His obtained through the standard matri multiplication [11] as shown in (1) at the bottom of this page. If, for some and, and R 11 R 1 R 1 R R 11 R 1 R1 R It is verified that and cos sin sin cos cos sin sin cos : (13) R 11 R 11 + R 1 R 1 R 1 R 1 + R R cos( + ) (14) (R 11 R 1 + R 1 R ) R 1 R 11 + R R 1 sin( + ) (15) + R 11 + R 1 + cos sin (1) + R 1 + R + sin + cos : (17) Therefore, H is closed under its composition operation. The inverse transformation of a homogeneous transformation T is also obtained as and q q 1 q 1 q +( cos sin ) cos ( sin cos ) sin +( cos sin ) sin +( sin cos ) cos cos sin + cos() sin() sin cos + sin() + cos() + (9a) T T R 11 R 1 R 1 R 1 R 11 R 1 R 1 R 1 R 11R 11 + R 1R 1 R 11R 1 + R 1R + R 11 + R 1 R 1 R11 + R R1 R 1 R1 + R R + R 1 + R 1 (1)

5 83 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 14, NO. 5, OCTOBER 1998 the inverse matri. Note that the determinant D of the homogeneous matri is alwas 1 T 1 R 11 R 1 R 1 R 1 1 R 11 R 1 R + R 1 R 1 R R 1 R 11 1 : (18) Thus, H is also closed under the inverse operation. Let f: T! H be a function that maps each heterogeneous transformation q (; ; ) T into a homogeneous transformation T Notice that f (q) cos sin sin cos 1 T H: (19) f (; ; ) f (; ; +n) () for an integer n, and hence, there is not a one-to-one correspondence between T and H. Furthermore, Proposition 3: For an heterogeneous transformations, q; q H, 1) f (q) f (q ) f (q q ). ) f (q 1 ) (f(q)) 1. 3) hh; i is a group. 4) hh; i is homomorphic to ht ; i, but is not isomorphic to it. This Proposition reveals that the homogeneous group is a subgroup of the heterogeneous group. B. Comparisons We observe several advantages of the heterogeneous transformation group ht ; i over the homogeneous transformation group hh; i, which are based on two facts: 1) the matri size difference; ) the use of raw rotation angle versus its trigonometric values. 1) Minimum Sizeness and Redundanc: To implement these theories, a heterogeneous transformation in T is represented b a 3 1 matri, while a homogeneous transformation theor needs a33matri. Although the homogeneous transformation uses a larger matri, it does not perform an additional functionalit above the heterogeneous theor. Thus, the larger matri is actuall contains less information and causes adversar effects. Obviousl the heterogeneous matri needs less memor space. The computation time for composition or inverse in the heterogeneous formulation is much less than that in the homogeneous theor. The redundant representation of rotation b a matri in the homogeneous transformation causes another problem. After a composition operation, the rotational matri [R ij ] i; j 1; is computed as polnomial functions of the two argument matri components (1), where these R-elements must satisf the trigonometric constraints, R11 + R1 1. However, because of the floating point arithmetic errors, this condition ma not alwas hold. This inconsistenc never occurs in the heterogeneous theor. ) Rotation Angles: There ma be a question on the use of the raw rotation angle (in the heterogeneous theor) rather than its trigonometric values, sin and cos (in the homogeneous theor). Our basic viewpoint is that the raw angle has more information than its trigonometric values (rotation of or 4 is eplicitl indicated). The following eamples illustrate this concept. Fig.. Counterclockwise and clockwise rotations. Eample 1: Fig. depicts two continuous motions of a vehicle (among other possible ones) from an initial positioning to a destination. The distance between the two points is a unit distance and the two orientations are opposite. Considering onl the first and last positionings, we represent the two motions b two heterogeneous transformations. One motion is represented as q a (1; ; ) T, because its net rotation is. The other one is represented b q b (1; ; ) T, because its net rotation is. Thus, these two distinct transformations are well represented b two distinct heterogeneous transformations. On the other hand, both heterogeneous transformations are mapped into the same homogeneous transformation 1 1 f (q a )f(q b ) 1 : 1 Therefore, differentiating a counterclockwise rotation from a clockwise rotation for a rigid bod is impossible in the homogeneous formulation. When we navigate a vehicle, whether the vehicle should turn left (counterclockwise) or right (clockwise) is the essential information. The ambiguit inherent in the homogeneous transformation is not appropriate in real navigation applications. Eample : Consider a heterogeneous transformation q c (; ; ) T, which includes a rotation of. If we appl this transformation two times, we obtain q c qc q c (1) which is a pure rotation of. Independent of and, it returns to the original point (, ) with the net rotation of (this fact itself is interesting). This result seems quite reasonable. The same transformation q c is translated into a homogeneous transformation cos sin M c f (q c ) sin cos 1 When we appl this transformation twice M c Therefore, the final result is an identit transformation without an rotational information. We interpret this result as a loss of important : :

6 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 14, NO. 5, OCTOBER Fig. 3. Positioning of rigid bodies as transformation. information. We have recognized the capabilit of the heterogeneous theor as more informative and natural. Eample 3: As discussed in the net section, the dead reckoning capabilit of a vehicle can be perfectl implemented b the heterogeneous transformation group theor. The vehicle s incremental motion is composed of the current transformations to obtain the net transformation (positioning). Therefore, b accumulating incremental angular changes, its value is not limited in the range of [; ] or [; ]. Actuall the value keeps the histor of rotations from the beginning of the world. This is implemented in the Yamabico robot control and we have seen this practice to be informative and useful. Angles ; ; ; 4; 111are considered distinct. This is related to the fact that the third component of the composite transformation in (3) is 1 + and can be algebraicall indefinitel large or small. If we use the homogeneous transformations for the dead reckoning task, the rotation angle information is partiall lost. In some applications, it does not matter. There are situations, however, where that loss is not allowable. IV. RIGID BODY MOTION ANALYSIS This heterogeneous transformation theor is a powerful tool to analze rigid bod motions in robotics. This section focuses on onl the results related to the concept of relative transformations. This is just one small portion of man useful applications of this theor. A. Relative Transformations When there is a robot vehicle in the global plane, we define a vehicle transformation q V ( V ; V ; V ) T () where a local Cartesian coordinate sstem is fied on the vehicle, and the origin s position ( V ; V ) and the direction V of its X-ais determines the transformation given in () (Fig. 3). This transformation completel represents the three degrees of freedom that a rigid bod possesses in a plane [5]. Suppose there eists another object or another vehicle in the plane. Each rigid bod object has its own transformation q ( ; ; ) T (3) in the global coordinate sstem. We compute the relative transformation q 3 ( 3 ; 3 ; 3 ) T (4) of the object with respect to the vehicle coordinate sstem. Note that there eists a relation q V q 3 q (5) : () between q V, q, and q 3. The eact values of the components of the relative transformation q 3 are obtained as follows: Proposition 4: If the transformation of a vehicle and an object are q V ( V ; V ; V ) T and q ( ; ; ) T, respectivel, the relative transformation q 3 of the object with respect to the vehicle is 3 q ( V )cos V +( V )sin V ( V ) sin V +( V ) cos V V Proof: Since b appling q V 1, we have q 3 q 1 V V V V q 1 q V q 3 q V cos V V sin V V sin V V cos V V : (7) ( V ) cos V +( V ) sin V ( V ) sin V +( V ) cos V V This result is useful in numerous situations, especiall when the vehicle is moving. The relative transformation q 3 provides the positional relations between the vehicle and the object. If 3 >, the object is in front of the vehicle and if 3 <, the object is in the rear of the vehicle. Similarl, if 3 >, the object is on the left

7 83 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 14, NO. 5, OCTOBER 1998 Fig. 5. Circular transformation. Fig. 4. Sequence of transformations. side of the vehicle and if 3 <, the object is on its right side. To determine the timing when the vehicle passes the object, the method of checking the sign of 3 is more informative than computing the Euclidean distance between the vehicle and the object, and finding its minimal point. B. Motion Description Let a transformation of a rigid bod robot be q (; ; ) T.If the bod moves, the transformation q(t) is a function of time t. One method to describe a rigid bod s motion over time is to specif a sequence of transformations [8], [9], [1]. In Fig. 4, a sequence (q ;q 1 ;q ;q 3 ;q 4 ;q 5 ;q ;q 7 ) (8) of eight transformations is shown. Instead of absolute transformations, we can use relative transformations. When the vehicle is at q i, instead of the net transformation q i+1, we can give a relative transformation r i1 that satisfies If q i and q i+1 are known, the equation q i r i+1 q i+1 : (9) r i+1 q i 1 q i+1 (3) allows us to solve for the unknown r i+1. From a vehicle pilot s viewpoint, specifing the net motion with a relative transformation with respect to the current positioning is more useful than specifing an equivalent global transformation. C. Trajector Generation The relative transformation concept can be applied to generate smooth trajectories or motions. The vehicle s positioning is defined b a transformation q (; ; ) T. When the vehicle is moving, q is a function of time q(t) ((t); (t); (t)) T : (31) If we know the incremental motion 1q (a relative transformation) in the last sampling-time interval, we can compose its current transformation q with 1q to obtain the net transformation q q q 1q: (3) Fig.. Cornu spiral (clothoid curve). To implement the odometr (or dead-reckoning) capacit for autonomous vehicles, we must evaluate 1q at ever sampling interval 1T (the same principle applies for drawing a curve whose curvature is known). The question is how to evaluate the incremental motion (incremental transformation) 1q. Assume that we can measure the traveling distance of the vehicle s reference point and the direction change of the vehicle in 1T. (For instance, on a differential-drive vehicle, and are obtained b the incremental traveling distances, 1l and 1r, of the left and right wheels.) Since an incremental transformation 1q has three degrees of freedom, precisel speaking, it is not possible to determine 1q from onl and. However, if we assume that the curvature along this short curve segment is constant (and hence, the curve is a circular arc), we can compute the eact value of 1q. The method to evaluate 1q 1q(; ) is shown below. In Fig. 5, the

8 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 14, NO. 5, OCTOBER Fig. 7. Cubic spiral. circular arc is placed with its beginning at the origin, since we are evaluating the relative incremental transformation 1q. Assume that as shown in Fig. 5. The radius r of the circular arc is r : (33) Therefore 1q(; ) 1 1 r sin r(1 cos ) sin 1 cos : (34) On the other hand, if, the circular arc becomes a straight line segment and combining these two cases if sin 1q(; ) (35) 1 cos if : Therefore, 1q(; ) is represented b (35). However, if we know empiricall jj is small, the Talor epansion unifies the two equations. Note that sin 3 5 3! + 5! ! (3) 5! and 1 cos 1 (1! + 4 4!! + 111) 1! 4! : (37)! With these formulas, we can rewrite (35) as follows: 1 3! + 4 5! 111 1q(; ) 1! 4! + 4 : (38) 111! Approimating a small curved motion b a circular arc was first proposed b Wang [13]. However, the method shown here (defining 1q and composing it to the vehicle transformation) is theoreticall more transparent and easier to implement. The effectiveness of this curve generation method is demonstrated with two eample curves in Figs. and 7. The first one is a cornu spiral (clothoid) and the second a cubic spiral. Their curvatures are a linear and a quadratic function of arc length s, respectivel (s) As (s) A s 4 s where A is a nonzero constant. Therefore, the cornu spiral has one inflection point and the cubic spiral has two inflection points. Both curve classes are useful for motion planning for autonomous vehicles [8], [14]. V. CONCLUSION We have proved that the set of a new -D heterogeneous transformations forms a group. This group theor provides an elegant algebraic structure that allow robot motion analsis to be more transparent. Furthermore, it has been demonstrated that this new formulation is superior to the conventional homogeneous transformation formulation, because of its small size, no redundanc, and eplicit angle representation. This theor can be applied to analze problems in several aspects on robotics. Onl a few eample areas are given in this paper: problems related to relative transformations and discrete generation of trajectories. The authors have successfull applied these algorithms in constructing the software sstem for the autonomous mobile robot Yamabico-11 at the Naval Postgraduate School. Furthermore, onl the composition and inverse functions were needed to implement. REFERENCES [1] R. P. Paul, Robot Manipulators: Mathematics, Programming, and Control. Cambridge, MA: MIT Press, [] N. Bloch, Abstract Algebra with Applications. Englewood Cliffs, NJ: Prentice-Hall, [3] J. D. Fole and A. Van Dan, Fundamentals of Interactive Computer Graphics. Reading, MA: Addison-Wesle, [4] Y. Kanaama, D. L. MacPherson, and G. W. Krahn, Two dimensional transformations and its application to vehicle motion control and analsis, in Proc. Int. Conf. Robot. Automat., Atlanta, GA, Ma 7, 1993, pp [5] T. Lozano-Perez, Spatial planning: A configuration space approach, IEEE Trans. Comput., vol. C-3, pp , Feb [] K. W. Spring, Euler parameters and the use of quaternion algebra in the manipulation of finite rotations: A review, Mech. Mach. Theor, vol. 1, no. 5, pp , 198.

9 834 IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. 14, NO. 5, OCTOBER 1998 [7] J. Funda, R. H. Talor, and R. P. Paul, On homogeneous transforms, quaternions, and computational efficienc, IEEE Trans. Robot. Automat., vol., pp , June 199. [8] Y. Kanaama and B. I. Hartman, Smooth local path planning for autonomous vehicles, in Proc. IEEE Int. Conf. Robot. Automat., 1989, pp [9] Y. Kanaama and M. Onishi, Locomotion functions in the mobile robot language, MML, in Proc. IEEE Int. Conf. Robot. Automat., 1991, pp [1] I. N. Herstein, Topics in Algebra, nd ed. New York: Wile, [11] G. Strang, Linear Algebra and Its Applications, 3rd ed. New York: HBJ College, 198. [1] Y. Kanaama, Two dimensional wheeled vehicle kinematics, in Proc. Int. Conf. Robot. Automat., San Diego, CA, Ma 8 13, 1994, pp [13] C. M. Wang, Location estimation and uncertaint analsis for mobile robots, in Proc. Int. Conf. Robot. Automat., Philadelphia, PA, Apr. 4 9, 1988, pp [14] Y. Kanaama and N. Miake, Trajector generation for mobile robots, in Robotics Research. Cambridge, MA: MIT Press, 198, vol. 3, pp Quadratic Normal Forms of Redundant Robot Kinematics with Application to Singularit Avoidance Krzsztof Tchoń Abstract We derive a rank condition under which the redundant robot kinematics around a corank 1 singular configuration can be given a quadratic normal form. This normal form is further eploited to introduce new sufficient conditions for local avoidabilit and unavoidabilit of singular configurations. Inde Terms Kinematics, redundant robotic manipulator, normal form, singular configuration, singularit avoidance. I. INTRODUCTION The problem of characterizing the behavior of robotic manipulators in a neighborhood of singular configurations affects significantl both trajector planning and control of robots. To cope with this problem we have initiated in [1] a normal form approach rooted in singularit theor. Benefits of the normal form approach lie in providing simple mathematical models of kinematics around singular configurations, called normal forms, that are locall equivalent to the original kinematics. Until now the normal form approach has delivered a fairl complete collection of mathematical models of singularities of nonredundant kinematics [], [3], [4] and proved to be capable of providing a new solution to the singular inverse kinematic problem [5] as well as to the problem of singularit avoidance in robot kinematics that have the redundanc degree 1 []. Manuscript received Januar 7, 1997; revised Ma 9, This research has been done in part within a research project Singular and nonholonomic robots: mathematical models, control algorithms and trajector planning, supported b the Polish State Committee of Scientific Research. This paper was recommended for publication b Associate Editor Y. F. Zheng and Editor A. Goldenberg upon evaluation of the reviewers comments. The author is with the Institute of Engineering Cbernetics, Wroc law Universit of Technolog, Wroc law 5-37, Poland ( tchon@ict.pwr. wroc.pl). Publisher Item Identifier S 14-9X(98) In this article, we shall derive a rank condition under which the redundant robot kinematics around a corank 1 singular configuration can be transformed to a quadratic normal form, generalizing the condition established in [] for the nonredundant case. Furthermore, as a peculiar b-product of the normal form approach we shall present new sufficient conditions for avoidabilit and unavoidabilit that etend results obtained in [] to the kinematics with arbitrar degree of redundanc. The idea of using normal forms of kinematics to solve the singularit avoidance problem comes from [] and seems to be more powerful than the dnamical sstem approach developed in [], [7], and [8]. Recentl, sufficient local avoidabilit conditions, conceptuall close to ours, have been obtained in [9]. There are two ke concepts underling this article: a normal form and a local singularit avoidance, that will be defined below. Let k() denote a coordinate representation of the kinematics of a robotic manipulator. We epose the kinematics to jointspace and taskspace coordinate changes! '();! () transforming the kinematics to the form k () k ' 1 (): The simplest map k () that can be produced in this wa is called the normal form of k, while the relation between k and an k defined above is referred to as RL-equivalence [1]. The local singularit avoidance will be comprehended in the following wa. A singular configuration is defined as locall avoidable if there eists in a neighborhood of a nonsingular configuration such that k 1 k(). Similarl, will be called locall unavoidable, if in a certain neighborhood of there are no nonsingular configurations verifing k 1 k(). The concepts of local avoidabilit/unavoidabilit are applicable if one needs to avoid a singular configuration b a small modification of the robot s trajector. Clearl, a singular configuration that cannot be avoided locall ma still appear to be globall avoidable. In this article, the avoidabilit/unavoidabilit will alwas be meant as local. This article is composed as follows. Section II presents main results concerned with the quadratic normal form and the singularit avoidance. These results are eemplified in Section III. Section IV concludes the article. II. MAIN RESULTS Let k : R n! R m ; k() (k 1();k (); 111;k m()) represent the kinematics of an unlimited robotic manipulator with respect to certain coordinate sstems in the joint and in the task manifolds. The map k is analtic. A vector ( 1; ; 111; n) R n denotes positions of the joints, called manipulator s configurations. A vector ( 1 ; ; 111; m ) R m stands for the position and orientation of the end-effector. We shall alwas assume that n m. The difference nm will be referred to as the redundanc degree of the kinematics. Let us recall that a configuration R n is called regular if the Jacobian matri of the kinematics () has maimum rank at, i.e., rank J() m () otherwise the configuration is singular. Singular configurations can be distinguished b their coranks defined as corank() l, rank J() m l: (3) 14 9X/98$ IEEE