Inverse Kinematics of Functionally-Redundant Serial Manipulators under Joint Limits and Singularity Avoidance

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1 Inverse Kinematics of Functionally-Redundant Serial Manipulators under Joint Limits and Singularity Avoidance Liguo Huo and Luc Baron Department of Mechanical Engineering École Polytechnique de Montréal P.O. 79, station Centre-Ville Montréal, Québec, Canada, HC A7 Abstract Kinematic redundancy of a pair of manipulatortask can be classified into intrinsic and functional redundancies. This paper focuses on the functional-redundancy resolution. Instead of working on the null-space of the Jacobian matrix, the instantaneous twist is rather decomposed into two orthogonal subspaces of the tool frame. The twist-decomposition approach and the augmented approach which projecting a secondary task onto the null space of the augmented Jacobian is compared numerically. The performance criterion function considering the both of joint-limits and singularity avoidances are proposed and tested too. The numerical simulations of an arc-welding operation shown that the twist-decomposition approach is an effective inverse kinematics resolution method on the functionally redundant manipulator. Index Terms Robotics, kinematics, redundancy, singularity, joint-limits Fig.. Graphic simulator of the arc-welding operation with a PUMA. I. INTRODUCTION The inverse kinematics of serial robotic manipulators has been in-depth studied for decades at both displacement and velocity levels. It is well known that the former is highly nonlinear and deserve dedicated solution procedures, while the latter is linear and requires the inversion of the Jacobian matrix of the manipulator. For a general -degrees-of-freedom (DOF) task, many research works have reported algorithms (e.g., [,,,,,, 7, 8]) allowing to choose an optimal solution when the manipulator has more joints than the corresponding DOF of its end-effector (EE). All of them use the Gradient Projection Method (GPM) to track the desired EE path, i.e., the main task, and solve a trajectory optimization problem, i.e., the secondary task. The latter is related to a performance criterion. The gradient of this function is projected onto the null space of the Jacobian matrix, namely J, in order to choose the best solution among the infinitely many that exist. Hence, the secondary task is performed under the constraint that the main task is realized. The concept of kinematic redundancy is also related to the definition of the task instead of being only an intrinsic property of the manipulator. Although this fact is still not well understood in practice, it has been recognized by several researchers. Samson [9] clearly presented that the redundancy depends on the task and may change with time. Siciliano [] said that the manipulator can be functionally redundant when only a number of components of its operational space are concerned with a specific task, even if the dimension of operational and joint spaces are equal. Although Siciliano proposed the concept of functional redundancy, he didn t develop the corresponding solution procedure. In the case of functional redundancy, J are often full rank square matrices, i.e., its null space doesn t exist, thus the general projection method working on the null space of J can not be applied here. In fact, many industrial tasks such as arc-welding, milling, deburing, laser-cutting, gluing and many other tasks, require less than six-dof, because of the presence of a symmetry axis or plane. For example, the general arc-welding task requires -DOF for the displacement of the end-point of the electrode, but requires only -DOF for its orientation. The rotation of the welding-tool around the electrode axis is clearly irrelevant to the view of the task to be accomplished, the so-called functional redundancy. Baron [8] proposed to add a virtual joint to the manipulator so that a column is added to J, rendering it underdetermined. This augmented approach in solving functional redundancy suffers from the potential ill-conditioning of the augmented J, and the additional computation cost required to solve this augmented J. Alternatively, Huo and Baron [] proposed the

2 twist decomposition algorithms (TWA) to solve the functional redundancy. Instead of using the projection onto the null space of J, the TWA is based on the orthogonal decomposition of the required twist into two orthogonal subspaces in the instantaneous tool frame. The TWA has great difference with the task-decomposition approach proposed by Nakamura [, ] on the theoretical base, although both of them consider the order of task priority. First, the TWA classifies the order of task priority in instantaneous tool frame instead of in the robot base frame. Second, the TWA is directly developed from the minimum-norm solution without considering the projection onto the null space of J, while the development of the taskdecomposition approach is based on the general equation using the null space of J. As some researchers [] have realized that the success of the GPM relies on the evaluation of the performance criterion on the joints position. For example, in the case of avoidance of the joint-limits [8, ], the performance criterion can be written as to maintain the manipulator as close as possible to the mid-joint position, i.e., z = W( ) with and W being defined as min, () ( max+ min ), W Diag(/( max min )). () However, if the task requires not only the avoidance of the joint-limits but also keeping the configuration as far as possible from singularities, then we need a different performance criterion relating to the joint-limits and singularities. Here, the Manipulability Measure (MM) proposed by Yoshikaw [] and condition number (Cond) of J are used together to generate a new performance index, named here Manipulability Parameter (MP). In this paper, the solutions for the two kinds of redundancy, i.e., intrinsic and functional redundancy, are reviewed, particularly the TWA for the functional redundancy. In order to avoid not only the joint limits but also the singularities at the same time, a new performance criterion is developed and applied. Finally, a numerical example with application to arc-welding is demonstrated. The numerical comparison between TWA with augmented approach and two different performance criteria are included in the numerical demonstration. II. BACKGROUND ON KINEMATIC REDUNDANCY Before formulating the problem, let us briefly recall the two basic sources of kinematic redundancy of a robotic manipulator with respect to a given task. A. Basic Definitions Let J denote, the joint space of a robotic manipulator having n+ rigid bodies serially connected by n joints, either revolute R or prismatic P. The posture of the manipulator in J is given by the n-dimensional vector, namely, and hence, n = dim(j ) = dim(). Moreover, let O denote, the operational space of the EE of the robotic manipulator resulting from the joint space J. Since any free-moving rigid body in space can have at most six degrees-of-freedom (DOFs), the dimension of O is also at most six, and hence, o = dim(o). Furthermore, let T denote, the task space such as required by the functional mobility of the EE, independently of the manipulator s architecture and hence, t = dim(t ). Now, let us recall the following three definitions: Definition : Intrinsic redundancy A serial manipulator is said to be intrinsically redundant when the dimension of the joint space J, denoted by n = dim(j ), is greater than the dimension of the resulting operational space O of the EE, denoted by o = dim(o), i.e., when n > o. The degree of intrinsic redundancy of a serial manipulator, namely r I, is computed as r I = n o. () Definition : Functional redundancy A pair of serial manipulator-task is said to be functionally redundant when the dimension of the operational space O of the EE, denoted by o = dim(o), is greater than the dimension of the task space T of the EE, denoted by t = dim(t ), while the task space being totally included into the operation space of the manipulator, i.e., T O, and hence, o > t. The degree of functional redundancy of a pair of serial manipulator-task, namely r F, is computed as r F = o t. () Definition : Kinematic redundancy A pair of serial manipulator-task is said to be kinematically redundant when the dimension of the joint space J, denoted by n = dim(j ), is greater than the dimension of the task space T of the EE, denoted by t = dim(t ), while the task space being totally included into the resulting operation space of the manipulator, i.e., T O, and hence, n > t. The degree of kinematic redundancy of a pair of serial manipulator-task, namely r K, is computed as r K = n t. () Upon substitution of eqs.() and () into eq.(), it becomes apparent that the kinematic redundancy come from two sources, the intrinsic and functional redundancies, i.e., r K = r I + r F. () In the literature, most of the research working on redundancyresolution of serial manipulators suppose that r F =, and thus, study r K = r I. In this paper, the opposite case is studied, i.e., supposing r I =, and thus, study r K = r F. B. Problem Formulation The inverse kinematics of serial manipulators is usually based on the linear relationship between the EE velocity, called twist and denoted t, and the joint velocities, denoted, given by t = J, (7) This definition of functional redundancy of serial manipulator-task is expandable to other types of manipulators such as parallel and hybrid ones.

3 with t and defined as t [ω T ṗ T ] T R, and the Jacobian matrix J being defined as [ n ] T R n, (8) J [ A B ] T, A, B R n, (9) where ω R is the angular velocity vector of the EE, ṗ R is the translation velocity vector of a point on the EE, while i is the velocity of joint i. It is noteworthy that t is not defined as a vector of R, but rather as a set of two vectors of R casted into a column array, and hence, t R R. This distinction is important and will be further used in section. For the sake of numerical computation, we replace velocities by small displacements in eq.(7), i.e., t t and. When J is square: The resolved motion-rate method proposed by Whitney [] can be used to compute as = J t, () where it is apparent that J must be non-singular. When J has more rows than columns: The mobility of the EE is less than -DOF, because the manipulator has either less than six joints and/or these joints do not allow to produce the full EE mobility. In both cases, the can be computed from t as = J t, () where J is the left generalized inverse of J such that J (J T J) J T. () This right generalized inverse is usually not required, because there always exist a time-invariant frame in which the constrained mobility of the EE are rows of zeros in J. Thus, the user can directly eliminate these rows from J in order to render it square and full rank. For example, planar and spherical serial manipulators belong to this case. When J has less rows than columns: For intrinsically-redundant serial manipulators, J always has more columns than rows, and hence, eq.(7) becomes an underdetermined linear algebraic system having infinitely many solutions. In this case, Liégeois[] proposed to compute as = (J ) t + ( J J)h, () minimum norm solution homogeneous solution where J is defined as the right generalized inverse of J such that J J T (JJ T ), () and h is an arbitrary vector of J allowing to satisfy a secondary task. The first term in the right-hand side (RHS) of eq.() is known as the minimum-norm solution of eq.(7), i.e., the M that minimizes among all the that are solutions of eq.(7). The second part of the RHS of eq.() is known as the homogeneous solution of eq.(7), i.e., the H that produce t =, i.e., no displacement of the EE. This joint displacement H is also known as the self-motion of the manipulator. It is symbolically computed as the projection of an arbitrary vector h onto the null space of J with the orthogonal projector ( J J). Equation () is used to solve the kinematic inversion of intrinsically-redundant manipulators by many researchers (e.g., [, 7]), including those who payed a special attention in avoiding the squaring of the condition number while solving eq.(). Sometimes, it exists a time-invariant frame in which the functional redundancy are rows of J. In this case, the eq.() can be used to solve the inverse kinematics with these rows previously eliminated from J. For example, an arc-welding task may require the -D positioning of the end-point of the welding electrode at a constant vertical orientation. In this case, the redundant task is the orientation around the electrode axis, i.e., the vertical axis, which is time-invariant in the base frame. Most of the times, it is not possible to find such a time-invariant frame to express the functional redundancy of a task. For the general arc-welding, the electrode axis needs to undergo arbitrary orientations in space, and hence, the electrode axis is time-varying in base frame. Therefore, eq.() can not be used. Below, a twist-decomposition approach to solve this problem is compared with the augmented approach. III. TWIST-DECOMPOSITION APPROACH Before proposing a functional-redundancy resolution algorithm, let us briefly review the orthogonal decomposition of vectors and twists. A. Orthogonal-Decomposition of Vectors Decomposing any vector ( ) of R into two orthogonal parts, [ ] M, the component lying on the subspace, M, and [ ] M, the component lying in the orthogonal subspace, M, using the projector M and an orthogonal complement of M, namely M, as follows: ( ) = [ ] M + [ ] M = M( ) + M ( ). () It is apparent from eq.(), that M and M are related by M + M = and MM = O, where and O are the identity and zero matrices, respectively. The orthogonal complement of M thus defined, M, is therefore unique, and hence, both M and M are projectors projecting vectors of R onto the subspace M and the orthogonal subspace M respectively. These projectors are given for the four possible dimensions i of subspaces of R as: M i = P L O, M i = O L P i = -D task i = -D task i = -D task i = -D task, () where the plane and line projectors, P and L, respectively, are defined as: P L, L ee T, (7) in which e is a unit vector along the line L and normal to the plane P. The null-projector O is the zero matrix that projects any vector of R onto the null-subspace O, while the identity-projector is the identity matrix that projects any vector of R onto itself.

4 B. Orthogonal-Decomposition of Twists Any twist array ( ) of R can also be decomposed into two orthogonal parts, [ ] T, the component lying on the task subspace, T, and [ ] T, the component lying on the orthogonal task subspace (also designated as the redundant subspace), T, using the twist projector T and an orthogonal complement of T, namely T, as follows: ( ) = [ ] T + [ ] T = T( ) + T ( ) (8) It is apparent from eq.(8), that T and T are projectors of twists that must verify all the properties of projectors. However, twists are not vectors of R, and hence, projectors of twists cannot be defined as in eqs.() and (7), e.g., T tt T, T tt T, (9) but must rather be defined as block diagonal matrices of projectors of R, i.e., [ ] Mω O T, () O M v [ ] T Mω O T =, () O M v where M ω and M v are projectors of R defined in eqs.() and (7) which allow the projection of the angular and translational velocity vectors, respectively. It is noteworthy that the matrices of eq.(9) do not verify the properties of projectors, and hence, can not be used for orthogonal decomposition. Finally, eq.(8) becomes t = t T + t T = Tt + ( T)t. () C. Functional-Redundancy Resolution For functionally-redundant serial manipulators, it is possible to decompose the twist of the EE into two orthogonal parts, one lying into task subspace and another one lying into the redundant subspace. Substituting eq.() into eq.() yields = (J T) t + J ( T)Jh, () task displacement redundant displacement where h is an arbitrary vector of J allowing to satisfy a secondary task. Vector h is often chosen as the gradient of a performance criterion function to minimize. The first part of the RHS of eq.() is the joint displacement required by the task, while the second part is the joint displacement in the redundant subspace (or irrelevant to the task). Clearly, eq.() does not require the projection onto the nullspace of J as most of the redundancy-resolution algorithms do, but rather requires an orthogonal projection based on the instantaneous geometry of the task to be accomplished. As shown in Algorithm I, eq.() is used within a resolved-motion rate method. At lines -, the joint position and the desired EE pose {p d, Q d } are first initialized, then the actual EE pose {p, Q} is computed with the direct kinematic model DK(). At lines -, an EE displacement t is computed from the difference between the desired and actual EE poses. The vect( ) at line is the function transforming a rotation matrix into an axial vector as defined in [7] (page ). At lines 7-8, the instantaneous orthogonal twist projector T is computed from DK(). At line 9, the orthogonal decomposition method is used to compute the corresponding joint displacement. Finally, the algorithm is stop whenever the norm of is smaller than a certain threshold ɛ. Algorithm I: Twist-Decomposition Algorithm initial joint position; {p d, Q d } desired EE pose; {p, Q} DK() Q Q T Q d p [ p d p ] Qvect( Q) t p 7 DK() [ e M] ω, f M v, J Mω O 8 T, O M v 9 J T t + J ( T)Jh if < ɛ then stop; else +, and go to. IV. PERFORMANCE CRITERION As presented by Yoshikawa [8], the manipulability ellipsoid represents an ability of manipulation, i.e., the EE can move at high speed in the direction of the major axis of the ellipsoid, while only with lower speed in the direction of the minor axes. If the ellipsoid is almost a sphere, the EE can move in all directions uniformly. On the other hand, the larger the volume of the ellipsoid is, the faster the EE can move. One of the representative measures for the ability of manipulation derived from the manipulability ellipsoid is the volume of the ellipsoid. This is given by c m w, where the coefficient c m is a constant when m is fixed, so the volume is proportional to the ω. Hence, we can regard ω as a representative measure, called the manipulability measure of the manipulator at configuration. The manipulability measure ω has the following properties: ω = det(jj T ) = σ σ σ m. () The scalars σ, σ, σ m in eq.() are called singular values of J, and they are equal to the m larger values of the n roots { λ i, i =,,..., n}, where λ i (i =,,..., n) are eigenvalues of the matrix J T J. Further, the singular-value decomposition of J is J = UΣV T, () where U and V are, respectively, m m and n n orthogonal matrices, and Σ is an m n diagonal matrix. Thus, if let the u i be the ith column vector of U, the principle axes of the manipulability ellipsoid are σ u, σ u,..., σ m u m. Another index that might be induced from the manipulability ellipsoid is the ratio of the minimum and maximum radius of the ellipsoid, i.e., ω = σ σ m. ()

5 ω is the condition number of the Jacobian matrix J, and is independent of the ellipsoid s size. ω is an index of the directional uniformly of the ellipsoid. The closer to unity this index is, the more spherical the ellipsoid is. Here, the ω and ω are combined together to generate another index ω mp as ω ω mp = kω = kσ σ σm, (7) where k is an arbitrarily selected constant factor, such as in the section V-C. The index ω mp is named as the Manipulability Parameter (MP). ω mp can represent the volume and the shape of the manipulability ellipsoid at the same time. The smaller the ω mp is, the more spherical the ellipsoid is or the larger the ellipsoid s volume is, the faster the EE can move in the direction of the minor axis of the ellipsoid. Thus, ω mp can be used to detect the singularity configuration, at the instant the EE lost the velocity in the direction of the minor axis of the ellipsoid. Thus, the performance criterion, which is maintaining the manipulator as close as possible to the mid-joint position and as far as possible to the singularities, could be written as z = W( )ω mp. (8) Vector h is thus chosen as minus the gradient of z, i.e., h = z. (9) (rad.) joint i a i b i α i.. π/ π/.. π/.. π/. unit rad. m m rad. TABLE I DH PARAMETERS OF THE PUMA st turn nd turn 7 Fig.. Joint positions of two consecutive turns of trajectory Λ as computed by the resolved-motion rate method (without considering the functional redundancy). A. Task Description V. NUMERICAL EXAMPLE When performing arc-welding operations, the electrode of the welding tool has an axis of symmetry around which the welding tool may be rotated without interfering with the task to be performed. This axis describes the geometry of the functional redundancy (or the redundant subspace of twists). The unit vector e denote the orientation of the axis of symmetry along the electrode. The projection of ω along e is the irrelevant component of ω, while its projection onto the plane normal to e is the relevant component of ω. For a general arc-welding task around the electrode axis e, the twist projector is defined as T weld [ ( ee T ) ] [, T ee T weld Now, substituting eq.() into eq.() yields = J T weld t + J [ ee T Ah ], () ], () where A is the upper part of J as defined in eq.(8). Equation () can be used as line 9 of Algorithm I in order to solve the inverse kinematics of serial manipulators while performing a general arc-welding task. A PUMA serial manipulator is used to perform a pipeto-bride welding task. Its Denavit-Hartenberg parameters are described in Table I. The EE must perform consecutively the trajectory Λ (T = 8 sec.), i.e.,. cos(ωt) p =. +. sin(ωt).9, () Q = cos α sin α cos β sin α sin β sin α cos α cos β cos α cos β sin β cos β, () with α = π + ωt, β = π, ω = π T, t T, where distances and angles are expressed in meter and radians, respectively. The orientation of the electrode axis, namely e, can be computed as e = Q Q Q k, k [ ] T. () Figure shows the joint positions to perform twice the trajectory Λ as computed by the resolved-motion rate method without considering the functional redundancy, i.e., using eq.() at line 9 of Algorithm I. Apparently, without taking advantage of the axis of symmetry of the electrode, the manipulator is able to perform the first turn of the trajectory Λ while the second consecutive turn is not possible without exceeding the joint limits. Thus, the joint-space trajectories need to be optimized by taking advantage of the functionally redundant DOF.

6 7 7 st turn nd turn appear only at the first turn, because of the arbitrarily chosen initial conditions, and not thereafter. rad.) (rad.) st turn (a) nd turn C. Joint-limits and Singularity Avoidance As the joint-limits and singularity avoidance are considered together as the secondary task, the h is written as h = W( )ωmp, () According to the comparison in section V-B, the TWA is able to minimize the joint-motion range and increase the accuracy than the augmented approach. Hence, only TWA is applied to the case of the joint-limits and singularity avoidance. With the substitution of eq.() into eq.(), the reached joint space trajectories are shown in Figure. Clearly, it is almost the same as Figure (b), i.e., the joint-limits avoidance can also be achieved with the performance index function of eq.(). The manipulability indices reached with or without the singularity avoidance are shown in Figure (a, b, c), and are compared in Table II. method Max. ω Min. ω Max. ωmp Without singu. avoid With singu. avoid Perc..%.% % TABLE II MANIPULABILITY INDICES WITH OR WITHOUT THE (b) g.. Joint position with respect to time for the augmented approach (a) d TWA(b) with the joint-limits avoidant secondary task. SINGULARITY AVOIDANCE, WHERE ω IS CONDITION NUMBER OF J, ω IS THE MANIPULABILITY MEASURE, ωmp IS THE MANIPULABILITY PARAMETER, AND with singu. avoid. Perc. = without singu. avoid. Two different secondary tasks are discussed in here, i.e., ly joint-limits avoidance, and joint-limits and singularity oidances together. st turn nd turn joint-limits Avoidance Firstly, the secondary task h is chosen to only avoid the int-limits such that: (rad.)

7 7 change. The ω mp is decreased from.87 to 8.88, i.e., the EE can move faster in the direction of the minor axis of the manipulability ellipsoid. VI. CONCLUSIONS Two different approaches have been compared to solve the functional redundancy of a serial robotic manipulator, while performing an arc-welding operation. A performance index function related to joint-limits and singularity avoidance is developed and tested to be effective. Our numerical implementation shows that the twist-decomposition approach is more effective than the non-redundant or augmented approaches. ACKNOWLEDGMENTS The financial support from the Natural Sciences and Engineering Research Council of Canada under grant OGPIN- 8 and the research contract CDT-P7 are gracefully acknowledged. REFERENCES [] Liegeois, A., Automatic supervisory control of the configuration and behavior of multibody mechanisms, IEEE Trans. on System, Man, Cybern., vol. SMC-7, no., pp., 977. [] Siciliano, B., Solving manipulator redundancy with the augmented task space method using the constraint Jacobian transpose, IEEE Int. Conf. on Robotics and Automation, vol., pp. 8, 99. [] Yoshikawa, T., Basic optimization methods of redundant manipulators, Lab. Robotics and Aut., vol. 8, No., pp. 9, 99. [] Whitney, D.E., The mathematics of coordinated control of prosthetic arms and manipulator, ASME J. Dynamics Syst., Measur. and Cont., vol. 9, no., pp. 9, 97. [] Klein, C.A. and Huang, C.-H., Review of pseudoinverse control for use with kinematically redundant manipulators, IEEE Trans. on Systems, Man, and Cyb., vol. SMC-, no., pp., 98. [] Angeles, J., Habib, M. and Lopez-Cajun, C.S., Efficient Algorithms for the Kinematic Inversion of redundant Robot Manipulators, Int. J. of Robotics and Automation, vol., no., pp., 987. [7] Arenson, N., Angles, J. and Slutski, L., Redundancy-resolution algorithms for isotropic robots, Advances in Robot Kinematics: Analysis and Control, pp., Salzburg, 998. [8] Baron, L., A joint-limits avoidance strategy for arc-welding robots, Int. Conf. on Integrated Design and Manufacturing in Mech. Eng., Montreal, Canada, May -9,. [9] Samson, C., Le Borgne, M. and Espiau, B., Robot control: the task function approach, Oxford [Angleterre]: Clarendon Press, pages, 99. [] Sciavicco, L. and Siciliano, B., Modelling and control of robot manipulators, Springer, London, 77 pages,. [] Huo, L., and Baron, L., Kinematic inversion of functionally-redundant serial manipulators: application to arc-welding, Trans. of the Canadian Society for Mech. Eng.,. [] Nakamura, Y., Hanafusa, H. and Yoshikawa, T., Task-priority based redundancy control of robot manipulators, Int. J. Robotics Research, Vol., No., 987. [] Nakamura, Y., Advanced robotics redundancy and optimization, Addison-Wesley Pub. Co., Massachusetts, 7 pages, 99. [] Chaumette, F., Marchand, É., A redundancy-based iterative approach for avoiding joint limits: application to visual servoing, IEEE Trans. on robotics and automation, vol. 7, no.,. [] Yoshikawa, T., Analyais and control of robot manipulators with redundancy, Robotics Research: The First International Symposium, pp. 7 77, 98. [] Whitney, D.E., Resolved motion rate control of manipulators and human prostheses, IEEE Trans. Man-Machine Syst., vol., no., pp. 7, 99. [7] Angeles, J., Fundamental of Robotic Mechanical Systems: Theory, Methods, and Algorithms, second edition, Springer-Verlag, New York, pages,. [8] Yoshikawa, T., Foundations of robotics: analysis and control, The MIT press, 8 pages, ω ω ω mp (a) (b) (c) Fig.. The manipulability indices with or without the singularity avoidant secondary task in the period of closing to the singularity. Where denotes the case without singularity avoidance; denotes the case with singularity avoidance. (a) Condition number ω ; (b) Manipulability measure ω; (c) Manipulability parameter ω mp.