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Inverse Kinematics of Functionally-Redundant Serial Manipulators under Joint Limits and Singularity Avoidance Liguo Huo and Luc Baron Department of Mechanical Engineering ´ Ecole Polytechnique de Montr´eal P.O. 6079, station Centre-Ville Montr´eal, Qu´ebec, Canada, H3C 3A7
[email protected],
[email protected]
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. 1.
Graphic simulator of the arc-welding operation with a PUMA500.
I. I NTRODUCTION 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 6-degrees-of-freedom (DOF) task, many research works have reported algorithms (e.g., [1, 2, 3, 4, 5, 6, 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
[10] 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 3-DOF for the displacement of the end-point of the electrode, but requires only 2-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 [11] proposed the
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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 [12, 13] 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 [14] 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, 11], the performance criterion can be written as to maintain the manipulator as close as possible to ¯ i.e., the mid-joint position θ, z=
1 ¯ → min , W(θ − θ) θ 2
(1)
¯ and W being defined as with θ ¯ ≡ 1 (θ max +θ min ), W ≡ Diag(1/(θ max −θ min )). (2) θ 2 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 [15] 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 K INEMATIC R EDUNDANCY 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 + 1 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 θ, and hence, n = dim(J ) = dim(θ). 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) ≤ 6. 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 ) ≤ 6. Now, let us recall the following three definitions: Definition 1: 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) ≤ 6, i.e., when n > o. The degree of intrinsic redundancy of a serial manipulator, namely rI , is computed as rI = n − o.
(3)
Definition 2: Functional redundancy1 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) ≤ 6, is greater than the dimension of the task space T of the EE, denoted by t = dim(T ) ≤ 6, 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 rF , is computed as rF = o − t.
(4)
Definition 3: 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 ) ≤ 6, 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 rK , is computed as rK = n − t.
(5)
Upon substitution of eqs.(3) and (4) into eq.(5), it becomes apparent that the kinematic redundancy come from two sources, the intrinsic and functional redundancies, i.e., rK = rI + rF .
(6)
In the literature, most of the research working on redundancyresolution of serial manipulators suppose that rF = 0, and thus, study rK = rI . In this paper, the opposite case is studied, i.e., supposing rI = 0, and thus, study rK = rF . B. Problem Formulation The inverse kinematics of serial manipulators is usually based on the linear relationship between the EE velocity, called ˙ given twist and denoted t, and the joint velocities, denoted θ, by ˙ t = Jθ, (7) 1 This definition of functional redundancy of serial manipulator-task is expandable to other types of manipulators such as parallel and hybrid ones.
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with t and θ˙ defined as T
T T
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t ≡ [ω p˙ ] ∈ 2 × < ,
θ˙ ≡ [θ˙1 · · · θ˙n ] ∈ < , T
n
and the Jacobian matrix J being defined as £ ¤T J≡ A B , A, B ∈
