The power manipulability – A new homogeneous performance index of robot manipulators
Introduction
Kinetostatic performance indices, based principally on the Jacobian matrix, require the evaluation of determinants, eigenvalues or singular values. This may be executed conveniently only if the Jacobian matrix elements have the same units [1], [2], [3]. Otherwise, any involved mathematical derivation such as determinant, eigenvalues or singular values evaluation will constitute a noncommensurable system [4], which is physically inconsistent, causing then the instability of results with respect to physical units’ choice. Furthermore, the use of these indices is restricted to mechanisms having only one kind of joint, i.e. either prismatic or rotational, but not a combination of both. Beside, translations and rotations should be separated in the task space, and giving in that case different indices for rotations and translations [5], ignoring the correlation between these two forms of motion.
The kinematic manipulability index has been at first introduced to evaluate the kinematic transmissibility of a robot manipulator as an index of the relationship between the linear and angular velocities at each joint and at the end-effecter of the manipulator [6]. Considering a robot manipulator with n DOF, the vector represents its joint or generalized velocities. Let vE, ωE be the translation and rotation end-effector velocities, respectively, with the following dependency relation:
The linear (6×n) operator J is the Jacobian. JT and JR are its components of translation and rotation, respectively [7]. The inverse relation can be obtained by introducing the pseudo-inverse of J (i.e., J+).
Performing the following normalisation on joints’ velocities vector
It can be noticed that it is more consistent if the real kinematics’ limit values are used instead of the unity in the right-hand side of Eq. (3). The unity limit method lacks a physical significance, while the real kinematics’ limit method is really unjustified, for we are looking for a natural index, which should be independent in terms of physical units’ choice and task specifications. Furthermore, if we were concerned by an optimal design problem, the kinematics limits are assumed to be unknown at this stage.
From Eqs. (2), (3), the following condition on the end-effector velocity is obtained
This inequality represents an ellipsoid in the task space (Fig. 1). It measures the capability of the manipulator to transmit a certain velocity to its end-effector. Concerning this point, two parameters are taken into consideration, the volume of the ellipsoid and its isotropy. The former may be formulated using the product J JT instead of , the manipulability or the ellipsoid volume is proportional to [3], [6], [7], [8]
For a non-redundant manipulator, this quantity will be equal to the determinant of the Jacobian |J|.
The manipulator can have a good manipulability volume in a given configuration, but it cannot produce the same velocity along all the directions. For this reason, as far as performance is concerned, the ellipsoid uniformity or isotropy is more suitable [3], [9], and it may be quantified using the singular value decomposition of the Jacobian; it’s the inverse of the condition numberwhere σMin and σMax are the maximal and the minimal singular values of J (these are the square roots of the maximal and minimal eigenvalue of J JT). A better isotropy index (close to unity) means that the manipulator can transmit velocity uniformly along all directions (this uniformity has a sense only if the velocity components are homogenous, i.e. having the same physical units). Thus, the isotropy concept has more significance than the ellipsoid volume in the performance analysis of manipulators. However, this approach is not without flaws and weaknesses.
The performance parameters cited are pose-dependant parameters, i.e. local parameters. Hence, for a design purpose, global parameters are sometimes needed; they can be obtained by integrating local measures over some region in configuration space [10].where W is a specific point of the manipulator workspace, and the denominator is the workspace volume. The concept of isotropic configuration in a Jacobian matrix, as well as its local and global measures are the most used tools in robotics [10], [11], [12]. These measures are also very convenient in determining the optimal postures for a particular task [13], [14], [15].
The manipulability force ellipsoid is a similar index that evaluates the static torque–force transmission from the joints to the end-effecter [3], [6], [8], while the dynamic manipulability ellipsoid was introduced as a measure of the acceleration capability of the end-effector by taking into account the dynamic parameters of the mechanism [3], [8], [16], even though this procedure manipulates modified forms of accelerations and generalized forces that are velocity and gravity dependant.
The normalisation condition on the articulations velocities (eq. (3)) does not have a physical meaning; therefore, the use of the real kinematic limits (weighted Jacobian) [17], or the polytopes [18], is more consistent. However, it’s preferable that no restriction will be made at this stage of the design. In addition, it’s simply possible to modify the ellipsoid volume and aspect by rearranging the kinematic limits. Therefore, the manipulability indices are not a natural performance measure of mechanisms. In addition, these indices are not homogeneous; they depend on the physical unit choice, particularly with different articulation types, where the normalisation condition in Eq. (3) loses its consistency. Even if there are articulations of the same type, it’s almost inadmissible to group the translational and rotational end-effector velocities in the same parameter [5], [7], since there are two different forms of movement and we cannot say that this translation is bigger than that rotation. Angeles [1], [19], [20] has tried to resolve this problem by introducing the characteristic length, but this concept lacks a direct geometric interpretation. In the case of parallel robots containing one type of actuators, complex degrees of freedom can be substituted by a homogeneous task space vector. This is possible by taking into account the translational movement of three points on the end-effector [21]. Still the choice of these three points remains arbitrary, this approach lacks generality.
One common and fatal mistake made in past approaches is to combine position and orientation information into a single scalar performance parameter, which makes it physically inconsistent. In the following analysis, we will develop a new approach, giving rise to a new parameter, which does not depend on the physical unit choice, i.e. natural performance index for mechanisms. This is possible when a common quantity between translation and rotation concept is manipulated, i.e. the power concept. Because of its scalar character, it was difficult to exploit the power directly in order to obtain a performance index; this is not the case for the possibly vectorial quantities like velocity or force, henceforth, further developments are based on the power components (force, velocity) behaviour study.
The rest of this paper is organized as follows: the principle of the dual manipulability is briefly described, which will be useful in the development of the new concept of the apparent power. Then, a vectorial representation of the power is presented. Using these developments, it will be possible to present the homogeneous performance indices. Finally, a dexterity analysis study is carried out on a 2 DOF serial manipulator and a 3 DOF planar parallel mechanism.
Section snippets
Duality between task-space and articulation-space ellipsoids
Here, the duality between the task-space and the articulation-space manipulability will be exposed. As it is possible that a sphere in the articulation space is transformed into an ellipsoid in the task space, a similar result can be obtained with the inverse transformation, i.e. a sphere in the articulation space is transformed into an ellipsoid in the task space. Considering the well-known robot kinetostatic equationswhere τ is the articulation effort vector (generalized
The notion of apparent power
The apparent power concept already exists in electricity as a complex quantity, being equal to the product of the complex expressions of voltage and current conjugate. In the following, it will be shown that an approach of non-regular representation of the power can be extended to a multi-body kinematic analysis. Before, a relationship has to be established between the manipulability kinetostatic performance and internal wrench/twist of a manipulator. This correlation is more perceptible with a
The power vector of the end-effector
It’s evident that the power is a scalar quantity. However, only an oriented or vectorial representation of it can be useful in performances evaluation. In order to acquire such a form, it is assumed at first that the end-effector transmits an absolutely active power P to the exterior (corresponding effort and velocity are collinear). This power is shared between translation and rotation.
An absolutely transmitted active translational power means that the end-effector force fE and velocity vE
The homogeneous expression of the apparent power
Using Eqs. (27), (39), (40), the apparent power of the link k can be written with respect to the power vector p, i.e. in a homogeneous form
The 6×6 matrix [Pk] has to be written in a bloc form (each bloc is a 3×3 matrix)
Then, we obtain for the first element of the quadrivector
Application for a 2-DOF serial manipulator
In order to point out the efficiency of the method, it is applied to a simple structure of two link manipulator (Fig. 10).
Assuming that: l1=l2=l; the end-effector point OE being attached to the x-axis. Then, q2=−2q1 and the Jacobian matrix takes the form
Let p be the power vector at the end effector, its associated kinematic parameters are
For the first link, the associated twist and wrench are
Application for a planar 3-DOF mechanism
A planar 3-DOF parallel manipulator has been extensively studied as the fundamental prototype of parallel manipulators [24], [25], principally due to the possible representation of the 3 degree of freedom in space [21]. In this example, a simple symmetrical planar 3-DOF mechanism with 3 prismatic legs is used (Fig. 13), in order to demonstrate the effectiveness of our approach and then derive the corresponding power manipulability ellipsoids and performance indices. Within a dexterity analysis
Conclusion and discussion
The power manipulability ellipsoid is a homogenous tensor defined in a 6th dimension space. It includes both translational and rotational components, but in a unique physical unit. Instead of using a purely kinematic or static parameter (velocity, force), a combination of these two quantities have been used and a new vectorial representation of power was introduced. That was only possible by imposing a useful condition on the end-effector wrench and twist; Since f and v are collinear, and m and
Acknowledgement
The authors express their sincere thanks to the editor for his continuous support; they would also like to thank the anonymous reviewers for helpful and constructive comments on this paper.
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