Optimal Task Placement of a Serial Robot Manipulator for Manipulability and Mechanical Power Optimization ()
Power consumption and accuracy are main aspects to be taken into account in the movement executed by high performance robots. The first aspect is important from the economical point of view, while the second is requested to satisfy technical specifications. Aiming at increasing the robot performance, a strategy that maximizes the manipulator accuracy and minimizes the mechanical power consumption is considered in this work. The end-effector is constrained to follow a predefined path during the optimal task positioning. The proposed strategy defines a relation between mechanical power and manipulability as a key element of the manipulator analysis, establishing a performance index for a rigid body transformation. This transformation is used to compute the optimal task positioning through the optimization of a multicriteria objective function. Numerical simulations regarding a serial robot manipulator demonstrate the viability of the proposed methodology.
1. Introduction
Minimization of production costs and maximization of productivity are some of the major objectives of industrial automation. In this scenario, serial robot manipulators have been proved to be very useful tools.
Due to the augmenting use of serial robot manipulators to perform a number of tasks in industry, requirements concerning higher precision, improved productivity, reduced costs, and better manufacturing quality become very important. To effectively explore all the potential of the robot, its path planning is a subject of major concern.
There are several works in the literature dedicated to different approaches concerning this subject. Regarding the off-line path planning approach, in Saramago [1] a solution of moving a robot manipulator with minimum cost along a specified geometric path in the presence of moving obstacles is presented. The optimal traveling time and the minimum mechanical energy of the actuators are considered together to form a multiobjective function to be minimized along the process.
In Santos [2] a strategy to determine the trajectory for a defined movement taking into account the requirements of torque, velocity, operation time and robot positioning along the movement is proposed. The analysis is performed as an off-line path planning through spline interpolation techniques in the manipulator joint space. In the paper of Chiddarwar [3] an off-line path planning for coordinated manipulation is proposed. The swept sphere volume technique is used to model multiple robots and static obstacles. In Santos [4] the trajectory of two manipulators while manipulating a single object in a collaborative task and the object placement are written as an optimization problem. End-effector positioning and torque requirements are considered together in an optimal control formulation. Accuracy and energy consumption are improved during the path planning. Also, the flexibility effects of manipulators working in a vertical workspace are taken into account, and joint limits are considered in a box-constrained objective function to ensure the movement feasibility at the optimal configuration.
In von Stryk [5] an optimal control path-planning strategy for which the dynamics of the robot and the total traveling time are considered as objectives to be optimized is proposed. In Santos [6] an optimal control strategy that considers the presence of moving obstacles in the workspace is presented.
The problem of time-optimal control along a specified path has been investigated for several authors (Bobrow [7], Slotine [8]). In Constantinescu [9] a smooth timeoptimal strategy for serial robot manipulators is presented.
The paper of Seraji [10] addressed a geometric approach to determine the appropriate base location from which a robot can reach a target point. The work of Zeghloul [11] presented some kinematic performance criteria for the optimal placement of robots, and proposed a general optimization method to determine the placement of manipulators automatically.
In Feddema [12], an algorithm for determining the optimal placement of a robotic manipulator for minimum time coordinated motion is proposed. In Park [13] a study that characterizes a set of desired goal positions and a left-invariant distance metric parametrized by length scale is studied.
The paper of Chirikjian [14] proposed several metrics to be used to generate interpolated sequences of motions of a solid body. In Martinez [15] an analysis regarding the metric of a rigid displacement obtained from the kinematic mapping is presented and other metrics for the set of spatial and planar displacements are proposed. The work of Tabarah [16] proposed an optimization strategy for determining the optimal base for a given manipulator, and two cooperating manipulators, following a continuous path with constant velocity. They considered the constant velocity as a constraint. A strategy for velocity minimization is proposed by Zhang [17], based on a neural network solver. A kinematic planning scheme is reformulated as a quadratic problem that is solved by using a real-time algorithm, and applied to a Puma 560 robotic arm.
In Abdel-Malek [18] a strategy regarding the placement of a robot manipulator aiming at reaching a specified set of target points is described. The placement of a serial manipulator in a given workspace is achieved by defining the position and orientation of the manipulator’s base with respect to a fixed reference frame. The strategy is based on characterizing the robot placement by adjusting a constrained cost function that represents the workspace to a set of target points.
With the aim of increasing productivity in the path following, industrial robotic applications have been addressed in the literature by determining path-constrained time-optimal motions, by taking into account the torque limits of the actuators. In these formulations, the joint actuator torques are considered as controlled inputs and the open loop control schemes result in bang-bang or bang-singular-bang controls (Bobrow [7], Chen [19], Shiller [20]). The paper of Xidias [21] proposed the generalization of the task scheduling problem for articulated robots in a constrained 2D environment. An algorithm for the optimum collision-free movement is proposed.
However, the increase of the speed may result (in some cases) in increasing the robot end-effector vibration or decreasing its accuracy. This is observed due to several factors, such as the bang-bang nature of the control, joint flexibility, joint friction, gear arrangement, or even a combination of them.
Vibration is not desired because it can both degrade the system performance and decrease the actuator lifetime. A related and negative effect of vibration is the poor accuracy of the system.
The present contribution discusses an alternative way to improve the overall performance of serial robot manipulators. The use of manipulability and mechanical power related indexes to achieve the best task placement is proposed.
As the basic idea of manipulability consists in describing directions in the task or joint space that maximizes the ratio between some measure of effort in joint space and a measure of performance in task space, the present contribution consists in the optimal positioning of the task to be performed, which results in the maximization of the robot manipulability and the minimization of the required mechanical power. This will lead to accuracy maximization and power consumption minimization, since manipulability can be understood as an ability measurement (Fu [22]) and mechanical power is related to power consumption (Saramago [1]).
In the present paper both kinematics and dynamics features are considered in the proposed task placement strategy. From the best of the authors’ knowledge, this problem was not previously addressed in the literature, despite its importance and applicability.
The position of the end-effector is fixed according to a Cartesian reference system, and described as a sequence of sucessive positions. The optimization process applies a translational and rotational transformation matrix with respect to a given reference frame. As a result, an optimal sequence of kinematical robot positions is determined.
The outline of the paper is the following. In Section 2, a review about manipulability is presented. In Section 3, the mechanical power concept is discussed. A strategy to describe the task and its positioning in the design space is proposed in Section 4. Additionally in this section the design variables are defined. In Section 5 the multicriteria programming formulation is presented. In Section 6, the optimization strategy is established and the numerical experiments are performed in Section 7. Finally, the conclusions are drawn in Section 8.
2. Manipulability
Some factors should be taken into account when the posture of the manipulator in the workspace is determined for performing a given task during operation. An important factor is the ease of arbitrarily changing the position and orientation of the end-effector at the tip of the manipulator.
As an approach for evaluating quantitatively the ability of manipulators from the viewpoint of kinematics, the concepts of manipulability ellipsoid and manipulability measure (Yoshikawa [23]) are presented.
Consider a manipulator with n degrees of freedom. The joint variables are denoted by an n-dimensional vector, q = [q1, q2, ..., qn]T. An m-dimensional vector r = [r1, r2, ..., rm]T, (m ≤ n) describes the position and orientation of the end-effector.
The relation between the velocity vector v corresponding to r and the joint velocity 
is given by:
(1)
where J(q) is the Jacobian matrix, computed from the kinematics description of the robot.
The set of all end-effector velocities v that are realizable by joint velocities such that the Euclidean norm of 
satisfies 
defines an ellipsoid in the m-dimensional Euclidean space. In the direction of the major axis of the ellipsoid, the end-effector can move at high speed, and in the direction of the minor axis, the end-effector can move only at low speed. Also, the larger the ellipsoid is, the faster the end-effector can move.
Since this ellipsoid represents an ability of manipulation, it is called the manipulability ellipsoid.
The principal axes of the ellipsoid can be found by making use of the singular-value decomposition of J(q).
One of the representative measures for the ability of manipulation derived from the manipulability ellipsoid is the volume of the ellipsoid. Since this volume is proportional to the eigenvalues of the Jacobian, it can be seen as a representative measure. Therefore, the manipulability measure, f1, for the manipulator configuration, q, is defined as
(2)
Generally f1 ≥ 0 holds, and f1 = 0 if and only if rank J(q) < m.
There is a direct relation between singular configuration and manipulability (through the Jacobian concept). According to this relationship, it can be shown that the larger the manipulability measure, the larger the ability of avoiding singular configurations.
3. Mechanical Power
The consideration of the dynamics behavior of a serial robot manipulator is of great importance for its path planning. This information allows a detailed analysis and consequently the development of a precise control specification.
Many efficient schemes have been proposed to model the dynamics of rigid multibody mechanical systems (Richard [24], Mata [25], Mata [26]). The dynamics model can be obtained explicitly through algebraically computation, or numerically through iterative computation.
The techniques based on the Newton-Euler method starts from the dynamics of all individual parts of the system. They look at the instantaneous or infinitesimal aspects of the motion, using vector quantities such as Cartesian velocities and forces.
Alternatively, the Euler-Lagrange based methods starts from the kinetic and potential energy of the entire system, by considering the states of the system during a finite time interval. This approach works with scalar quantities, namely the energies.
Independently of the approach used, at the end of the process the generalized forces are determined:
(3)
where 
and 
are the joint position, velocity and acceleration, respectively. M(q) is the joint space mass matrix and 
is the vector of Coriolis and centrifugal forces. Vector g(q) is the vector of gravitational forces and τ is the generalized force vector.
The energy that is necessary to move the robot is an important design issue, because in real applications energy supply is limited and any energy reduction leads to smaller operational costs. Due to the relationship that exists between energy and force, the minimal energy can be estimated from the generalized force, 
, that is associated to each joint i at the time instant t0 ≥ t ≥ tf.
In the present contribution, for a manipulator with n degrees of freedom, the interval between the initial time t0 and the final time tf is represented by a set of N points and the computation is numerically carried out through a recursive Newton-Euler formulation (Craig [27]). Then, the mechanical power is used for design purposes as defined by:
(4)
This expression is representative of the phenomenon under study because it considers both the kinematics and the dynamic aspects of the trajectory, simultaneously (Saramago [1]).
4. Task Specification
The task is specified as a set S of N Cartesian points (x, y, z) and the respective time step. Hence,
. (5)
In the case of non-redundant manipulators there exist a finite number of configurations that satisfies the end-effector positioning specification. This path-following problem is a highly constrained task, and the inverse kinematics problem will have four solutions or less.
Considering the need of movement continuity together with a time-fixed specification at each step and a relative small step size, the point-to-point path planning strategy may fail due to the complexity of the constraints.
Therefore, a task positioning optimization is proposed as an alternative to increase the robot performance without changing the position of the task reference points with respect to each other. In other words, all the points will be moved simultaneously through a rigid body transformation to find the best position from the point of view of the ability of the manipulator.
To achieve this goal, the use of a homogeneous transformation A is proposed, which in this case is defined as
(6)
where β is a rotation angle in the z axis (given in rad) and δx , δy and δz are translations along the reference axes (given in meters), respectively. If desired, it is possible to extend this formulation to consider the rotation with respect to the reference axes.
The physical meaning of considering a rotation angle with respect to x or y is that the robot’s base plane will not be parallel to the ground’s reference plane. In other words, the robot’s z axis (its height) is not normal to the ground’s reference plane (xy plane). As positioning two planes according to a precise angle may not be possible in a number of practical situations (even when it is possible, it is not an easy procedure to be done, in general terms), the rotation along the z axis only is considered, maintaining the robot’s base plane parallel to the task’s base plane.
After defining the set S that describes the task, the inverse kinematics with respect to each Cartesian point is computed.
The inverse kinematics computation can be achieved algebraically, by considering the analytical model of the manipulator and its geometry, or numerically. With the aim of presenting a general procedure, in the present contribution the inverse kinematics problem is solved numerically through an inverse problem formulation. This approach results an efficient and general procedure that works satisfactorily for manipulators exhibiting arbitrary complexity (Santos [28]).
The obtained path planning presents an end-effector positioning error, while the robot executes the path. The interest is focused on errors due to task positioning, as the inverse kinematics can generally be solved with a sufficient precision.
Once the objective is specified, it is necessary to define the optimization domain, i.e., the design space. As the design variables are defined in a theoretical space (defining a rigid body transformation) and the positioning constraints are defined in the Cartesian space (the workspace), a constraint parameter is included in the formulation.
Defining the theoretical positioning error at each point as
(7)
where Pbase, Pend and 
are the robot base point, the end-effector point, and the kinematics representation of the manipulator, respectively. The penalty parameter used as a constraint in the following optimization formulation is given by
(8)
and represents the sum of all end-effector positioning errors during the movement.
Equation (7) means that the end-effector error is considered null when the accuracy is better than the one specified by ε (given in meters). Otherwise, this error corresponds to the distance from the end-effector specified position to the position obtained from the direct kinematics calculation.
In some robotic implementations it is necessary to plan the layout of the workspace, i.e., it is required to locate the robot base in such a way that dexterity is maximized at (or around) given targets. It is worth mentioning that such cases are also covered by the present formulation. Since the optimal design ((β, δx, δy, δz) in Equation (6)) means a coordinate transformation of the end-effector path regarding the fixed robot base, it is also possible to adjust the location of the robot base regarding the fixed end-effector reference through the relation (-β, -δx, -δy, -δz).
5. Multicriteria Programming
In problems with multiple criteria, one deals with a design variable vector x, which satisfies all constraints and makes the scalar performance index that is calculated from the m components of an objective function vector f(x) as small as possible. This goal can be achieved by the vector optimization problem:
(9)
where Ω
is the function domain (the design space).
An important feature of such multiple criteria optimization problem is that the optimizer has to deal with objective conflicts (Eschenauer [29]). Other authors discuss the same topic by defining compromise programming, because there is no unique solution to the problem (Vanderplaats [30]).
5.1. The Proposed Formulation
To consider together the manipulability and the total mechanical power, the following scalar objective function is proposed
(10)
where f1 and f2 are given by Equations (2) and (4), respectively. The reference values 
and 
are obtained from the computation of the manipulability and mechanical power sum at the initial task positioning. Therefore, in the ideal case the objective function value is lower than 2, where 
(the manipulability was increased) and 
(the mechanical power was decreased).
The specification of a reference value to both the manipulability measure and the mechanical power are of paramount importance in the construction of the multicriteria objective function.
The inverse kinematics error is included in the problem as an equality constraint in the optimization formulation, through the expression sE = 0 (Equation (8)).
It should be pointed out that the proposed formulation is able to handle the case in which the task is outside the robot workspace. This configuration reflects in the inverse kinematics error, which is taken into account as a constraint (Equation (8)) of the optimization formulation.
Additionally, initial and final velocity specifications are considered as
(11)
(12)
for each joint j in the initial and final times t0 and tf, respectively. To represent a rest-to-rest motion, zero initial and final velocities are considered above.
6. Optimization Strategy
An important feature of the proposed analysis is the computation of highly nonlinear equations. Despite the good performance of classical nonlinear programming techniques, the global optimal design is hardly reached due to the existence of local minima in the design space.
With the aim of improving the robustness of the optimization process, the use of a global methodology is considered. The tunneling algorithm (Levy [31,32]) is a heuristic methodology designed to find the global minimum of a function. It is composed of a sequence of cycles, each cycle consists of two phases: a minimization phase having the purpose of lowering the current function value, and a tunneling phase that is devoted to finding a new initial point (other than the last minimum found) for the next minimization phase. This algorithm was first introduced by Levy [31], and the name derives from its graphic interpretation.
In summary, the computation evolves according to the following phases:
a) Minimization phase: Given an initial point q0, the optimization procedure computes a local minimum q* of f(q). At the end, it is considered that a local minimum is found.
b) Tunneling phase: Given q* found above, since it is a local minimum, there exists at least one q0
, such that
(13)
In other words, there exists q0 
Z = {q 
Ω – {q*} | f(q) ≤ f(q*) }. To move from q* to q0 along the tunneling phase, it is defined a new initial point q = q* + δ, q 
Ω that is used in the auxiliary function
(14)
which has a pole in q* , for a sufficient large value of η By computing both phases iteratively, the sequence of local minima may lead to the desired global minimum..
Different values for η are suggested for being used in Equation (14) to avoid undesirable points and prevent the search algorithm to fail (Levy [31,32]). In the present case the stop criterion is five consecutive iterations without further improvement in the minimal objective value.
The minimization phase is performed by using a constrained formulation, where the objective function is given by Equation (10) and the constraint function is defined by Equation (8). The velocity specification (Equations (11) and (12)) is achieved through the determination of a suitable interpolation polynomial function using the joint position coordinates.
In Equation (14), during the tunneling phase, only the unconstrained formulation given by Equation (10) is considered. This unconstrained nonlinear programming problem is solved by using the BFGS variable metric method (Luenberger [33], Vanderplaats [30]). The constrained formulation of the minimization phase (Equations (10), (11) and (12)) is solved by using the modified method of the feasible directions (Vanderplaats [34]).
The algorithms are implemented in FORTRAN by using the optimization library DOT (Vanderplaats [35]).
7. Numerical Simulations
The numerical simulations performed to illustrate the presented methodology use two robot models, namely the Scara and the Puma 560 robot manipulators. These two types of serial robot manipulators are classical configurations in the field. The case studies have been proposed due to their complexity as compared to those normally found in the literature. The following results considers only optimal profiles whose sum of the end-effector error is zero (Equation (8)), given ε = 0.001 m (Equation (7)). This means that the inverse kinematics solution is found for all positions with the required precision. Different tolerance values may result from different optimal profiles.
As different values for the weighting factor k lead to different optimization problems (Equation (10)), the value of the global minimum is normalized according to the following equation
(15)
This performance index enables comparing the improvement of each objective without the weighting effect, and is used in the subsequent analysis.
In summary, the global optimization for each weighting factor (k = 0, 0.1, 0.2, 0.3, ..., 1) is performed according to Equation (10) and the comparison among the optimal designs obtained is performed by using Equation (15). Thus, eleven global optimizations for each task have been carried out. The maximum inverse kinematics error was defined as being lower than ε = 0.001 m in Equation (7).
A Scara manipulator was used in the first and second numerical experiments. The third numerical experiment refers to a Puma 560 manipulator. By considering the ability of describing a complex path, the data for a set of Cartesian specification points (Equation (5)) is determined through a cubic spline interpolation of the parametric coordinates of the desired path. Therefore, all the values are obtained from a suited set of Cartesian points.
The path for each experiment is arbitrarily defined with the aim of exploring the efficiency of the methodology in performing a complex task. All the obtained results consider only configurations for which inverse kinematics errors are lower than ε, i.e., sE = 0 (Equation (8)).
Each value k in Equation (10) defines a new optimization problem. To compare the values given by different optimization formulations, the normalization of the results, despite the weighting factor, is taken into account as established by f4 (Equation (15)).
7.1. First Experiment
In this experiment, a subset of 7 reference points is defined: (0, 0.4, 0, –0.3), (1, 0.4, –0.1, –0.3), (2, 0.2, –0.1, –0.3), (3, 0.2, –0.2, –0.3), (4, 0.4, –0.25, –0.3), (5, 0.4, –0.2, –0.3) and (6, 0.2, –0.25, –0.3). Comparing these values with Equation (5), the initial task is inside a rectangle defined by 0.2 ≤ x ≤ 0.4, –0.3 ≤ y ≤ –0.1 and z = –0.3. The corresponding traveling time is defined by 0 ≤ t ≤ 6. These data are computed by interpolating cubic splines for t, x, y and z, respectively. The interpolated data encompass a set of N = 27 points (Equation (5)) that describes the path-following problem, as presented in Figure 1.
At the original position, the values of the design variables are: β= 0 rad, δx = 0 m, δy = 0 m, and δz = 0 m. These values are used as an initial guess for the optimization process (Equation (10)).
The global minimum values obtained are presented in Figure 2.
The optimal values of manipulability and mechanical power indexes demonstrate that there is no clear dependence between the variables. Good results are provided for k = 0.2, 0.5, 0.6, and 0.9.
Figure 1. The proposed path-following problem from two different perspectives.
Details about the minimal performance index, k = 0.9, are presented in Table 1. The overall performance was improved 28%, while the minimal manipulability and the mechanical power were improved 62% and 18%, respectively.
Some of the values provided by other weighting factors may result in further improvement of the manipulability or mechanical power, separately. However, the present configuration is a global minimum of the performance index f4.
The improvement obtained during the tunneling process is shown in Figure 3.
After the 6th iteration of the tunneling process, no further improvement was achieved during the next five consecutive iterations. This behavior defines the stop criteria for the global optimization search. The above result highlights the good performance of the method, as satisfactory results are obtained after a small number of iterations.
Figure 4 shows the robot performing the path following task and Figure 5 presents the robot optimal positioning, according to the optimal design variable values