In this paper an analysis of productivity will be carried out from the resolution of the problem of trajectory planning of industrial robots. The analysis entails economic considerations, thus overcoming some limitations of the existing literature. Two methodologies based on optimizationsimulation procedures are compared to calculate the time needed to perform an industrial robot task. The simulation methodology relies on the use of robotics and automation software called GRASP. The optimization methodology developed in this work is based on the kinematics and the dynamics of industrial robots. It allows us to pose a multiobjective optimization problem to assess the tradeoffs between the economic variables by means of the Pareto fronts. The comparison is carried out for different examples and from a multidisciplinary point of view, thus, to determine the impact of using each method. Results have shown the opportunity costs of non using the methodology with optimized time trajectories. Furthermore, it allows companies to stay competitive because of the quick adaptation to rapidly changing markets.
Time needed to perform a trajectory for industrial robots is a very important issue in order to improve productivity in many economic activities. Specifically, most algorithms seek to find the minimum time trajectory in order to increase the working time and subsequently to reduce the unproductive time. The existing literature shows a lack of studies that consider both the economic issues and the motion of industrial robots.
In this paper, the working times of industrial robots are compared between two different approaches while taking into account the corresponding economic impacts. The comparison is applied to several examples, which covers a wide range of parameters that govern the kinematics and dynamics of the industrial robots. The first methodology is based on a robotic simulation program called GRASP (BYG System Ltd) and the second on optimization techniques.
When the working times have been calculated, the assembly line productivity is estimated by means of the time difference, so that we can quantify the impact of each method. Productivity is quantified by conducting an economic study based on the working times of robotic tasks and, more specifically, the time needed to manufacture and assemble a certain product.
A multiobjective optimization problem is posed to assess the tradeoffs between the economic variables by means of the Pareto fronts (see Section
Those times will enable us to set conclusions about which method is more useful in order to increase the productivity of the robotic system.
The paper is organized as follows. Initially, we will explain in detail the main characteristics of the trajectory planning methodology and how the time is obtained. Consequently, the economic analysis will provide insight on the productivity of assembly lines. Finally, the conclusions will be discussed in the last section.
Currently there are a great number of methodologies to solve the trajectory planning problem for industrial robots which give the time needed to perform a task. But few papers tackle the analysis of productivity related to the working times obtained.
Over the years, the algorithms have been polished and the working assumptions of the robotic systems have been increasingly adjusted to real conditions. This fact has been achieved by analysing the complete behaviour of the robotic system, particularly the characteristics of the actuators and the mechanical structure of the robot. To tackle this problem other important working parameters and variables have been taken into account, such as the input torques, the energy consumed, and the power transmitted. Furthermore, the kinematic properties of the robot’s links, such as the velocities, accelerations, and jerks, must be also considered. The aforementioned algorithms provide a smooth robot motion for the robotic system.
To obtain the best trajectories in terms of minimum times, some of the working parameters have been included in the appropriate objective function of the optimization procedure (input torques, the energy consumed, and the power transmitted). The optimization criteria most widely used can be sorted as follows.
Minimum time required, which is directly bounded to productivity.
Minimum jerk, which is bounded to the quality of work, accuracy, and equipment maintenance.
Minimum energy consumed or minimum actuator effort, both linked to savings.
Hybrid criteria, for example, minimum time and energy.
In the past, the early algorithms that solved the trajectory planning problem tried to minimize the time needed for performing the task (see [
The early algorithms in trajectory planning sought to minimize the time needed for performing the task. The dynamics properties of actuators were neglected. A recent example of this type of algorithm can be found in [
Later, the trajectory planning problem was tackled by searching for jerkoptimal trajectories. Jerks are highly important for working with precision and without vibration. They also have an effect on the control system and the wearing of mobile parts such as joints and bars. These methods allow a reduction in errors during trajectory tracking, the stresses in the actuators and also in the mechanical structure of the robot, and the excitement of resonance frequencies. Jerk restriction is introduced by other authors [
In [
In [
Another different approach to solving the trajectory planning problem is based on minimizing the torque and the energy consumed instead of the trajectory time or the jerk. This approach leads to smoother trajectories. An early example is seen in [
Similarly, in [
The work in [
In [
Afterwards, new perspectives appear for solving the trajectory planning problem. The main point was to use a weighted objective function to optimize the working parameters [
In this paper we will introduce two methods to solve the trajectory planning problem for industrial robots working in complex environments. The time will be used in the economical study.
In the first method, the procedure calculates the optimal trajectory by neglecting initially the potential presence of obstacles in the workspace. By removing the obstacles (real or potential) from the optimization problem, the algorithm will calculate a minimum time trajectory as a starting point. Then the procedure must take into account the real obstacles presented in the workspace. When obstacles are considered, the initial trajectory will not be feasible and will have to evolve so that it can become a solution. The way this initial trajectory evolves until a new feasible collisionfree trajectory is obtained is presented in this paper. It is a direct algorithm that works in a discrete space of trajectories, approaching the first solution to the global solution as the discretization is refined. The solutions obtained are efficient trajectories (i.e., the minimum time trajectories). All the trajectories obtained meet the physical limitations of the robot. The solution also avoids collisions and takes into account the constraint of energy consumed.
The second method calculates the times using the kinematic properties of the robotic system by means of a simulation program called GRASP.
Our objective is to calculate the minimum time trajectory
We calculate the time needed to go from
This optimization problem can be stated as in [
The optimization problem is subject to:
the robot dynamics
unknown boundary conditions (position, velocity, and acceleration) for intermediate configurations a priori
boundary conditions for initial and final configurations
collision avoidance within the robot workspace
actuator torque rate limits
maximum power in the actuators
maximum jerk on the actuators
energy consumed
Here, the main definitions and processes used to obtain the freecollision trajectories of the robot (and subsequently the time needed to perform the trajectory) are detailed.
It is expressed in joint coordinates
Given a feasible configuration of the robot
The robot endeffector occupies a position corresponding to a node of the discretized workspace in Cartesian coordinates and its distance to the endeffector position in the configuration
Given a sequence of
To ensure continuity, the following conditions associated with the given configurations are considered.
(a) Position: for each interval
(b) Velocity: the initial and final velocities of the trajectory must be zero, obtaining
When passing through each intermediate configuration, the final velocity of previous interval must be equal to the initial velocity of the next interval; that gives
(c) Acceleration: for each intermediate configuration, the final actuator acceleration of the previous interval must be equal to the initial acceleration of the next, resulting in
Knowing the time required to perform the trajectory between the different configurations, using the above equations, the coefficients of the cubic polynomials can be obtained efficiently by means of the calculation of the normal time [
In addition, the minimum time trajectory s must meet the following four types of constraints:
(d) maximum torque in the actuators,
(e)
(f) maximum power in the actuators,
(g)
(h) maximum jerk on the actuators,
(i)
(j) energy consumed
where
To obtain the minimum time, an optimization problem is solved using variables defined in time increment at each interval (see [
Let
So the trajectories of a certain generation will have one passing configuration more than the previous generation, but they will keep the same initial and final configurations.
The problem of obtaining a feasible and efficient trajectory for a robot in an environment with static obstacles allowing the motion between two given configurations (
The proposed process for solving the problem involves the following steps which are implemented in the algorithm.
Obtaining the minimum time trajectory: using the procedure described in Section
Search for collisions: the first configuration from
Obtaining adjacent configurations: up to six new adjacent configurations to
Obtaining offspring trajectories: for each one of the
Trajectory selection: the generated trajectories are introduced in previous section (d) on the set of trajectories ordered by time
Refining the trajectory: in case that the collisionfree trajectory
Finally the trajectory
The proposed solution to the problem is
The simulation program GRASP10 for robotic environments is used to obtain the time needed to perform the motion between two given configurations. Among other tasks, GRASP10 can model and simulate the robot kinematic behavior. In this paper we have used the original model of PUMA 560 robot that comes with the program as a comparator. It is assumed that the point to point trajectory calculation procedures of GRASP10 correspond to the real robot. The robot kinematic characteristics are shown in Table
Kinematic characteristics of PUMA 560 robot.
Joint  1  2  3  4  5  6 

Minimum angle (°)  −160.0  −215.0  −45.0  −140.0  −100.0  −266.0 
Maximum angle (°)  160.0  35.0  225.0  140.0  100.0  266.0 
Maximum velocity (°/s)  82.0  54.0  122.0  228.0  241.0  228.0 
It should be noted that GRASP10 does not perform dynamic calculations but only kinematic ones. It is therefore very important to indicate the maximum working velocities, which have been obtained from actual robot by considering the properties of each actuator, primarily its maximum power, and the working torque.
When the trajectory is generated by GRASP10, the working actuators act simultaneously and at least one of them is moved to its maximum speed, calculated as follows for each actuator: (see Table
This methodology (use of GRASP10 for calculating the time required to perform a trajectory) has been applied to the same examples that have been resolved by the optimization algorithm explained in Section
For each example, the initial data are the initial and final configurations and the kinematics of the robot. The obstacle has been incorporated after the generation of the path, so that it burdens the previously calculated one.
In this section the productivity will be quantified by conducting an economic study based on the working times of robotic tasks.
The aim is to increase the profitability of production lines by designing flexible manufacturing systems. This allows companies to stay competitive because of the quick adaptation to rapidly changing markets. For instance, by adjusting the working hours in assembly lines or by deciding which products are more suitable to be manufactured according to the current demand.
This is performed by posing a multiobjective optimization problem, which makes use of the optimization algorithm, above presented, to solve the kinematics and dynamics of robot arms. The optimization method finds the minimum time trajectory to perform industrial tasks in production lines while taking into consideration the physical constraints of the real posed problem and then economic issues are also considered in the process.
Furthermore, Pareto fronts will be introduced, which will serve to determine those variables which mainly affect the improved productivity. To be more precise, the multiobjective optimization problem allows obtaining the Pareto frontiers, which provides information about the tradeoffs between the competing variables (i.e., execution times and benefits for the different products that can be manufactured at the production line).
Therefore, the economic study starts by defining the economic objective function to be used. It is formulated as follows:
Each one of these tasks is performed by the robot arm, which uses a certain time to describe the optimal trajectory. As above mentioned, the developed algorithm (Section
Besides, the amount that an additional item adds to a company’s total revenue during a period is called the marginal revenue of the product (
This factor is defined as the additional products manufactured per hour because of reducing the time used by the robot arm (
The marginal revenue product can be obtained by multiplying the marginal product (
Therefore, for companies in perfect competition, the marginal revenue product
The law of diminishing marginal returns tells us that if the quantity of a factor is increased while other inputs are held constant, its marginal product will eventually decline. If marginal product is falling (
The marginal revenue of a product (calculated in Section
Many realworld problems face two different types of mathematical difficulties. Those difficulties are the existence of multiple and conflicting objectives and a highly complex search space. Contrary to a single optimal solution, competing goals entail a set of compromise solutions generally denoted by the Paretooptimal, for example, [
A solution is defined as Pareto optimal if no improvement in one objective can be accomplished without adversely affecting at least one other objective. In the objective space, the hypersurface that represents all possible Paretooptimal solutions is termed as the Pareto front or frontier. A design that is located along the Pareto front is neither better nor worse than any other solution along the Pareto front. Hence, the solutions that compose the Paretooptimal set are equivalently optimal. The objective of multiobjective optimization using this technique is to generate as many Paretooptimal solutions as possible to adequately represent the Pareto front. This allows obtaining sufficient information for a tradeoff decision between competing variables. The Pareto front can be discontinuous, concave, or convex and, in general, is not known a priori.
The concept of domination enables the comparison of a set of designs with multiple objectives. Such a concept is not required for single objective optimization on account of the fact that the value of the objective function is the only measure of the quality of the design.
Then, in a direct comparison of two designs, if one design dominates another, the dominating design is superior and nearer to the Pareto front. Instead, if neither design dominates the other, the designs are nondominant to each other. Therefore, the best designs (with equally good objective vectors) in an arbitrary set of solutions can be distinguished because they are not dominated by any other design in the set; they compose the nondominated subset. Similarly, the designs that compose the Paretooptimal set are the nondominated set associated with the entire feasible space and are located along the Pareto front.
Consequently, the trajectory planning optimization problem in robotics can be defined as finding a motion law along a given geometric path, taking into account some predefined requirements, while generating minimum trajectory time of the robot arm. The inputs of the trajectory planning problem are the geometric path and the kinematic and dynamic constraints, while the output is the trajectory of the joints (or of the endeffector). The trajectory is expressed as a time sequence of position, velocity, and acceleration values. The optimized trajectory should also meet the physical limitations of the robot, the constraint of energy consumption and collisions avoidance.
The times obtained from the simulationoptimization procedures lead to different benefits. Therefore the Pareto fronts can be determined, thus showing the tradeoffs between the benefits and the obtained times. They show the opportunity cost of time higher than the minimum (see Section
The proposed multiobjective optimization methodology and the Pareto optimality have been applied to different examples in order to set the above mentioned tradeoffs between the benefits and the obtained times. Therefore, now the objective is conducting an economic study to quantify the productivity of the assembly line by comparing the minimum time trajectory to perform certain industrial tasks. It is obtained using two methods: an optimization algorithm and the simulation software GRASP.
These times are obtained while taking into consideration the physical constraints of the real working problem and the economic issues involved in the process. Twenty examples have been solved. The examples differ in the initial and final configurations of the robot, that is, the location of the endeffector (see Table
Working times needed to perform the trajectory for each example.
Optimization algorithm  Example 










Working time (s)  0.7488  0.5909  0.7511  1.5557  0.7488  0.5115  1.3920  0.7118  0.4529  1.0000  
Example 











Working time (s)  0.7488  0.5909  0.7511  1.5557  0.7488  0.5115  1.3920  0.7118  0.4529  1.0000  


Simulation 
Example 










Working time (s)  0.7500  0.6200  0.7700  1.6000  0.7600  0.5300  1.4000  0.7200  0.4600  1.2000  
Example 











Working time (s)  0.7500  0.6000  0.7600  1.5600  0.7500  0.5200  1.4300  0.7400  0.4600  1.3000 
This industrial robot arm is probably the most common robot in university laboratories. It is a 6R (revolution) type robot, with 6 degrees of freedom, and uses direct current (DC) servo motors as its actuators. This robot is a compact computercontrolled robot not only to perform service tasks, but also to carry out mediumtolightweight assembly, welding, materials handling, packaging, inspection applications, personal care, and so forth. The Series 500 is the most widely used model in the PUMA line of electrically driven robots.
With a 36inch reach and 5pound payload capacity, this robot is designed with a high degree of flexibility and reliability. The range of these angles from
Table
With regard to the economic issues associated with the robot industrial tasks we suppose for all examples the following quantities and considerations:
unitary cost to produce a certain item: 0.8 € (without considering the cost of the energy consumed);
item price: 1 €. For the sake of simplicity, we assume that only one product is produced at this point.
When the cost of the energy consumed is considered, the different examples have different costs because of the different working times. Therefore cost of the consumed energy: 0.0676 €/kWh (it is an average cost). This cost has been added to the cost of 0.8 €.
For reasons of clarity, the manufactured products are obtained in only one shift of 8 hours (365 working days in a year). The benefits
The time of the other industrial tasks needed to produce the item has been defined as 90 s, that is, the summation of times shown in (
The optimization algorithm presents different working times for the different examples, thus leading also to different benefits.
For instance, the case number 19, which has no constraints in both the jerk and the energy consumed, presents the maximum annual revenue (Figure
Annual revenue for each example based on the current demand.
The benefits for all examples obtained by means of the optimization algorithm are depicted in Figure
The industrial tasks are carried out in only a few seconds, so that the time scale is based on seconds. Note that the GRASP working times are similar to those obtained with the developed optimization algorithm, although slightly higher.
Then total benefits are also similar to those reported in Figure
Consequently, a lower mean value of the benefits is obtained, that is, 23,132.84 €/year with a standard deviation of 96.02.
Note that different prices, costs, and number of items produced by year may lead to higher differences in benefits between the optimization algorithm and GRASP. However, the selected values are only intended to illustrate the importance of using efficient algorithms of robot trajectory optimization for saving time and reducing costs in production lines.
In addition, a new analysis is carried out using Pareto optimality to illustrate the loss of benefits on account of not using optimization algorithms.
For that, we consider that three different products can be manufactured and assembled in the same production line. The loss of benefits is represented by the Pareto fronts for three different products. They differ in their cumulative time to be manufactured and assembled but share the same working time
Then the minimum trajectory time for case 4 is used for the three products, in this example, 1.55 s. These products also differ in the total costs (without considering the energy costs), prices, and values of the parameter
Consequently, if the market conditions do not change and the optimization algorithm is not used, the minimum trajectory time is not obtained.
In this scenario, there is a benefit loss due to the fact that robot arm may present higher working times.
Moreover, the multiobjective optimization problem allows obtaining the Pareto frontiers, which provides information about the tradeoff of the decision variables. One main tradeoff is between the benefits and the working time (i.e., the Pareto frontier). Results for example 4 are shown in Figure
Pareto frontiers obtained with the optimization algorithm for “example 4” and the three different products manufactured.
It is worth pointing out that for the analyzed examples, the differences between their annual energy costs are almost negligible compared with the other costs.
Furthermore, the concept of nondomination sorting can be used to categorize each design within a set into a hierarchy of nondominated levels or fronts. Each different level of nondomination represents a relative distance from the Pareto front. The best nondominated front is closest to the Pareto front and each subsequent front lags further behind and is, hence, increasingly inferior.
Through this sorting, each design is associated with a front that defines the quality of the design relative to the rest of the group. To isolate the various fronts, the designs that belong to the nondominated subset of the entire group are first identified. These designs are the best in the group, the closest to (or members of) the Pareto front. For instance, in our multiobjective trajectory optimization, the nondominated subset (i.e., the best solution in terms of greater benefits) is represented by the Pareto frontier of Product 2 for times higher than 1.96 s (see Figure
Any design belonging to this front is then temporarily set aside and another comparison process determines the next level of nondominated designs from the remaining population.
This nondominated subset is the front of Product 3 and the procedure is repeated until the entire population has been sorted into the appropriate level, that is, the Pareto frontier of Product 1.
Note, however, that for working times lower than 1.76 s the Pareto frontier of Product 1 dominates the frontier of Product 2. That is, higher benefits are expected to be achieved in the assembly line for Product 1.
Additionally, for the twenty examples analyzed in this work, a normal distribution function of the cost
Benefits obtained for Product 1 and “example 4” versus costs.
Benefits obtained for Product 1 and “example 4” versus price.
Figure
Figure
For the sake of conciseness, the cost and price fluctuations have been considered only for Product 1. Its current market values are
These functions are sampled, with these values being used in the multiobjective optimization problem to obtain the results presented in Figures
These figures show a hypothetic fluctuation in the market conditions with regard to costs and prices for case 4.
The fluctuations can be directly translated into benefits using the algorithm, thus allowing managers in the decision making process regarding which products should be manufactured, and also to define an efficient scheduling. Therefore, the design and planning of the robot tasks are considerably improved.
Finally, the algorithm has also been run for example 4 to simulate an increase in the future demand of a 20% compared with the current demand (i.e., a total of 137,778 products manufactured per year). The aim is to answer the question about how many extra working hours are needed to respond to that increase in the demand. The best solution is given by the optimization algorithm, since it reports the minimum trajectory time. The solution is that we need an additional 3,504 hours per year to meet such demand. This information can be used during the decision making process to design an efficient scheduling.
This work deals with trajectory planning of industrial robots for assembly lines in a costefficient way, thus overcoming limitations of the economic analysis methods which are currently available. It has been demonstrated that the multiobjective optimization algorithm finds the minimum time trajectory of industrial robots and the maximum annual revenue. This means greater annual revenue and better adaptation to market fluctuations in terms of costs, prices, and product demands. This is carried out by taking under consideration the physical constraints of the real working problem and the economic issues involved in the process. The proposed procedure has been successfully validated in different examples of robotic industrial tasks, where a better planning and design of production lines have been found.
The results from different examples have been compared using two methodologies, an optimization procedure and a simulation technique.
We have checked that the results obtained with the optimization procedure lead to lower working times and therefore greater annual revenues in comparison with those obtained with the simulation technique. Consequently the number of products manufactured and profits are increased while the number of the shifts required is reduced. The core of this paper is the procedure to obtain the best working times in real complex industrial robots with many degrees of freedom and mechanical constraints.
This has shown the worth of the methodology, with the overall objective of improving the profitability of production lines by designing flexible manufacturing systems. Furthermore, an entire set of equally optimal solutions for each process, the Paretooptimal sets, are generated.
This provides information about the tradeoffs between the different competing variables of the multiobjective optimization problem (i.e., working times and profits for the different products that can be manufactured at the production line).
Once the optimal time to perform each process is obtained in a costeffective manner the results can be used for improving a wide variety of robotic industrial tasks. This can help managers in the decision making process regarding which products should be manufactured and to define an efficient scheduling to produce them. This is because it allows adjusting the number of shifts needed according to the existing demand of the products manufactured. Then companies may stay competitive because the algorithm allows a quick adaptation to rapidly changing markets.
As a further research this methodology will be extended to deal with new decision variables in the multiobjective optimization problem such as the energy consumed and time simultaneously since they are conflicting variables.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This paper has been possible due to the funding from the Science and Innovation Ministry of the Spanish Government by means of the Researching and Technologic Development Project DPI201020814C0201 (IDEMOV).