Energy Minimization in Time-Constrained Robotic Tasks via Sequential Quadratic Programming Marco Faroni, Domenico Gorni, Antonio Visioli To cite this version: Marco Faroni, Domenico Gorni, Antonio Visioli. Energy Minimization in Time-Constrained Robotic Tasks via Sequential Quadratic Programming. IEEE International Conference on Emerging Technolo- gies for Factory Automation 2018, Dec 2018, Turin, Italy. hal-03046893 HAL Id: hal-03046893 https://hal.archives-ouvertes.fr/hal-03046893 Submitted on 8 Dec 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Energy Minimization in Time-Constrained Robotic Tasks via Sequential Quadratic Programming Marco Faroni Domenico Gorni Antonio Visioli Dipartimento di Ingegneria Dipartimento di Ingegneria Dipartimento di Ingegneria Meccanica e Industriale Meccanica e Industriale Meccanica e Industriale University of Brescia University of Brescia University of Brescia Brescia, Italy Brescia, Italy Brescia, Italy Email: [email protected] Email: [email protected] Email: [email protected] Abstract— Reduction of the energy consumption in robotized joining methods are gluing, arc welding and stud welding. In processes is a key issue in nowadays manufacturing. In this paper, this kind of plants hundreds of robots that work unceasingly we propose a simple approach to energy minimization of robotic are involved [6], [7]. These data point out that, in order to tasks with assigned cycle time based on sequential quadratic programming. The method aims at re-shaping a given timing law satisfy economic efficiency criteria, industries are more and in the sense of energy saving, without modifying the desired path more interested in robot with lower energy consumption for and the given cycle time. Thanks to the iterative linearization of comparable process time and accuracy. the nonlinear time-constraint, the resulting minimization problem The problem of computing optimal energy consumption is solved by only using common quadratic programming solvers, paths and trajectories has been subject of research since the making the method suitable for a direct implementation in robot industrial controllers. At first, the method is devised by only late 1960s [8]. In the last years, many researches have been considering the kinematics of the manipulator. The dynamic carried out trying to solve it, however there is still the lack of model is then straightforwardly included, without significantly a simple methodology that can be easily implemented. First of increasing the complexity of the method. Validation in simulation all, the energy minimization problem can be addressed from environment is provided in order to show the effectiveness of the two different points of view based on the robotic application. methodology. Index Terms—Industrial robotics, energy minimization, trajec- The first approach concerns about finding optimal geometrical tory optimization, quadratic programming. paths which can globally decrease the total energy consump- tion. For instance, if the robot has to execute a task like pick and place, the only geometrical constrains are the starting I. INTRODUCTION and the ending positions (and a few intermediate waypoints Nowadays energy efficiency as well as the conservation of at most). Therefore, the geometrical path can be devised in energy and natural resources are key aspects in the economic such a way that the energy consumption is minimized. The sector. The target of reduction of CO2 emission is strictly second approach is followed in case of predefined geometrical correlated to the reduction of energy waste. In fact, companies paths. In this case, a decrease in the energy consumption belonging to many different sectors have been stimulated by can be obtained by acting on the timing law applied to the the European Energy Management Standard (EN 6001) to manipulator actuators by means of change in jerk, acceleration reduce and monitor their energy consumption. Furthermore, and velocity reference values. during the last years, the price of energy is continuously This paper deals with this second class of strategies. In increasing [1], [2]. Companies therefore react moving from the this framework, most of the proposed approaches set up reduction of production time to the optimal trade-off between a nonlinear optimization problem, which minimizes a cost energy consumption and time. Recent statistics show that one function proportional to the energy consumption and takes into of the major consumers of energy is actually the manufacturing account the manipulator limits (e.g., configuration, velocity, industry. acceleration and torque limits) in the constraints [4], [9], [10], During the last decades, the number of robots implemented [11]. However, these methodologies usually require heavy in manufacturing has exponentially increased, moving compa- computational burden and their implementation may often nies to optimize their energy consumption in order to reduce result to be cumbersome, as also pointed out in [12]. Heuristic the total amount of consumed energy [3]. For example, mea- approaches are also often adopted by practitioners [13], [14]. surements show that an average 200-kg-payload robot in body In this case, the timing law is devised by using smooth func- shop yearly consumes around 8 [MWh] and, obviously, the tions and by using an iterative approach to minimize velocity most part of this energy is spent during the movement [4], [5]. or acceleration peaks given the total task time and checking In a typical automotive car body, all the sheet metal parts are that the resulting motion satisfies the robot limits. These joined together by up to 4000 weld spots. Furthermore, other approaches are easily implementable, computationally light, and do not require optimization solvers. However, compliance q(t) can be defined by means of the path-velocity decomposi- with the robot limits can be checked only a posteriori, and the tion approach as the composition of the following functions: resulting timing law is suboptimal with respect to the original s : [0;t f ] ! [0;s f ]; t 7! s(t); problem. There is still the need of easily-implementable and (1) q : [0;s ] ! n; s 7! q(s); effective techniques which can attract practitioners and boost f R the industrial application of optimization-based methods. where s represents the Euclidean distance from the beginning of the path, and q is a vector that contains all the joint coor- This was also pointed out in [12], where a nonlinear T optimization approach was developed to minimize a simplified dinates respect to time q(t) = [q1(t);:::;qn(t)] . Furthermore energy index (namely, the squared sum of the joint accel- t f is the total traveling time and s f = s(t f ) is the total length erations). Such approach still requires the use of nonlinear of the path. As this paper deals with path tracking problems, solvers and does not take into account the dynamics of the we reasonably assume thats ˙(t) > 0 8 t 2]0;t f [ ands ˙(t) ≥ 0 in manipulator. Nevertheless, it gives very good results, with a the extreme points t = 0 and t = t f . reduction of the energy consumption in the order of 30%. For a given path the joints velocities and accelerations can be rewritten as a function of the defined scalar path coordinate In this paper, we therefore follow a similar approach, aiming trajectory, as follows: at tackling the need for nonlinear solvers in the resolution q˙(t) = q0(s(t))s˙(t); of the optimization problem and the lack of inclusion of (2) the dynamics of the robot in the problem. First of all, the q¨(t) = q0(s(t))s¨(t) + q00(s(t))s˙(t)2; energy minimization problem is set up in a different way with wheres ˙ = ds=dt,s ¨ = d2s=dt2, q0(s) = dq=ds, and q00(s) = respect to [12]: the energy consumption index is written as a d2q=ds2. quadratic cost function and all the robot limits are written as In case the dynamic model of the manipulator is not avail- linear constraints in the optimization variable, which is given able, it is possible to have an approximate measure of the by the curvilinear abscissa along the nominal path. The only energy consumption as a weighted sum of squared angular nonlinear constraint is therefore given by the imposition of the accelerations for all the joints [17]. Therefore the problem of total execution time of the task. The optimization problem is minimizing the energy consumption can be written as: then solved by means of a Sequential Quadratic Programming Z t f (SQP) approach [15], which consists in iteratively linearizing minimize q¨(t)T q¨(t)dt the nonlinear constraint around the solution obtained at the s 0 previous iteration. The proposed method therefore presents subject to s(0) = 0; the great advantage of requiring just the implementation of s(t f ) = s f Quadratic Programming (QP) techniques, which could also be s˙(0) = s˙0 (3) easily implemented on industrial robot controllers. Moreover, s˙(t ) = s˙ we address the inclusion of the dynamic model of the manip- f f ulator and we show that, by following the approach presented s˙(t) ≥ 0 in [16], this does not significantly increase the computational q˙ ≤ q˙(t) ≤ q˙ complexity of the problem. q¨ ≤ q¨(t) ≤ q¨ The paper is organized as follows. In Section II, the wheres ˙ ands ˙ are the initial and final velocities (typically basic energy-optimal time-constrained path tracking problem 0 f equal to zero), while q˙, q˙, q¨, and q¨ are, respectively, the lower is defined.