A Review on Energy-Saving Optimization Methods for Robotic and Automatic Systems <s> Point-to-Point Trajectory Optimization <s> The motion profile for point-to-point control under acceleration, which is applied to many applications in electro-mechanical systems, should be planned so as to maximize the energy conversion efficiency. Initially, a theoretically ideal profile was derived by adopting Pontryagin's minimum principle. The performance of this energy efficient motion profile is compared to those of three conventional profiles widely used in practice: trapezoidal, variable-rate transymmetric, and exponential. Finally, a selection guide for motion profile is presented for applications in which energy conversion efficiency and peak acceleration become critical, such as the material handling machine, copying machine, various dedicated purpose machines, electric motor cars, and space vehicles. <s> BIB001 </s> A Review on Energy-Saving Optimization Methods for Robotic and Automatic Systems <s> Point-to-Point Trajectory Optimization <s> This paper proposes a technique of iterative dynamic programming to plan minimum energy consumption trajectories for robotic manipulators. The dynamic programming method is modified to perform a series of dynamic programming passes over a small reconfigurable grid covering only a portion of the solution space at any one pass. Although strictly no longer a global optimization process, this iterative approach retains the ability to avoid some poor local minima while avoiding the curse of dimensionality associated with a pure dynamic programming approach. The algorithm has an inherent parallel structure, allowing for reduced computation time on parallel architecture computers. No limiting assumptions are made about the performance index, or function to be optimized. As such, extremely complex functions and constraints are easily handled. Joint actuator and time constraints are considered in this work. The modified dynamic programming approach is verified experimentally by planning and executing a minimum energy consumption path for a Reis V15 industrial manipulator. <s> BIB002 </s> A Review on Energy-Saving Optimization Methods for Robotic and Automatic Systems <s> Point-to-Point Trajectory Optimization <s> This paper considers the control problem of a robotic manipulator with separately excited dc motor drives as actuators. An innovative method is proposed which achieves robot speed-control requirements, with simultaneous minimization of total electromechanical losses, while the drives follow the desired speed profiles of the robot joints under various loads and random load disturbances. If there is no demand for a specific speed profile, the optimal speed trajectory is determined by minimizing an electromechanical losses criterion. Controllable energy losses, such as armature copper losses, armature iron losses, field copper losses, stray load losses, brush load losses, friction and windage losses, can be expressed proportionally to the squares of the armature and the field (exciting) currents, the angular velocity and the magnetic field flux. The controllable energy loss term is also included in the optimal control integral quadratic performance index, defined for the whole operation period. Thus the appropriate control signals required for following the desired trajectory by simultaneous energy loss minimization for the whole operation interval are achieved. Two case studies of optimal robot control with and without minimization of actuator energy losses are presented and compared, showing the energy savings that can be achieved by the proposed methodology. <s> BIB003 </s> A Review on Energy-Saving Optimization Methods for Robotic and Automatic Systems <s> Point-to-Point Trajectory Optimization <s> An optimized path-planning approach is presented for flexible dynamic systems. Feedforward command profiles are determined for rest-to-rest large angle maneuvers that keep vibrations to a minimum. The performance index included both minimum energy and minimum time optimization problems. The feedfoward command profiles were generated by solving a discrete-time optimal control problem via discrete dynamic programming (DDP). A simple planar two-link flexible robot arm was utilized to demonstrate the DDP path-planning scheme. Numerical simulation results included: 1) minimum effort, 2) minimum effort with bounds, 3) minimum time, and 4) minimum torque-rate or power, respectively. The DDP approach demonstrated an effective alternative for numerical optimization, to generate optimal feedforward trajectory profiles. <s> BIB004 </s> A Review on Energy-Saving Optimization Methods for Robotic and Automatic Systems <s> Point-to-Point Trajectory Optimization <s> In this paper, the optimal time-energy trajectory planning problem for a three degrees-of-freedom planar manipulator is considered using a multi-level fuzzy-neural system. First, a neural network is achieved to solve the inverse kinematics problem (IKP). After pre-processing steps characterizing the minimum time trajectory and the corresponding torques, a fuzzy-neuro controller is developed to deal with the minimum energy trajectory planning. The basic function of this controller is to select the minimum energy link actuator torques allowing the robot end-effector to move from a starting point S to a target point T. This optimization is performed under actuators and workspace constraints. It starts with a fuzzy clustering of input-output data, uses Tsukamoto fuzzy inference and back-propagation algorithm to train the resulting adaptive neural network. The premises parameters (antecedent membership functions parameters) as well as rule-consequence parameters are then learned and optimized, giving the optimal energy torques. Simulation results are provided to prove the efficiency of the proposal. <s> BIB005 </s> A Review on Energy-Saving Optimization Methods for Robotic and Automatic Systems <s> Point-to-Point Trajectory Optimization <s> Optimal trajectory planning for robot manipulators is a very important issue in the research field of robotics. Many applications require smooth trajectories and the minimization of a performance index, usually the traveling time or the mechanical energy of the actuators. This paper presents a novel method that uses eighth-degree polynomial functions to generate smooth trajectories for the parametric representation of a given path. The optimization algorithm presented in this paper minimizes the mechanical energy consumed in the robot manipulator. To solve the optimization model, a genetic algorithm is implemented. A software platform has been developed to test this optimal trajectory-planning algorithm. The software includes modules to solve the direct kinematics, the inverse kinematics, and the dynamics of the robot manipulator. <s> BIB006 </s> A Review on Energy-Saving Optimization Methods for Robotic and Automatic Systems <s> Point-to-Point Trajectory Optimization <s> A new method of minimum-energy point-to-point (PTP) trajectory planning is proposed for an LCD glass-handing robot, which is driven by a permanent magnet synchronous motor (PMSM). The variable structure controller (VSC) is designed to track the trajectories and compensate LuGre model of frictional torque effects. The robot system is described by a mechanical equation and an electrical equation. To generate the minimum-energy trajectory, we employ a high-degree polynomial with suitable end-point conditions. The real-coded genetic algorithm (RGA) is used to search for the coefficients of the polynomial with the fitness function of minimum-energy input. Finally, numerical simulations of the minimum-energy inputs are compared for various degrees of polynomials. It is concluded that the proposed methodology can effectively design the minimum-energy trajectory, and the nonlinear VSC can track the designed trajectories for the robot system driven by a PMSM. <s> BIB007 </s> A Review on Energy-Saving Optimization Methods for Robotic and Automatic Systems <s> Point-to-Point Trajectory Optimization <s> The main objective of this paper is to propose a novel minimum-energy trajectory planning for an LCD glass-handing robot, which is driven by a permanent magnet synchronous motor (PMSM). The system is described by a mathematical model of the electromechanical system. To generate the minimum-energy trajectory, we employ a high-degree polynomial with suitable conditions of angular displacement, speed, acceleration and jerk at the start and end points. The real-coded genetic algorithm (RGA) is used to search for the minimum-energy point-to-point (PTP) trajectory for the motion profile. The coefficients of the polynomial are determined by the RGA with the fitness function of minimum-energy input. The results are compared for various degrees of polynomials and to obtain the minimum energy input. It is concluded that the proposed methodology with minimum-energy input can also be applied to any electromechanical system driven by a PMSM. <s> BIB008 </s> A Review on Energy-Saving Optimization Methods for Robotic and Automatic Systems <s> Point-to-Point Trajectory Optimization <s> In this paper, we propose a novel trajectory planning method for a precision positioning table driven by a permanent magnet synchronous motor (PMSM). To generate the mechatronic trajectory we develop a real-coded genetic algorithm (RGA) to search for the optimal trajectory for the motion. In this method, we design a high-degree polynomial to compute minimum-energy trajectories, and consider the problem of initial and final constraints in the algorithm. The RGA determines the parameters, which are to formulate the polynomial, and satisfy the desired fitness function of the minimum input energy. The adaptive control (AC) are developed based on mathematical knowledge of the precision positioning table and the PMSM. The proposed adaptive control is robust not only to parametric uncertainties of mass variations but also to external disturbances. The effectiveness of the proposed trajectory planning and adaptive control are demonstrated by numerical simulations in the point to point motion processes for the precision positioning table. <s> BIB009 </s> A Review on Energy-Saving Optimization Methods for Robotic and Automatic Systems <s> Point-to-Point Trajectory Optimization <s> In this paper, we propose a novel minimum-energy point-to-point (PTP) trajectory planning method for a simple mechatronic system, which is a one-dimensional precision positioning table driven by a permanent magnet synchronous motor (PMSM). To generate the mechatronic trajectory we develop a real-coded genetic algorithm (RGA) to search for the optimal trajectory for the PTP motion. In this method, we design a high-degree polynomial to compute minimum-energy point-to-point trajectories, and consider the problem of initial and final constraints (position, velocity, acceleration and jerk) in the algorithm. The RGA determines the parameters, which are to formulate the polynomial, and satisfy the desired fitness function of the minimum input energy. The algorithm is tested in numerical simulations results and yields good results for the comparisons between the seven-degree to twelve-degree polynomial during motion operations. <s> BIB010 </s> A Review on Energy-Saving Optimization Methods for Robotic and Automatic Systems <s> Point-to-Point Trajectory Optimization <s> In this paper, an minimum-energy point-to-point (PTP) trajectory is designed for a mechatronic system, which is a toggle mechanism driven by a permanent magnet synchronous motor (PMSM). To generate the PTP trajectory, we employ the real-coded genetic algorithm (RGA) to search for the minimum-absolute-energy trajectory for the mechatronic system. In this paper, a high-degree polynomial is used and the initial and final conditions (position, velocity, acceleration and jerk) are considered as the constraints for the trajectory. In the RGA method, the coefficients of the polynomial are determined by satisfying the desired fitness function of input absolute electric energy. The numerical simulations are compared between several degrees' polynomials during motion operation time for the toggle mechanism. <s> BIB011 </s> A Review on Energy-Saving Optimization Methods for Robotic and Automatic Systems <s> Point-to-Point Trajectory Optimization <s> In this paper, we propose a novel minimum-energy point-to-point (PTP) trajectory planning method for a motor-toggle servomechanism, which is described by the mathematical model of a mechatronic system. To generate the mechatronic trajectory, we employed the real-coded genetic algorithm (RGA) to search for the minimum-energy trajectory for the PTP motion profile, which is described by a polynomial with suitable conditions of position, velocity, and acceleration at the start and end points. The RGA algorithm determines the coefficients of polynomials with the fitness function of minimum-energy input, and is performed in numerical simulations and experiments. The results are compared with those of the S-curve by a traditional proportional-integral control. The proposed methodology can also be applied to any mechatronic system driven by a motor. <s> BIB012 </s> A Review on Energy-Saving Optimization Methods for Robotic and Automatic Systems <s> Point-to-Point Trajectory Optimization <s> This paper proposes a novel cost function formulation for minimization of the energy consumption of industrial robots by trajectory optimization. Besides the dynamics of the robot including friction losses, the model especially takes into account the losses of servo drives and inverters. Furthermore, the ability of energy exchange between the robot axes via the coupled DC-bus is included, since the servo drives support generator mode during deceleration. The utilized energy-based robot model is applicable to different manipulator types. For the energy-efficient motion planning, point-to-point trajectories are defined by B-spline functions. The given nonlinear optimization problem is solved using gradient-based methods, considering kinematic and dynamic constraints. Several simulation results are presented, demonstrating the intense effect of energy exchange in the robot controller's power electronics. Furthermore, a comparative study is given showing that the proposed method is able to outperform existing torque-based approaches. <s> BIB013 </s> A Review on Energy-Saving Optimization Methods for Robotic and Automatic Systems <s> Point-to-Point Trajectory Optimization <s> This paper proposes a minimum energy trajectory planning method with residual vibration reduction for a robot manipulator mounted on flexible base, in which a point-to-point motion task of the manipulator is considered. In the proposed method, the joint angle of the robot manipulator is generated by radial basis function networks (RBFNs). The maximum residual vibration amplitude and operating energy are adopted as the objective functions, and then, the RBFNs are tuned by an elitist non-dominated sorting genetic algorithm (NSGA-II). The trajectory obtained using the proposed method can suppress the residual vibration of the flexible base in energy conservation. Results obtained from simulations reveal a trade-off relationship between the residual vibration amplitude and the operating energy. Furthermore, the validity of the proposed method is confirmed experimentally. <s> BIB014 </s> A Review on Energy-Saving Optimization Methods for Robotic and Automatic Systems <s> Point-to-Point Trajectory Optimization <s> The improvement of energy efficiency in automation and robotic systems still has high significance in engineering research. This paper presents experimental results of minimum energy trajectory optimization for multi-axis manipulators using electrical energy exchange via the shared inverter DC link. Thus, existing simulation findings demonstrating the potential of the proposed minimum energy optimization approach are validated. In this context, a global optimization method is applied utilizing an advanced cost function formulation that is based on an energy flow model including all servo drive components. After the validation of the utilized system model, increased efficiency improvements are proven and compared to established methods. The presented approach as well as the experimental results are fully transferable to any kind of multi-axis system comprising DC link energy supply. <s> BIB015 </s> A Review on Energy-Saving Optimization Methods for Robotic and Automatic Systems <s> Point-to-Point Trajectory Optimization <s> Abstract The elevator system driven by a permanent magnet synchronous motor (PMSM) is studied in this paper. The mathematical model of the elevator system includes the electrical and mechanical equations, and the dimensionless forms are derived for the purpose of practicable upward and downward movement. In this paper, the trapezoidal, cycloidal, five-degree (5-D) and seven-degree (7-D) polynomial and industry trajectories are designed and compared numerically in various motion and the absolute input energies. From numerical simulations, it is found that the trapezoidal trajectory consumes the minimum energy; the 7-D polynomial trajectory consumes the maximum one. The less end-point constraints are required, the less energy is consumed. Finally, the proposed sliding mode controller (SMC) is employed to demonstrate the robustness and well tracking control performance numerically. <s> BIB016 </s> A Review on Energy-Saving Optimization Methods for Robotic and Automatic Systems <s> Point-to-Point Trajectory Optimization <s> The energy demand of applications using in- verter controlled electrical servo drives is considerably de- pendent on the applied trajectory profiles. In this paper, the improvement potential in supply energy consumption is an- alyzed with respect to the standard servo controller func- tionality. The optimal choice of trajectory profiles and ex- ecution times for point-to-point movements offers distinct energy savings. From the presented results, a number of general trajectory parametrization advices of even roughly known system characteristics can be derived. Even with lit- tle effort and without knowledge of exact system parameters, considerable energy savings are achievable. <s> BIB017 </s> A Review on Energy-Saving Optimization Methods for Robotic and Automatic Systems <s> Point-to-Point Trajectory Optimization <s> Analyzing the mechatronic behavior of a manufacturing cell used in a customized manufacturing process is a difficult task with numerous obstacles. Therefore, a method that can be easily used for developing and optimizing a customized manufacturing cell, i.e., universal contacting module (UCM) cell, for the in-circuit testing of electronic modules is desired. In this paper, we present a convenient method using multi-domain simulation tools for analyzing the mechatronic behavior of the UCM cell. The UCM cell, which consists of mechatronic components such as a six-axis industrial robot and conveyor systems, were successfully modeled, simulated, and validated under several payloads. This work also presents a modeling procedure that can be applied by system engineers with a basic background in control systems for analyzing the mechatronic behavior of manufacturing cell components. <s> BIB018 </s> A Review on Energy-Saving Optimization Methods for Robotic and Automatic Systems <s> Point-to-Point Trajectory Optimization <s> Abstract The main idea of this paper is to design a novel point-to-point (PTP) trajectory based on minimum absolute input energy (MAIE) for an LCD glass-handing robot, which is driven by a permanent magnet synchronous motor (PMSM). The mechatronic system is described by a mathematical model of electrical and mechanical coupling equations. To generate the MAIE PTP trajectory, we employ a high-degree polynomial and compare with the trapezoidal, cycloidal and zero-jerk trajectories for various constraint conditions, which satisfy their corresponding desired constraints of angular displacement, speed, acceleration and jerk at the start and end times. The real-coded genetic algorithm (RGA) is used to search for the coefficients of high-degree polynomials for the PTP trajectories, and the inverse of absolute input electrical energy is adopted as a fitness function. From numerical simulations, it is found that either increasing the degree number of polynomials or decreasing the constraints at the start and end times will decrease the absolute input electrical energy. The proposed methodology for designing the MAIE PTP trajectory can also be applied to any mechatronic system driven by a PMSM. <s> BIB019 </s> A Review on Energy-Saving Optimization Methods for Robotic and Automatic Systems <s> Point-to-Point Trajectory Optimization <s> Abstract In this paper, energy-saving trajectories are planned for a toggle mechanism driven by a permanent magnet synchronous motor (PMSM). The point-to-point (PTP) trajectory is described by a high-degree polynomial, which satisfies the end conditions of displacement, velocity, acceleration and jerk at the initial and final times. The real-coded genetic algorithm (RGA) method is employed to obtain the mechatronic system’s parameters, and to determine the coefficients of the polynomial, and its fitness function is the inverse of various input energies. The numerical simulations and experiments are compared among several degrees’ polynomials during the whole operation motion. Finally, it is found that the input absolute electrical energy (IAEE) occurs when the highest-degree polynomial is chosen. From the percentage of relative error with respect to the absolute input energy of a 7-degree polynomial, it is found that the percentage of relative error achieves 32% as the degree is 24 with enough for minimum input energy degree. The proposed methodology described in this research has the contribution. It can be applied to any mechatronic system which is required to design a minimum-energy point to point trajectory. <s> BIB020 </s> A Review on Energy-Saving Optimization Methods for Robotic and Automatic Systems <s> Point-to-Point Trajectory Optimization <s> Reducing the energy consumption of industrial robots (IR) that are used in manufacturing systems has become a main focus in the development of green production systems. This is due to the reality that almost all automated manufacturing processes are using IR as the main component. Thus, reducing the energy consumption of IR will automatically reduce operating costs and CO2 emissions. Therefore, a method for reducing the energy consumption of IR in manufacturing systems is desired. Firstly, this paper presents a literature survey of the research in energy consumption analysis of IR that is used in manufacturing processes. The survey found that current research in this field is focused on the development of simulation models of IR that are able to be used to predict its energy consumption. Secondly, a modular model to analyze power consumption and dynamic behavior of IR is developed. Afterward, an experimental investigation is carried out to validate and estimate the accuracy of the model developed. The investigation shows that the developed modular model can be conveniently used to optimize the industrial robot’s operating parameters, which are commonly needed for production planning and at the process optimization stage. In addition, the investigation shows that the process constraints, environment layout, productivity requirement, as well as the robot payload and operating speed are the key factors that must be considered for optimizing the productivity and efficiency of IR. <s> BIB021 </s> A Review on Energy-Saving Optimization Methods for Robotic and Automatic Systems <s> Point-to-Point Trajectory Optimization <s> This paper considers real-time energy-optimal trajectory generation for a servomotor system that performs a single-axis point-to-point positioning task for a fixed time interval. The servomotor system is subject to acceleration and speed constraints. The trajectory generation is formulated as a linear-constrained optimal control problem (LCOCP), and the Pontryagin's maximum principle is applied to derive necessary optimality conditions. Instead of solving multipoint boundary value problems directly, this paper proposes a novel real-time algorithm based on two realizations: solving the LCOCP is equivalent to determining an optimal time interval of the speed-constrained arc and solving a specific acceleration-constrained optimal control problem (ACOCP), and solving an ACOCP is equivalent to determining optimal switch times of acceleration-constrained arcs and solving a specific two-point boundary value problem (TBVP). The proposed algorithm constructs sequences of time intervals, ACOCPs, switch times, and TBVPs, such that all sequences converge to their counterparts of an optimal solution of the LCOCP. Numerical simulation verifies that the proposed algorithm is capable of generating energy-optimal trajectories in real time. Experiments validate that the use of energy-optimal trajectories as references in a servomotor system does not compromise tracking performance but leads to considerable less energy consumption. <s> BIB022 </s> A Review on Energy-Saving Optimization Methods for Robotic and Automatic Systems <s> Point-to-Point Trajectory Optimization <s> Abstract Among the techniques to improve energy efficiency of mechatronic systems, the synthesis of optimized motion profiles has been proved to be an effective and almost costless method. In order to boost the use of this approach, this paper proposes an analytical method for improving energy efficiency in rest-to-rest motion through optimal planning, by selecting both the optimal motion law and the optimal motion time (duration). The proposed technique is suitable for constant inertia systems. As a matter of fact, this kind of mechatronic systems covers a wide range of applications in production, packaging or logistic plants. The method practical application is straightforward, since it just relies on the knowledge of the task specifications and of some system parameters to compute the optimal motion time for each selected trajectory and to compare analytically different motion profiles. The limitations due the servo-actuator (torque, speed, bandwidth), to machine throughput requirements, as well as the smoothness specifications (acceleration and jerk limitations, degree of continuity) are explicitly accounted for through bounds, to ensure the feasibility of the motion profiles developed. Since the proposed method proposes an analytical algebraic equation, it does not require either time-consuming simulations and trial-and-error evaluations, or numerical optimizations. Hence, it is well suited for industrial mechatronic applications. <s> BIB023 </s> A Review on Energy-Saving Optimization Methods for Robotic and Automatic Systems <s> Point-to-Point Trajectory Optimization <s> Abstract This paper presents a proposal for a minimum energy feedforward control technique for flexible structures to suppress residual vibrations in point-to-point (PTP) motion. In the proposed method, the trajectory profile of the PTP motion is generated through a cycloidal function whose input is the output of a polynomial function. The obtained trajectory is dependent upon the coefficients of the polynomial function. To achieve the suppression of the residual vibration as well as the operating energy of this PTP motion, the coefficients are tuned by metaheuristic algorithms. In the numerical simulations, we investigated the PTP motions of a single-link flexible manipulator and a robotic arm attached to a flexible link. The simulation results were compared with those of previous studies, revealing the effectiveness of the proposed method. <s> BIB024 </s> A Review on Energy-Saving Optimization Methods for Robotic and Automatic Systems <s> Point-to-Point Trajectory Optimization <s> Abstract: This paper presents an optimization technique to develop minimum energy consumption trajectories for redundant/hyper-redundant manipulators with pre-defined kinematic and dynamic constraints. The optimization technique presents and combines two novel methods for trajectory optimization. In the first method, the system’s kinematic and dynamics constraints are handled in a sequential manner within the cost function to avoid running the inverse dynamics when the constraints are not satisfied. Thus, the complexity and computational effort of the optimization algorithm is significantly reduced. For the second method, a novel virtual link concept is introduced to replace all the redundant links to eliminate physical impossible configurations before running the inverse dynamic model for the trajectory optimization. The method is verified on a three-degree-of-freedom redundant manipulator. <s> BIB025
In accord with BIB023 , PTP trajectory optimization methods can be classified into inverse and direct approaches. In the inverse approach , the mechatronic and robotic system is formulated through a mathematical model, generally simplified in order to reduce the computational effort. The trajectory, usually expressed as a polynomial function, is then synthesized taking into account the energy content and is minimized in an optimization. On the contrary, the direct approach  BIB016 BIB017 BIB023  BIB021 BIB018 ] models the mechatronic and robotic system in a more complete way without neglecting the contribution of secondary loss terms (e.g., motor losses, motor-drive losses, auxiliaries losses, etc.) and taking into account also the non-ideal behavior of losses, as some of these are motion parameter-dependent. The model is then exploited to test different motion planning strategies (i.e., different pre-defined motion profiles) and to assess the impact of different parameters on the energy consumption through a sensitivity analysis. Both the approaches have been applied to various electro-mechanical systems. In principle, they could be applied to systems with a different level of complexity and number of degrees of freedom. However, the direct approach is generally applied only on test cases with a low number of degrees of freedom: simple one-degree-of-freedom systems (e.g., systems driving a constant inertia such as an elevator BIB016 BIB023 , or a generic system BIB017 ) or two-degrees-of-freedom systems (e.g., storage and retrieval vehicles   ). This is because the direct approach involves a higher computational effort. However, an exception is given by the  BIB021 BIB018 works, for which an extended model of a six-degrees-of-freedom anthropomorphic industrial manipulator is created, exploiting a model-based simulation tool. Indeed, the inverse approach has been adopted on a broad variety of systems with one degree of freedom-such as a liquid-crystal display glass-handling robot BIB007 BIB008 BIB019 ; a toggle mechanism BIB011 BIB020 ; a precision positioning table BIB009 BIB010 BIB012 ; a linear axis  ; generic, one-degree-of-freedom systems BIB001 BIB003 BIB022 ; and one-degree-of-freedom manipulators mounted on a flexible base BIB014 BIB024 -with two degrees of freedom with rigid BIB015 and flexible links BIB004 ; with three degrees of freedom, such as a redundant planar manipulator BIB025 BIB005 ; and with six degrees of freedom, such as an anthropomorphic industrial manipulator BIB006 BIB002 BIB013 . Considering the inverse approach, the mathematical model is simplified to reduce the computational effort for calculating the optimal trajectory. Usually, only the kinematic and dynamic properties (e.g., length, mass and inertia of links, gear ratios, friction coefficients, etc.) of the mechanical system and electrical parameters (e.g., motor winding resistance, motor torque and voltage constants) are taken into account and the linear behavior is modeled. Thus, the mechanical model can be expressed as where ϑ is the vector of the independent coordinates; τ is the vector of the input control torque; J(ϑ) is the inertia matrix; H is a matrix that includes viscous friction, Coriolis and centrifugal terms; and G is a matrix that includes the gravitational and the Coulomb friction terms. The electrical equations of DC motor drivers at the jth axis are usually written as follows: where e j is the supply voltage; i j is the current flowing in the winding; R j and L j are the resistance and the inductance of the motor, respectively; K t,j is the motor constant, and K e,j is the motor velocity constant. An exception to this common way to operate is presented in BIB013 , where the losses of servo drives and inverters are taken into account as the bus-DC sharing is considered. After the model development, the second step of the inverse approach is the definition of the trajectory type and the parameters to be optimized, that is, design variables. Often, for serial robots, the trajectory is formulated in the joint space exploiting either a high-degree polynomial BIB006 BIB007 BIB008 BIB011 BIB019 BIB020 BIB009 BIB010 BIB012 or a B-spline formulation BIB025 BIB013 BIB015 . In BIB002 , the phase-space representation (i.e., joint velocity versus joint angle interpolation with B-splines for each joint) of the trajectory between two points is considered, while in BIB014 BIB024  and in BIB001 BIB003 BIB022 , a cycloidal trajectory and a modified parabolic velocity profile are adopted, respectively. When a high-degree polynomial formulation is chosen, the joint angular displacement is expressed as where each coefficient a i (with i = 0, 1, ..., n; n being the degree of the polynomial) is a real number and a n = 0. Under this condition, n + 1 coefficients are to be determined. In a PTP motion, the initial and final position, speed, and acceleration are usually imposed. Therefore, at least a fifth-degree polynomial should be defined to satisfy all the constraints. In order to obtain a smoother motion and to apply an optimization process, a polynomial with a degree higher with respect to the number of imposed values has to be chosen. However, increasing too greatly the degree of the polynomial does not lead to significant increment in saved energy, as shown in BIB019 . When a B-spline formulation of joint-space profiles for serial robots is considered for trajectory optimization, the joint position profile ϑ i (t) of the ith axis is expressed as where N j,d (t, k i ) are the basis functions and c i,j are the m + 1 control points BIB013 . The next step of the approach is to define a function to be minimized, referred to as the cost or objective function. There have been several function formulations proposed in literature; these include the following: • minimum effort (i.e., sum of squared torque) BIB025 BIB004 ; • minimum torque-rate (i.e., sum of squared-derivative torque) [ 