Path placement optimization of manipulators based on energy consumption: application to the orthoglide 3-axis
Raza Ur-Rehman, Stéphane Caro, Damien Chablat, Philippe Wenger
aa r X i v : . [ c s . R O ] O c t Path Placement Optimization of Manipulators Based on EnergyConsumption:Application to the Orthoglide 3-axis
Raza UR-REHMAN, St´ephane CARODamien CHABLAT, Philippe WENGER
Institut de Recherche en Communications et Cybern´etique de NantesUMR CNRS n ◦ { ur-rehman, caro, chablat, wenger } @irccyn.ec-nantes.fr Abstract
This paper deals with the optimal path placement for a manipulator based on energy con-sumption. It proposes a methodology to determine the optimal location of a given test pathwithin the workspace of a manipulator with minimal electric energy used by the actuatorswhile taking into account the geometric, kinematic and dynamic constraints. The proposedmethodology is applied to the Orthoglide 3-axis, a three-degree-of-freedom translational par-allel kinematic machine (PKM), as an illustrative example.
Keywords:
Path placement, Energy consumption, Optimization, Parallel manipulators.
Placement optimal de trajectoires de manipulateurs pour laminimisation de la consommation d’´energie :application `a l’Orthoglide 3-axesR´esum´e
L’objet de cet article est le placement optimal de trajectoires en utilisant la consommation´energ´etique comme crit`ere. Ce travail propose une mthodologie pour d´eterminer la locali-sation optimale de la trajectoire dans l’espace de travail d’un manipulateur en minimisantla consommation d’´energie ´electrique utilis´ee par les moteurs, tout en tenant compte descontraintes g´eom´etriques, cin´ematiques et dynamiques. La m´ethodologie propos´ee est ap-pliqu´ee `a l’Orthoglide 3-axes, manipulateur d’architecture parall`ele `a trois degr´es de libert´ede translation.
Mots cl´es:
Placement de trajectoires, Consommation ´energ´etique, Optimisation, Manipu-lateur parall`eles.
INTRODUCTION
Optimal trajectory planning of manipulators has been a relevant area for roboticists formany years. Indeed, several authors have worked on trajectory planning based on differentoptimization objectives. A review of trajectory planning techniques is given in [1]. Trajectoryplanning deals with the determination of the path and velocity/acceleration profiles (or thetime history of the robot’s joints), the start and end points of the trajectory being predefinedand fixed in the workspace. As a matter of fact, this approach is suitable for most of roboticapplications. A path is a continuous curve in the configuration space connecting the initialconfiguration of the manipulator to its final configuration [2]. Trajectory planning usuallyaims at minimizing the travel distance [3, 4], travel time [5, 6, 7] and/or the energy consumed[8, 9, 10, 11], while satisfying several geometric, kinematic and dynamic constraints.Another less explored aspect of trajectory planning is the placement of a given pathwithin the workspace. It aims at determining the optimum location of a predefined path tobe followed by the end-effector of the manipulator within its workspace with respect to oneor many given objective(s) and constraint(s). This path can be the shape of a componentto be machined, a welded profile or an artistic/decorative profile etc. In such situations,the trajectory planner cannot alter the shape of the path but he/she can only play with thelocation of that path within the workspace of the manipulator in order to optimize one orseveral criterion(a). Such an approach can be very interesting in many robotic applications.For example, in machining, a workpiece can be better located within the workspace of therobot to perform a given operation more efficiently with respect to the energy consumed.The path placement problem has not been extensively studied in the past. Nevertheless,some researchers proposed to solve it with respect to various optimization objectives. Severalperformance criteria for path location problems can be considered simultaneously (multiob-jective) or individually, such as travel time, different kinetostatic performance indices (manip-ulability or conditioning number), kinematic performance (velocity, acceleration), avoidingobstacles, reduced wear or vibration, energy consumption etc. In the following paragraphs,a brief survey of the work of different researchers to solve the problem of path placementoptimization for various applications is presented.Nelson and Donath [12] proposed an algorithm for the optimum location of an assemblytask in the manipulator workspace while taking the manipulability measure as the optimiza-tion criterion. They considered that the location of the assembly task within the workspacethat results in the highest manipulability is a locally optimal position for performing theassembly. However, Aspragathos [13, 14] considered that the manipulability index and thedexterity, usually quantified by the condition number of the Jacobian matrix of the manipu-lator, can characterize the motion ability of the manipulator, but these criteria cannot depictthe ability of a manipulator to move in a given direction. Hence they introduced a criterionto characterize the best velocity performance of the robot end-effector with the path location.They used the concept of the orientation of the manipulability ellipsoid relative to the desiredpath and used genetic algorithm to come up with an optimal solution.Fardanesh et al [15] proposed an approach for optimal positioning of a prescribed taskin the workspace of a 2 R -manipulator. Optimal location of the task is considered to be thelocation that yields the minimum cycle time for the task. In another study, Feddema [16]formulated and solved a problem of robot base placement for minimum time joint coordinated2otion within a work cell. The proposed algorithm considers only the kinematics and themaximum acceleration of each joint in order to obtain a 25% cycle time improvement for atypical example.Hemmerle [17] presented an algorithm for optimum path placement of a redundant ma-nipulator by defining a cost function related to robot joints motion and limits. The proposedapproach did not consider the path as a whole but points along the entire path, hence costfunction considers the performance only at the node points and not the path in-between thenodes.Chou and Sadler [18] developed an optimization technique for the optimum placement ofa robotic manipulator based on the actuators torque requirements. Pamanes and Zeghloul[19, 20] considered multiple kinematic indices to find the optimal placement of a manipulatorby specifying the path with a number of points and then assigning an optimization criterionto each point. The objective was to find the path location in order to have optimal values ofall the criteria assigned to the path’s points. In [21], the problem of optimal placement withjoint-limits and obstacle avoidance is addressed. Lately, a general formulation was presentedto determine the optimal location of a path for a redundant manipulator while dealing withmono- and multi-objective problems [22]. The goal of this research work was to keep thejoint variables within their limits and to minimize the magnitude of their displacements[23, 24, 25, 26, 27, 28, 29].With a general literature survey, it comes out that although several performance indicesare introduced or considered, there is very little emphasis on the dynamic aspects reflectingthe energy consumption by the robot actuators. The increasing number of the robotic ap-plications emphasize the importance of the energy saving not only to enhance efficiency butalso to face the world energy problems. Therefore, it is pertinent to locate well the path tobe followed by the end-effector of a robot within its workspace in order to minimize the en-ergy consumed by its actuators. Accordingly, the major contribution of this paper are i ) theminimization of the energy requirements by optimum path placement and ii ) use of electricenergy consumed by the actuators instead of treating mechanical energy relations. Hence,we propose an approach to optimize the location of a given path within the workspace of amanipulator in order to minimize the electric energy consumed by its actuators. It shouldbe noted that energy optimal poses can affect various performance indices such as manipula-bility, dexterity, stiffness etc. In the scope of this paper, we have only considered the electricenergy consumption as an optimization criterion (objective function) in order to highlightits influence on motion planning. However, others optimization criteria , such as the ma-nipulability, dexterity, stiffness, the motors torque required, will be taken into account infuture work. Accordingly, we will come up with multiobjective path placement optimizationproblem that may not be convex. For that matter, we will use other optimization tools suchas Genetic Algorithms to solve it.The paper is organized as follows. In section II, we propose a minimum energy pathplacement optimization problem that includes basic problem formulation, electric energycalculations and the algorithm proposed to solve the problem. In section III, the Orthoglide 3-axis: a three-degree-of-freedom translational Parallel Kinematics Machine (PKM), is used asa case study to implement the proposed optimization methodology and results are presentedfor rectangular test paths. 3 PATH PLACEMENT OPTIMIZATION
The problem aims at determining the optimal location of a predefined path in order to min-imize the electric energy used by the actuators. The optimization problem is subject togeometric, kinematics and dynamics constraints. Geometric constraints include joint limitsand the boundaries of the workspace. Kinematic constraints deal with the maximum actu-ator velocities whereas dynamic constraints are related to actuator wrenches. Contrary tothe trajectories usually defined with start and end point configurations, the entire path issupposed to be known within the framework of this research work. The path location can bedefined in a similar way as to define the location of a workpiece with respect to a manipulatorreference point.
In order to formulate and describe the problem, two reference frames are defined: i ) the pathframe F p and ii ) the base frame F b , as shown in Fig. 1. The path frame F p , is attachedto the given/required path at a suitable point such as geometric center of the path. As F p is attached to the path, the end-effector trajectory parameters remains constant in thisreference frame, no matter where it is located. In other words, the path is fully defined andconstant in F p . It can also be named workpiece frame as it characterizes the position andthe orientation of the workpiece within the manipulator workspace. The base frame F b canalso be called global or manipulator frame. It is attached to the manipulator base and isused to locate a workpiece (or F p ) with respect to the manipulator coordinate system. Thelocation and orientation of F p with respect to F b can be defined in such a way that thewhole path lies within the workspace. The position of F p with respect to F b is defined with x PSfrag replacements X b Y b Z b O b F b F b X p Y p Z p O p F p P p F p P ( a ) PSfrag replacements X b Y b Z b O X p Y p Z p u vw φφ ψψ θ θ ˙ φ ˙ ψ ˙ θ ( b ) Figure 1: (a). Path P to be followed by the end-effector P of a manipulator, F b and F p beingthe base and path frames, (b). Euler anglesthe Cartesian coordinates of the origin of F p and the relative orientation of the two framesis characterized by means of Euler angles. However, keeping in view the constraints of themanipulator wrist, Euler angles are uniquely defined in the context of milling operation withthe parameterization given in Fig. 1(b). It allows to avoid the singularity of Euler parameters.4s a matter of fact, any trajectory defined in F p can be transformed in the base frame F b by a transformation matrix. For instance, point P , of Cartesian coordinates x P p , y P p , z P p in F p can be expressed in F b as follows: (cid:2) p (cid:3) F b = b T p (cid:2) p (cid:3) F p (1)namely, (cid:2) x P b y P b z P b (cid:3) T F b = b T p (cid:2) x P p y P p z P p (cid:3) T F p (2) b T p being the transformation matrix from F p to F b . Let O p ( x O p , y O p , z O p ) be the originof the path frame expressed in frame F b and ( φ, θ, ψ ) the Euler angles characterizing theorientation of frame F p with respect to frame F b . Accordingly, b T p = cos φ cos θ cos φ sin θ sin ψ − sin φ cos ψ cos φ sin θ cos ψ + sin φ sin ψ x O p sin φ cos θ sin φ sin θ sin ψ + cos φ cos ψ sin φ sin θ cos ψ − cos φ sin ψ y O p − sin θ cos θ sin ψ cos θ cos ψ z O p (3)The path placement is specified with b T p . Let x = [ x O p y O p z O p φ θ ψ ] T define thepath placement within the workspace of the manipulator, in the reference frame F b . Conse-quently, the components of x are the decision variables of the optimization problem at hand.In the context of a general machining process like milling operation, the feature to be ma-chined on the workpiece is defined with respect to the frame attached to the workpiece,namely F p . Likewise, the machining operation conditions such as machining velocity andacceleration are fully defined in F p . Finally, the part to be machined is defined by the de-signer and located in F b whereas the machining operation conditions and robot trajectoryplanning are defined by the production engineer with respect to the the corresponding part,namely F p . Here, we introduce a methodology to help the production engineer well locatethe workpiece, namely F p , within the robot base frame F b in order to minimize the actuatorselectric energy consumption. The goal of this research work is to help the path planner find the best location of the pathto be followed by the robot in order to minimize the energy used by its actuators. It can beformulated as an optimization problem, namely,“
For a predefined path in F p , find the optimum location and orientation of F p with respectto F b , defined by the decision variables x , in order to minimize the electric energy used bythe manipulator actuators to generate that path, while respecting the geometric, kinematicand dynamic constraints of the manipulator. ” It can also be formulated mathematically asfollows: min x E t = n X i =1 E i ( x ) subject to: q il ≤ q i ≤ q iu | ˙ q i | ≤ ˙ q iu ( i = 1 · · · n ) | τ i | ≤ τ iu (4)5 is the path placement vector corresponding to the transformation matrix b T p . E t is the to-tal electric energy required by the n actuators whereas E i is the total electric energy requiredby the i th actuator to follow the path. q i , ˙ q i , τ i are respectively the i th actuator displacement,rate and torque. q il is the lower bound and q iu (resp. ˙ q iu and τ iu ) is the upper bound of i th actuator displacement (resp. rate and torque). For a given path placement vector x , theseconstraints can be evaluated by means of the manipulator kinematic, velocity and dynamicmodels.It is noteworthy that the manipulator geometric constraints guarantee that the whole pathlies inside the prescribed workspace. Similarly, the bounds on actuator rates ( ˙ q iu ) and torques( τ iu ) ensure that the manipulator will not go through any singular configuration while follow-ing the path. The electric energy used by the actuators is formulated in the next section. The energy used by the motors depends on their corresponding velocities and torques. Asa matter of fact, the electric current in the motors varies with motor velocities and torques.Accordingly, the motor’s self-inductance phenomenon appears. The current I drawn by themotors and the motor electromotive potential V e can be calculated as a function of therequired torque τ and the angular velocity ω of the actuators, namely, I = τK t (5) V e = K e ω (6) K t being the torque sensitivity factor or motor constant expressed in [Nm/A] and K e theback electromotive force constant expressed in [V.(rad/sec) − ].The total electric power P T is composed of [30]: • The resistive power loss (Joule effect): P J = RI (7) • the inductive power loss: P L = LI dIdt (8) • the power used to produce the electromotive force: P EM = V e I (9)Accordingly, the total electric power P T can be expressed as follows: P T = P J + P L + P EM (10) R being the motor winding resistance expressed in Ohm [Ω] and L the motor inductancecoefficient expressed in Henry [H]. 6inally, the energy E consumed by a motor can be evaluated by integrating P T over the totaltrajectory time T , namely, E = Z T P T dt (11) P T being the instantaneous electric power at time t , defined in Eq. (10).It should be noted that Eq. (5) allows us to consider the energy used by the actuators whenthey do not move but still produce a torque to keep the manipulator at a certain stationaryconfiguration (with respect to that particular direction or actuator), like resisting the gravity.It is noteworthy that energy calculation model presented in this section is suitable forthe brushless motors, which are generally used for PKMs. However, depending on the mo-tors/derives in application, energy calculation model can be developed accordingly. To solve the problem, a general optimization approach is proposed as illustrated in Fig. 2.The approach can be summarized in three constituent elements or phases:1. Preparation Phase: Manipulator geometric, dynamic and electric parameters along withthe definition of the required path are used as the known input data of the optimizationproblem. The base frame F b is defined. The path to be followed by the end-effectorof the robot is defined in the path frame F p . The terms of the transformation matrixbetween F p and F b are the decision variables of the optimization problem.2. Evaluation Phase: At this stage, the inverse kinematic model (IKM), the inverse ve-locity model (IVM) and the inverse dynamic model (IDM) of the manipulator aredetermined for each set of design parameters obtained from the optimization routine.Accordingly, the objective function and the constraints of the optimization problem areevaluated.3. Optimization Phase: The objective function and constraints evaluated at the previousstep are analyzed by means of the optimization algorithm. Once the constraints aresatisfied, the objective function is tested for its optimum value. The convergence criteriaare checked. If latter are respected, the optimization algorithm stops. Otherwise, otheriterations are run as long as the convergence criteria are not satisfied.Finally, the optimum path placement is obtained by means of the position and orientationof the origin of F p with respect to F b defined by the terms of the transformation matrixbetween F p and F b , namely, the location vector x ∗ =[ x ∗ O p y ∗ O p z ∗ O p φ ∗ θ ∗ ψ ∗ ] T . The Orthoglide is a Delta-type PKM [31] dedicated to 3-axis rapid machining applicationsdeveloped in IRCCyN [32]. It gathers the advantages of both serial and parallel kinematic7Sfrag replacementsPreparation PhaseGeometric & Dynamic ParametersDefinition of base frame F b Electric motors parametersPath definitionDefinition of Path frame F p Trajectory Definition in F p Initial Guess x x = [ x Op y Op z Op φ θ ψ ] T EvaluationPhaseTransformation Matrix b T p Trajectory Definition in F b Inverse Geometric ModelInverse Kinematic ModelInverse Dynamic ModelObjective and Constraints EvaluationOptimization Phase q il ≤ q i ≤ q iu | ˙ q i | ≤ ˙ q iu | τ i | ≤ τ iu E t = E min x j = x j +1 Optimum value x ∗ = x j YesYesNoNoIKM, IVM and IDM Figure 2: Flowchart of the path placement optimization process8rchitectures such as regular workspace, homogeneous performances, good dynamic perfor-mances and stiffness. The Orthoglide is composed of three identical legs, as shown in Fig. 3(a).Each leg is made up of a prismatic joint, two revolute joints and a parallelogram joint. Pris-matic joints of the legs, mounted orthogonally, are actuated which result the motion of themobile platform in the Cartesian space with fixed orientation.The Orthoglide 3-axis geometric parameters are function of the size of the prescribed cubic (a) Orthoglide (cour-tesy: CNRS Pho-toth´eque/CARLSONLeif) z10
PSfrag replacements X [mm] Y [mm] Z [ mm ] C O b Q + Q − X b Y b Z b -100-50050 -100 -50 0 50 -120-100-80-60-40-2002040 (b) Cubic workspace (200 × ×
200 mm ) Figure 3: A snap and workspace of the OrthoglideWorkspace size L workspace = 0 . F b [m] O b (0 , , C ( − . , − . , − . Q + (0 . , . , . Q − ( − . , − . , − . L workspace [33]. Thebase frame F b is defined with the unit vector e i in the direction of the i th prismatic joint,namely, X b , Y b and Z b , the origin O b of F b being the intersecting point of e i . Two points Q + and Q − are defined in such a way that the velocity transmission factor is 1 / Q + Q − as its diagonal. It should benoted that the cubic workspace center, i.e., point C , and the origin O b of the reference frame F b do not coincide, as shown in Fig. 3(b). In the scope of this study, L workspace is equal to0 . Q + , Q − and C for the said workspace are9iven in Table 1. Similarly, the prismatic actuator bounds, ρ min and ρ max , can be calculated[33]. Table 2 shows the lower and upper bounds of the prismatic joints displacements andtheir maximum allowable velocity and torque for the Orthoglide. The geometric, kinematicand dynamic parameters of the Orthoglide are defined in [32, 33, 34, 35].Electric energy E i used by each actuator is calculated by means of Eqs. (5) to (11). Asthe Orthoglide 3-axis has three synchronous servo motors ( ref erence : P B D ), Eq. (10) is multiplied by 3 to cater for the power consumed by the each phaseof the motor in order to calculate the electric power P T i used by each actuator, i.e., P T i = 3( RI + LI dIdt + V e I ) (12) ρ i min .
126 m ρ i max .
383 m v i max .
00 m.s − τ i max .
274 NmTable 2: Orthoglide actuators parameters ( i = x, y, z ) In order to apply the methodology proposed for path placement optimization, a rectangulartest path is considered. The test path is defined by the length L and the width W of therectangle, as shown in Fig. 4( a ). Path reference frame F p is located at the geometric centerof the rectangle. This type of path can be the example of the generation of a rectangularpocket like that of Fig. 4( b ). The position of F p in the base frame F b is defined with theCartesian coordinates of the origin of F p , O p ( x O p , y O p , z O p ) and the orientation of F p withrespect to F b is given by Euler’s angles, as depicted in Fig. 1( b ). For the sake of simplicity,only one of the three rotation angles is considered i.e, rotation about Z b -axis while X b Y b and X p Y p planes are considered to be always parallel. Accordingly, there are four path placementvariables, i.e., x O p , y O p , z O p and φ , as illustrated in Fig. 4. The magnitude of the end-effectorvelocity is supposed to be constant along the path. Hence, for given path dimensions, po-sition vector p F p = [ x P p y P p z P p ] T and velocity vector v F p = [ ˙ x P p ˙ y P p ˙ z P p ] T in thepath frame can be evaluated as a function of time. Figure 5 shows the position and velocityprofiles in F p for a 0 . × .
10 m rectangular path and for a constant end-effector velocityof 1 . − . Position and velocity vectors defined in F p can be expressed in F b by means ofthe transformation matrix defined in Eq. (3), namely, x P b y P b z P b = cos φ − sin φ x Op sin φ cos φ y Op z Op x P p y P p z P p
70 −60 −40 -20 0 20 40−40−20020406080
PSfrag replacements X b Y b O b p F b F b X p Y p O p P p F p F p A BCD L W φ P P ath P X b [mm] Y b [ mm ] ( a ) PSfrag replacements ( b ) Workpiece Rectangular Pocket
Figure 4: Rectangular test path ˙ x P b ˙ y P b ˙ z P b = cos φ − sin φ φ cos φ ˙ x P p ˙ y P p ˙ z P p with x = [ x O p y O p z O p φ ] T being the decision variables vector of the optimization prob-lem. For a matter of simplicity and not to deal with tangent and curvature discontinuities,we consider that the path is composed of four independent line segments. Therefore, we donot pay attention to the discontinuities between the segments.In order to analyze the effect of external cutting/machining forces in the generation of agiven path, a groove milling operation is considered as shown in Fig. 6, [36]. With constantfeed rate or end-effector velocity v p of magnitude 0 .
66 m.s − , i.e, 40 m.min − , the followingcomponents of cutting forces are considered: F f =component in the feed direction = 10 N F a =component along the axis of cutting tool = 25 N F r =component perpendicular to F f and F a = 215 N The path placement optimization problem for the Orthoglide 3-axis is formulated in orderto minimize the total electric energy used by the three prismatic actuators. The kinematic,velocity and dynamic models of the manipulator are used to evaluate the required actuatordisplacements, velocities and torques. The constraints of the optimization problem are thegeometric, kinematic and dynamic ones. It should be noted that within the prescribed11Sfrag replacements [ m ] End-effector position vs time in F p Time [sec] [ m . s − ] End-effector velocity vs time in F p z P p x P p y P p z P p ˙ z P p ˙ x P p ˙ y P p ˙ z P p -1-0.500.51-0.15-0.1-0.0500.050.1 Figure 5: Test trajectory for a rectangular path of size 0 . × . F p and theorientation angle of F p with respect to F b .The optimization problem can be formulated as follows,min x E t = n X i =1 E i ( x ) subject to: ρ min ≤ ρ x,y,z ≤ ρ max | v x,y,z | ≤ v max | τ x,y,z | ≤ τ max (13)where x = [ x Op , y Op , z Op , φ ] T . The subscripts x , y and z are used for three prismatic actuatorsor three Cartesian directions. ρ min and ρ max are respectively the minimum and maximumdisplacements of the prismatic joints as presented in Table 1The optimization problem was solved by using the MATLAB fmincon function, which is ageneral constrained optimization solver using the derivative-based search algorithms. Theoptimization process was performed with different starting points and it turned out thatMATLAB fmincon function always converges to the same solution no matter the startingpoint. Furthermore, to study the variation pattern of energy requirements at different pointswithin the workspace, a workspace discretisation is carried out with respect to the pathplacement variables and the energy is calculated for each of the discrete point for a givenpath while verifying the constraints. 12Sfrag replacements X b Y b Z b F a F r F f V p Figure 6: Cutting forces
The path placement process introduced in this paper is highlighted by means of the rectan-gular path shown in Fig. 4. As a matter of fact, the path placement process is performedfor different rectangles with constant aspect ratio of 2, i.e.,
L/W = 2. With the help of theoptimization algorithm, the location of the path corresponding to minimum and maximumenergy consumption is obtained, i.e., the best and the worst path locations with respect tothe electric energy consumed. Figures 7 and 8 show the location of different rectangularpaths with the minimum and maximum energy consumption in the Orthoglide 3-axis cu-bic workspace. The magnitude of the energy used for both best and worst cases and thecorresponding gain of energy is given in Table 3 and is illustrated in Fig. 9. In Fig. 9, %saving is the percent energy saving between the best(minimum) and the worst(maximum)energy consumption. It can be seen from Figs. 7 and 8 that the energy consumption is aRectangular path dimensions [mm] E min [J] E max [J] % gainWidth ( W ) Length ( L )20 40 15 .
26 44 .
46 65 . .
88 61 .
35 62 . .
41 76 .
31 60 . .
55 89 .
80 57 . .
83 102 .
11 54 . .
82 113 .
46 49 . .
94 121 .
17 46 . X b [mm] Y b [mm] Z b [ mm ] C O b Q + Q − Y b Z b X b × × × × × ×
160 50 × -100-50050 -100 -50 0 50-120-100-80-60-40-200204060 Figure 7: Locations of rectangular path of different sizes ( W mm × L mm) that yield aminimum energy consumptionPSfrag replacements X b [mm] Y b [mm] Z b [ mm ] C O b Q + Q − X b Y b Z b × × × × × × × -100-50050 -100 -50 0 50-120-100-80-60-40-200204060 Figure 8: Locations of rectangular path of different sizes ( W mm × L mm) that yield amaximum energy consumptionand force transmission factors equal to one, i.e. point O b , with φ = 0 ◦ and is a maximumwhen the path is located in the vicinity of point Q − with φ = 45 ◦ . From Fig. 9, it can be no-ticed that the smaller the path, the higher the energy saving. This higher gain for the smallerpath is due to the higher range of displacement of the path within the manipulator workspace.In order to view the energy variation trends in the workspace, a test path of size 30mm × E [ J ] Energy ( E ) used for different rectangular path widthsPath widths W [mm] % s a v i n g Percentage saving for different rectangular path widths E min E max
20 30 40 50 60 70 8020 30 40 50 60 70 80
Figure 9: E min and E max and percentage saving as a function of the rectangular path width( L = 2 W )PSfrag replacements x Op [mm] z Op [mm] E [ J ] E vs x Op z Op ( φ = 0 ◦ , W = 30 mm, L = 60 mm) y Op = 0 mm y Op = −
102 mm y Op = 48 mm -100 -50 0 50 Figure 10: Energy as a function of x Op and z Op for a 30mm × x Op and z Op for a constant orientation φ of 0 ◦ and for three different values of y Op . Figure 11illustrates the isocontours of the energy required by the motors with respect to x Op and y Op for given values of z Op and φ , namely, z Op = 0 and φ = 0 ◦ . From Fig. 10, it is apparentthat the energy required is not sensitive to variations in z Op . The reason being that thepath lies in the X b Y b -plane and the actuator displacements along Z b -axis is not significant.Figures 10 and 11 also show that the energy required by the motors is a minimum when thepath is located in the neighbourhood of the isotropic configuration and is a maximum whenthe latter is located in the neighbourhood of singularities.Figures 12 and 13 show the variations in the energy used with the path orientation in15Sfrag replacements Energy [J] variation with x Op and y Op ( z Op = 0, φ = 0 ◦ ) x Op [mm] y O p [ mm ] -80 -60 -40 -20 0 20 -100-80-60-40-2002040 Figure 11: Energy vs x Op and y Op for 30mm × y Op [mm] x Op [mm] E [ J ] E vs x Op y Op ( z Op = 0) φ = 60 ◦ φ = 45 ◦ φ = 30 ◦ φ = 0 ◦ φ = 90 ◦ -100 -50 0 50 Figure 12: Energy as a function of x Op and y Op for different orientations ( z Op = 0)16ifferent areas of the cubic workspace. It can be seen that the energy used is usually a maxi-mum when φ = 45 ◦ . However, the energy consumption is a maximum for a path orientationdifferent than 45 ◦ , for some path locations. For example, the energy required at the upperright corner of the workspace is higher for φ = 30 ◦ than φ = 45 ◦ . Figure 14 shows thePSfrag replacements x Op [mm] φ [deg] E [ J ] E vs x Op and φ ( y Op = z Op = 0) -100-80 -60 -40 -20 0 20 40 Figure 13: Energy as a function of x Op and φ for y Op = z Op = 0PSfrag replacements [ m ] Max actuator displacements ( a ) [ m / s ] Max actuator velocities ( b ) [ N m ] Max actuator torques ( c ) [ J ] Energy used by each actuator ( d ) E max E min E max E x E y E z τ x τ y τ z V x V y V z X Y Z
Figure 14: Comparison of 30mm × E min and E max locations17omparison of trajectory parameters of 30mm × E = E max the range of actuator displacementsis larger than when E = E min . Similarly actuators experience higher values of maximumvelocities and torques when E = E max as shown in Fig. 14(b-c), which results in higherenergy consumption for each actuator, as shown in Fig. 14(d). These results mean that theactuators may reach their performance limits due to an inappropriate location of the path inthe workspace. This work has been partially funded by the European projects NEXT, acronyms for NextGeneration of Productions Systems, Project no IP 011815.
A methodology for path placement optimization was proposed in this paper. The electricenergy required by the actuators to follow a predefined path is considered as the optimizationcriterion. Electric energy requirement is calculated with the help of the required actuatortorques and velocities along with motors electric parameters. To verify the feasibility of thesolutions, actuators performance limits such as their joint limits, maximum velocities andtorque were used as the constraints of the optimization problem. The kinematic, velocityand dynamic models were used to come up with the objective function and constraints.The proposed methodology was applied to the Orthoglide 3-axis, a three-degree-of-freedomtranslational parallel manipulator with a quasi-cubic workspace. Rectangular test paths ofdifferent sizes were considered as illustrative examples. These paths are similar to those usedto realize pocketing operations.The use of the electric energy instead of mechanical energy as an optimization criterion ispertinent. Although actuator electric energy consumption depends on the mechanical energyrequirements, the electric energy evaluation is more comprehensive than its mechanical coun-terpart. General approach to calculate the mechanical energy with the help of manipulatorvelocity and dynamic models, i.e., by using actuator torques and velocities, may lead to anunder estimation of the energy requirements in the case where actuators are experiencingtorques with zero velocities. Besides, usual mechanical energy calculations do not considerthe resistive energy loss in the motor windings as well as the energy loss due to the variationsin the actuator velocities. Those variations affect the current requirements and hence induceelectromotive forces in the actuators. Accordingly, the electric energy formulation takes intoaccount all these energy losses.The energy required to perform a given task depends on the position and the orientationof the task within the workspace of the manipulator. Accordingly, some electric energy can besaved by properly selecting the position and the orientation of the task. Indeed, a misplacedtask can cause excessive energy consumption and can force the actuators to go over theirperformance limits.For the Orthoglide 3-axis, the optimum path location is found to be in the neighbourhoodof the isotropic configuration but there is no general rule to predict the exact optimal position18nd orientation of a task particularly for a complicated three dimensional task or for an irreg-ular workspace. However, a detailed analysis of the energy variation within the workspace fora given task can lead to the optimal position/orientation of that particular task. Numericaloptimization algorithms are useful for such a comprehensive analysis in which all the problemconstraints and performance measures can be considered simultaneously.In the future work, the path placement optimization problem will be dealt as a multi-objective one. For example, along with energy requirements, the manipulator dexterity andstiffness can be considered as optimization objectives.
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