Wrench capability in redundant planar parallel manipulators with net degree of constraint equal to four, five or six

Mechanism and Machine Theory 105 (2016) 58–79

Contents lists available at ScienceDirect

Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmt

Wrench capability in redundant planar parallel manipulators with net degree of constraint equal to four, five or six L. Mejia* , H. Simas** , D. Martins** Department of Mechanical Engineering, Federal University of Santa Catarina, Florianópolis, Santa Catarina CEP: 88040-900, Brazil

A R T I C L E

I N F O

Article history: Received 23 December 2015 Received in revised form 13 June 2016 Accepted 20 June 2016 Available online xxxx Keywords: Force capability Optimization Screw theory Davies method

A B S T R A C T This paper presents three mathematical closed-form solutions to obtain the maximum force with a prescribed moment in redundant planar parallel manipulators with a net degree of constraint equal to four, five or six. The proposed mathematical models are obtained integrating the screw theory, Davies method and classical optimization methods as primary mathematical tools. The greatest impact of the mathematical closed-form solutions proposed in this study lies in the fact that the force capability of manipulators with a net degree of constraint equal to four, five or six can be obtained in a direct way and without the use of a method or a numerical algorithm, representing an easier, faster, more direct and more versatile way to solve the force capability problem in manipulators satisfying the established net degree of constraint. The proposed mathematical closed-form solutions are general and can be applied to any redundant planar parallel manipulators with a net degree of constraint equal to four, five or six and they can be used in applications requiring a real-time response in terms of force capability. Some force capability polygons and polytopes are obtained for different manipulators in order to exemplify the use of the mathematical closed-form solutions. © 2016 Elsevier Ltd.All rights reserved.

1. Introduction Creating autonomous robots that can learn to act in unpredictable environments has been a long-standing goal of robotics, artificial intelligence, and cognitive sciences [1]. An important step towards the autonomy of robots is the need to provide them with a certain level of independence in order to face quick changes in the environment surrounding them. To get robots operating outside research centres or university facilities and beyond the supervision of engineers or experts, it is necessary to face different technological challenges. Among those challenges, the development of strategies that allow robots to interact with the environment as fast as possible or even in real time is crucial [1]. In this context, when a physical contact with the environment is established, a process-specific force needs to be exerted and this force has to be controlled in relation to the particular process in order to prevent overloading or damaging the manipulator or the objects to be manipulated. The wrench capability of a robotic mechanism is defined as the maximum wrench that can be applied (or sustained) for a given pose, based on the limits of its actuators [2] and it is dependent on its design, posture, actuation limits and redundancies [3]. This

* Principal corresponding author. ** Corresponding author. E-mail addresses: [email protected] (L. Mejia), [email protected] (H. Simas), [email protected] (D. Martins).

http://dx.doi.org/10.1016/j.mechmachtheory.2016.06.020 0094-114X/© 2016 Elsevier Ltd.All rights reserved.

L. Mejia, et al. / Mechanism and Machine Theory 105 (2016) 58–79

59

paper studies the wrench capabilities of planar redundantly-actuated parallel manipulators with a net degree of constraint (CN ) equal to four, five or six by using three new proposed mathematical closed-form solutions. Some important issues regarding the closed-form solutions proposed herein are related to its simplicity, generality and versatility. Aditionally, as the effort necessary to evaluate the mathematical closed-form solutions is reduced to the time needed to evaluate some finite operations, the computing time is much lower than that associated with other methodologies using heuristic or meta-heuristic approaches described in the literature. The very fast response of the proposed mathematical models allows us to contemplate applications that require a real-time response in terms of the manipulation of the force, such as grasping, polishing, and milling. The main characteristics associated with the proposed mathematical closed-form solutions will be discused in detail in Section 8. A manipulator is termed redundantly-actuated when an infinite number of resultant force/moment combinations can span the system of external forces [4]. Thus, there is an infinite number of solutions to the inverse static force problem [4]. In order to obtain a complete static analysis of robotic mechanisms several methodologies can be applied; however, the formalism presented by Davies [5] is the main tool used in order to solve the static problem of mechanisms and manipulators in this paper, and the net degree of constraint (CN ) is a property obtained from the Davies method that is defined as the number of primary variables needed to solve the static problem. The net degree of constraint (CN ) allows us to have an idea of the redundancy in planar parallel manipulators because this property is greater than three (CN > 3) in this kind of manipulators. In this paper redundant parallel manipulators with CN = 4, CN = 5 and CN = 6 are addressed. The Davies method and the net degree of constraint will be discussed in Sections 2.1 and 2.2 respectively in order to provide theoretical support for this study. Redundant actuation in parallel manipulators can be divided into three categories, namely branch, actuation, and a combined redundancy [2,6]. In the first category (branch redundancy), redundantly-actuated manipulators are those that feature additional branches beyond the minimum necessary to actuate the device. In the second category (actuation redundancy), redundantlyactuated manipulators are those in which the number of actuated joints by actuating one or more of the passive joints without any change to the architecture of the manipulator is increased. In the third category (combined redundancy), redundantlyactuated manipulators are devices that are a hybrid of the first two categories [2,6]. The mathematical closed-form solutions presented in this paper can be used in the three types of redundancies as long as the net degree of constraint is equal to four, five or six. The idea of measuring the manipulating ability of robotic mechanisms was first introduced in Ref. [7], where the velocity manipulability ellipsoid and the force manipulability ellipsoid were defined. Several studies related to the force capability and the force distribution in manipulators can be found in the literature. Nahon and Angeles [8] described the problem of a hand grasping an object as a redundantly-actuated kinematic chain, by minimizing the internal forces in the system using quadratic programming (QP). Wu et al. [9] optimized the driving force of a planar 3-DOF parallel manipulator with actuation redundancy by utilizing the least-square method and incorporated such mechanism into a 4-DOF hybrid machine tool. Tao and Luh [10] developed an algorithm to determine the minimum torque required to sustain a common load between two joint-redundant cooperating manipulators. Weihmann et al. [3] and Mejia et al. [11–14] proposed methodologies to evaluate the force capability of planar parallel manipulators using differential evolution algorithms (DE). Wu et al. [15] proposed a method to obtain the maximum dynamic load-carrying capacity of the conveyor by optimizing the internal forces of the redundantly actuated parallel manipulators. Nokleby et al. [2] and Zibil et al. [16] presented the scaling factor method and the explicit method to evaluate the force capabilities of parallel manipulators. Nokleby et al. [17] used these methods to obtain results for 3-RRR, 3-RPR and 3-PRR parallel architectures with redundant and non-redundant actuation. Some improvements on the scaling factor method presented by Nokleby et al. [2] were addressed by Mejia et al. [18] in order to avoid the use of optimization algorithms, resulting in the modified scaling factor method. Frantz et al. [19] and Mejia et al. [18] used the modified scaling factor method in order to obtain the wrench capabilities in cooperative manipulators. Mejia et al. [20] presented a generalized mathematical closed-form solution to obtain the maximum force with a prescribed moment Fapp in robotic mechanisms with 3 degrees of freedom (DoF). The generalized mathematical closed-form solution presented by Mejia et al. [20] has a very fast response and allows us to contemplate applications that require a real-time response in terms of the manipulation of the force, such as grasping, polishing, and milling. Firmani et al. [21] presented the wrench polytope concept and introduced the wrench performance indices for planar parallel manipulators. Firmani [22] extended the concept of wrench capabilities to redundant manipulators and analyzed the wrench workspace of different planar parallel manipulators. The maximum available value (MAV) presented by Finotello et al. [23] and the maximum force with a prescribed moment (Fapp ) presented by Firmani et al. [21] are similar force capability performance indices and they are defined as the maximum force that can be applied (or sustained) by a robotic mechanism with a prescribed moment [3]. If this prescribed moment is zero, it yields a pure force analysis. For a given direction, the maximum force that can be applied with zero moment will be denoted as Fm [3]. Other wrench performance indices for planar parallel manipulators have been presented in [21], but herein only the Fapp and Fm will be considered. The principal aim of this paper is to present generalized mathematical closed-form solutions to obtain the maximum force with a prescribed moment Fapp in planar redundantly-actuated mechanisms and manipulators with a net degree of constraint equal to four, five or six. The proposed mathematical models are obtained applying classical optimization methods, considering the cases in which the net degree of constraint is equal to the number of actuated joints in the mechanism or manipulator and their values are equal to four, five or six. In robotics, closed-form solutions are often very desirable, because they are faster than numerical solutions and readily identify all possible solutions [24]. The novelty of the study described herein lies in the fact that our main results are not methods or numerical algorithms, but mathematical closed-form solutions to obtain the force capability in planar redundantly-actuated mechanisms and

60

L. Mejia, et al. / Mechanism and Machine Theory 105 (2016) 58–79

manipulators with net degree of constraint equal to four, five or six. An equation is said to be a closed-form solution if it solves a given problem in terms of functions and mathematical operations from a given generally-accepted set [25]. This means that the main results reported in this paper are functions that can be used directly without the use of a method, numerical algorithm or optimization process, implying that their use represents simpler, faster and more direct solutions to obtain the force capability of manipulators compared with other solutions for the same problem found in literature. To validate the mathematical closed-form solutions proposed in this paper four cases were studied, the first is a redundantlyactuated 4RRR planar parallel manipulator, the second is a redundantly-actuated 5RRR planar parallel manipulator, the third is a redundantly-actuated 6RRR planar parallel manipulator and finally, the fourth is a redundantly-actuated 3RRR planar parallel manipulator. 2. Statics of robotic mechanisms In the static analysis of robotic mechanisms, the goal is to determine the force and moment requirements for the joints in relation to the wrenches applied at the end effector. It is possible to apply forces and moments at the joints of the mechanism to analyze the wrenches obtained at the end effector, or to apply external wrenches at the end effector to calculate the forces and moments required at the joints to balance these external forces. There are several methodologies which allow us to obtain a complete static analysis of robotic mechanisms; however, in this paper the formalism presented by Davies [5] is used as the primary mathematical tool to analyze the mechanisms statically. The Davies method appears in many publications in the literature and further explanations regarding its use can be found in [3,5,26,27,28,29,30], a brief resume of the Davies method is presented in Section 2.1. In this paper the Davies method was used because the obtention of the static model of a manipulator or mechanism is simple, easily adaptable and it is not necessary to use a pseudo-inverse as in other methodologies. Additionally, the Davies method together with the proposed mathematical closed-form solutions, constitutes a powerful tool that can be used in order to solve the force capability problem in redundantly-actuated parallel manipulators with net degree of constraint equal to four, five or six (CN = 4, CN = 5 or CN = 6). 2.1. The Davies method The Davies method provides a systematic way to relate the joint forces and moments in closed kinematic chains [30]. This method is based on graph theory, screw theoryand the Kirchhoff cut-set law and it can be used to obtain the statics of a robotic mechanism as a matricial expression [30]. The Davies method for static analysis can be described in a simplified way through the following steps: 1. Given a mechanism, draw its kinematic chain identifying all of its links “n” and direct couplings “e”. 2. Draw the coupling graph “GC ” for the mechanism with the links of the mechanism as the vertices of the graph, and the joints of the mechanism as the edges of the graph. 3. Generate the action graph “GA ” from “GC ” through unfolding single actions from direct couplings. In this step, each edge of “GC ” representing a coupling is replaced in “GA ” by c constraint edges in parallel. • Assign positive directions to each edge with an arrow pointing from the minor to major vertex. • Locate the number of cuts (k = n − 1) and chords (l = e − n + 1) in the action graph and depict them. 4. Write the cut-set matrix [Q N ]k,e with suitable signs. 5. Write a wrench $J for each edge from GA as follows: ⎡

⎤ ⎡ ⎤ ⎡ ⎤ −y x 1 $J = ⎣ 1 ⎦ JFx + ⎣ 0 ⎦ JFy + ⎣ 0 ⎦ JMz 0 1 0

(1)

6. Replace each wrench $J in the cut-set matrix [Q N ]k,e in order to obtain the generalized action matrix [AN ]3k,e 7. Operate algebraically the generalized action matrix [AN ]3k,e in order to statically solve the system.

2.2. Net degree of constraint (CN ) The net degree of constraint (CN ) is an intrinsic property obtained from the Davies method and is defined as the number of primary variables needed to solve the static problem [5,26,27]. The net degree of constraint can be directly obtained from the action graph of the mechanism, and it can be computed as shown in Eq. (2). CN = C − kk

(2)

L. Mejia, et al. / Mechanism and Machine Theory 105 (2016) 58–79

61

Fig. 1. (a). Redundantly-actuated 4RRR planar parallel manipulator. (b). Action graph of the redundantly-actuated 4RRR planar parallel manipulator.

where C represents the total number of internal constraints of the mechanism, k represents the dimension of the task space in which the mechanism works (in planar mechanisms k = 3) and k is the number of cuts in the action graph. In order to exemplify the computation process of the net degree of constraint, consider the redundantly-actuated planar parallel manipulator shown in Fig. 1 (a) and depicted by the action graph shown in Fig. 1(b). In the action graph, the vertices of the graph represent the links of the manipulator and the edges of the graph represent the actions at the joints of the manipulator. Based on the action graph shown in Fig. 1(b) it can be observed that the total number of internal constraint is equal to thirty-one (C = 31), represented as the thirty-one arrows in the graph, the number of cuts in the graph is equal to nine k = 9 (k1 , . . . , k9 ), and, since the manipulator is a planar system, the dimension of the task space is k = 3. With the use of Eq. (2), it is possible to obtain the net degree of constraint as shown below: CN = C − kk = 31 − 3 • 9 = 31 − 27 = 4

(3)

Based on the result obtained in Eq. (3), where the net degree of constraint of the evaluated manipulator is equal to four (CN = 4), it can be concluded that the static problem of the manipulator can be solved only if four internal forces are known. 2.3. Static models of mechanisms and manipulators with CN = 4, CN = 5 and CN = 6 In planar robotic mechanisms with CN = 4, once the Davies method has been applied in order to obtain its static model, it is possible to represent the wrenches at the end effector (Fx , Fy , Mz ) as a generalized function of a coefficient matrix (A3×4 ) and the four primary actions (t1 , t2 , t3 , t4 ), as shown in Eq. (4). In a similar way, in robotic mechanisms with CN = 5 the wrenches at the end effector (Fx , Fy , Mz ) can be represented statically by using the Davies method as a generalized function of a coefficient matrix (A3×5 ) and the five primary actions (t1 , t2 , t3 , t4 , t5 ) as shown in Eq. (5). Finally, robotic mechanisms with CN = 6 can have the wrenches at the end effector (Fx , Fy , Mz ) represented as a generalized function of a coefficient matrix (A3×6 ) and the six primary actions (t1 , t2 , t3 , t4 , t5 , t6 ) as shown in Eq. (6). ⎡ ⎤ ⎡ ⎤ t1 Fx a1 a 2 a 3 a 4 ⎢ ⎣ Fy ⎦ = ⎣ a 5 a 6 a 7 a 8 ⎦ • ⎢ t 2 ⎣ t3 Mz a9 a10 a11 a12 t4 ⎡

⎡

⎤

⎡

a 1 a2 a3 a4 Fx ⎣ Fy ⎦ = ⎣ a 6 a 7 a 8 a 9 Mz a11 a12 a13 a14

⎤ ⎥ ⎥ ⎦

⎡

t1 ⎢ t2 a5 ⎢ a10 ⎦ • ⎢ ⎢ t3 ⎣ t4 a15 t5 ⎤

(4)

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

(5)

62

L. Mejia, et al. / Mechanism and Machine Theory 105 (2016) 58–79

⎡

⎡

⎤ ⎡ Fx a 1 a2 a3 a4 a5 ⎣ Fy ⎦ = ⎣ a7 a8 a9 a10 a11 Mz a13 a14 a15 a16 a17

t1 ⎤ ⎢ t2 ⎢ a6 ⎢ ⎢t a12 ⎦ • ⎢ 3 ⎢ t4 ⎢ a18 ⎣ t5 t6

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(6)

 In Eqs. (4), (5) and (6) the coefficient matrix A3×CN represents kinematic expressions as a function of the manipulator joint positions, the t1 , . . . , tCN terms represent the primary actions of the actuated joints and the terms Fx , Fy , Mz represent the wrenches at the end effector. Eqs. (4), (5) and (6) can also be obtained using classical methods to solve the statics of robotic mechanisms, for example using the method of the transposed Jacobian, as cited in [21,22,31] and represented in a simplified way in Eq. (7). Eqs. (4), (5), (6) and (7) are equivalent static models under the same conditions (e.g. loads only at the end effector, no friction, no internal forces, negligible masses, among other.). F = J−T t

(7)

At this point it should be emphasized that the proposed closed-form solutions presented in this paper work only when the manipulator evaluated has a net degree of constraint equal to four, five or six (CN = 4, CN = 5 or CN = 6). Manipulators with CN values which are not four, five or six (CN = 4, CN = 5 or CN = 6) are not treated in this paper. To validate the mathematical closed-form solutions proposed in this paper four study cases satisfying the condition CN = 4, CN = 5 and CN = 6 will be considered as shown in Fig. 2. The first study case, shown in Fig. 2(a), is a redundantly-actuated 4RRR planar parallel manipulator (CN = 4), the second study case, shown in Fig. 2(b), is a redundantly-actuated 5RRR planar parallel manipulator (CN = 5), the third study case, shown in Fig. 2(c), is a redundantly-actuated 6RRR planar parallel manipulator (CN = 6), and finally, the fourth study case, shown in Fig. 2(d), is a redundantly-actuated 3RRR planar parallel manipulator (CN = 6). 3. Procedure to optimize the force capability in robotic mechanisms with CN = 4 The principal aim of this paper is to present generalized mathematical closed-form solutions to obtain the maximum force with a prescribed moment (Fapp ) in robotic mechanisms with CN = 4, CN = 5 and CN = 6. In other words, the force capability optimization problem is defined as the maximization of the force magnitude with a given direction and with a prescribed moment at the end effector in parallel manipulators satisfying the condition CN = 4, CN = 5 or CN = 6. In this section the procedure to obtain the mathematical closed-form solutions defining the maximum force with a prescribed moment (Fa pp) in robotic mechanisms with CN = 4 is shown in detail, and the resulting closed-form solution obtained for this kind of manipulators by using this procedure will be shown in Section 4. The optimization process used to obtain the mathematical closed-form solutions for manipulators with CN = 5 and CN = 6 is very similar to such as used in this section, and because of this, only the closed-form solutions for this kind of manipulators will be presented in Sections 5 and 6. In order to solve the force capability problem in parallel manipulators with CN = 4, the first consideration that must take into account is that the direction of the application of the force must be known. The imposition of a force direction allows us to determine the relations between the forces Fx and Fy at the manipulator end effector, as shown in Fig. 3. These relations can be expressed mathematically by Eqs. (8) and (9).

F=

Fx 2 + F y 2

Fy = Fx tan(h)

(8) (9)

Eqs. (8) and (9) can be rewritten with the substitution of the elements in Eq. (4), as shown in Eqs. (10) and (11), respectively.

F=

(a1 t1 + a2 t2 + a3 t3 + a4 t4 )2 + (a5 t1 + a6 t2 + a7 t3 + a8 t4 )2

va (h)t1 + vb (h)t2 + vc (h)t3 + vd (h)t4 = 0

(10) (11)

where: va (h) = a5 − a1 tan(h)

(12)

L. Mejia, et al. / Mechanism and Machine Theory 105 (2016) 58–79

63

Fig. 2. (a). Redundantly-actuated 4RRR planar parallel manipulator (CN = 4). (b). Redundantly-actuated 5RRR planar parallel manipulator (CN = 5). (c). Redundantly-actuated 6RRR planar parallel manipulator (CN = 6). (d). Redundantly-actuated 3RRR planar parallel manipulator (CN = 6).

vb (h) = a6 − a2 tan(h)

(13)

vc (h) = a7 − a3 tan(h)

(14)

vd (h) = a8 − a4 tan(h)

(15)

The formulation of the force capability optimization problem can be generalized as shown in Eqs. (16) to (22). For convenience, the objective function is represented in Eq. (16) as the negative of the square of the force, in order to maximize the force. Eq. (17) represents the equality constraint function of the force direction. Eq. (18) represents the equality constraint function of the prescribed moment (Miz ). Eqs. (19) to (22) represent the inequality constraint functions of the maximum admissible torque in each actuated joint t1 , t2 , t3 and t4 . minimize

 f (t1 , t2 , t3 , t4 ) = −F 2 = − (a1 t1 + a2 t2 + a3 t3 + a4 t4 )2 + (a5 t1 + a6 t2 + a7 t3 + a8 t8 )2

(16)

subject to: h1 (t1 , t2 , t3 ) : va (h)t1 + vb (h)t2 + vc (h)t3 + vd (h)t4 = 0

(17)

h2 (t1 , t2 , t3 ) : Mz − Miz = a9 t1 + a10 t2 + a11 t3 + a12 t4 − Miz = 0

(18)

g1 (t1 ) : −t1max  t1  t1max

(19)

64

L. Mejia, et al. / Mechanism and Machine Theory 105 (2016) 58–79

Fig. 3. Prescribed force direction at the manipulator end effector.

g2 (t2 ) : −t2max  t2  t2max

(20)

g3 (t3 ) : −t3max  t3  t3max

(21)

g4 (t4 ) : −t4max  t4  t4max

(22)

Note that Eqs. (19), (20), (21) and (22) represent eight inequality constraints, because each equation can be decomposed into two independent inequality constraints (e.g. Eq. (19) can be decomposed into g1a (t1 ) : −t1max − t1  0 and g1b (t1 ) : t1 − t1max  0). To solve the force capability optimization problem it is necessary to define a Lagrangian function as a combination of the objective function, the equality constraint functions, the inequality constraint functions, the Lagrange multipliers and the slack variables. For convenience, in this paper the Lagrange multipliers for the equality constraints are represented as k1 and k2 , the Lagrange multipliers for the inequality constraints are represented as l 1 , . . . , l 8 and the slack variables are represented as s1 , . . . , s8 . Eq. (23) shows the Lagrangian function used to solve the Fapp optimization problem herein.  L = − vm t12 + vn t22 + vo t32 + vp t42 + vq t1 t2 + vr t1 t3 + vs t1 t4 + vt t2 t3 + vu t2 t4 + vv t3 t4 + . . .

 . . . + k1 [va (h)t1 + vb (h)t2 + vc (h)t3 + vd (h)t4 ] + k2 a9 t1 + a10 t2 + a11 t3 + a12 t4 − Miz + . . .         . . . + l1 −t1max − t1 + s1 2 + l2 t1 − t1max + s2 2 + l3 −t2max − t2 + s3 2 + l4 t2 − t2max + s4 2 + . . .         . . . + l5 −t3max − t3 + s5 2 + l6 t3 − t3max + s6 2 + l7 −t4max − t4 + s7 2 + l8 t4 − t4max + s8 2

(23)

where: v m = a1 2 + a5 2

(24)

vn = a2 2 + a6 2

(25)

vo = a3 2 + a7 2

(26)

vp = a4 2 + a8 2

(27)

vq = 2a1 a2 + 2a5 a6

(28)

vr = 2a1 a3 + 2a5 a7

(29)

vs = 2a1 a4 + 2a5 a8

(30)

vt = 2a2 a3 + 2a6 a7

(31)

L. Mejia, et al. / Mechanism and Machine Theory 105 (2016) 58–79

65

vu = 2a2 a4 + 2a6 a8

(32)

vv = 2a3 a4 + 2a7 a8

(33)

Differentiating the Lagrangian function shown in Eq. (23) with respect to t, k, l and s, Eqs. (34) to (55) are obtained. These equations allow us to construct a mathematical system whose solution will solve the force capability optimization problem. The partial derivatives of the Lagrangian (L) as a function of the actuated joints (t1 , t2 , t3 and t4 ) are obtained as:

∂L = − (2vm t1 + vq t2 + vr t3 + vs t4 ) + k1 va (h) + k2 a9 − l1 + l2 = 0 ∂ t1

(34)

∂L = − (2vn t2 + vq t1 + vt t3 + vu t4 ) + k1 vb (h) + k2 a10 − l3 + l4 = 0 ∂ t2

(35)

∂L = − (2vo t3 + vr t1 + vt t2 + vv t4 ) + k1 vc (h) + k2 a11 − l 5 + l 6 = 0 ∂ t3

(36)

∂L = − (2vp t4 + vs t1 + vu t2 + vv t3 ) + k1 vd (h) + k2 a12 − l7 + l8 = 0 ∂ t4

(37)

The partial derivatives of the Lagrangian (L) as a function of the Lagrange multipliers for equality constraints (k1 and k2 ) are obtained as:

∂L = va (h)t1 + vb (h)t2 + vc (h)t3 + vd (h)t4 = 0 ∂ k1

(38)

∂L = a9 t1 + a10 t2 + a11 t3 + a12 t4 − Miz = 0 ∂ k2

(39)

The partial derivatives of the Lagrangian (L) as a function of the Lagrange multipliers for inequality constraints (l 1 , l 2 , l 3 , l 4 , l 5 , l 6 , l 7 and l 8 ) are obtained as:

∂L = s1 2 − t1max − t1 = 0 ∂ l1

(40)

∂L = s2 2 − t1max + t1 = 0 ∂ l2

(41)

∂L = s3 2 − t2max − t2 = 0 ∂ l3

(42)

∂L = s4 2 − t2max + t2 = 0 ∂ l4

(43)

∂L = s5 2 − t3max − t3 = 0 ∂l 5

(44)

∂L = s6 2 − t3max + t3 = 0 ∂l 6

(45)

∂L = s7 2 − t4max − t4 = 0 ∂ l7

(46)

∂L = s8 2 − t4max + t4 = 0 ∂ l8

(47)

Finally, the partial derivatives of the Lagrangian (L) as a function of the slack variables (s1 , s2 , s3 , s4 , s5 , s6 , s7 and s8 ) are obtained as:

∂L = 2s1 l1 = 0 ∂ s1

(48)

∂L = 2s2 l2 = 0 ∂ s2

(49)

∂L = 2s3 l3 = 0 ∂ s3

(50)

66

L. Mejia, et al. / Mechanism and Machine Theory 105 (2016) 58–79

∂L = 2s4 l4 = 0 ∂ s4

(51)

∂L = 2s5 l5 = 0 ∂ s5

(52)

∂L = 2s6 l6 = 0 ∂ s6

(53)

∂L = 2s7 l7 = 0 ∂ s7

(54)

∂L = 2s8 l8 = 0 ∂ s8

(55)

The mathematical system shown in Eqs. (34) to (55) is solved through the imposition of hypothetical values for sn and l n satisfying the conditions shown in Eqs. (48) to (55), but avoiding conflicting solutions. The case in which s1 = 0 and s2 = 0 can be considered as an example of such conflicting solutions, because although they are possible solutions for Eqs. (48) and (49), the solutions obtained for Eqs. (40) and (41), where t1 = −t1max and t1 = t1max , are physically impossible to put into practice because the maximum positive torque and the maximum negative torque cannot be applied in an actuator at the same time. 4. Mathematical closed-form solution to obtain the force capability in robotic mechanisms with CN = 4 As mentioned above, the mathematical closed-form solution to obtain the force capability in manipulators with CN = 4 is obtained from the solution of the mathematical system shown in Eqs. (34) to (55). This solution allows us to obtain the maximum force with a prescribed moment (Fapp ) in planar robotic mechanisms with CN = 4 as a function of the desired force direction (h) and the prescribed moment (Miz ) at the end effector of the manipulator. This mathematical closed-form solution is shown in Eqs. (56) to (75) and represents one of the most important results obtained in this study. It should be noted that the twelve kinematic variables (a1 , . . . , a12 ) were obtained from the solution of the forward static problem presented in Eq. (4), and the terms va (h), vb (h), vc (h) and vd (h) were previously obtained as shown in Eqs. (12), (13), (14) and (15).

Fapp (h, Miz ) = max [fe (−t1max , −t2max , t3 , t4 ) • kkt1 ; fe (−t1max , t2max , t3 , t4 ) • kkt1 ; · · · · · · fe (t1max , −t2max , t3 , t4 ) • kkt1 ; fe (t1max , t2max , t3 , t4 ) • kkt1 ; fe (−t1max , t2 , −t3max , t4 ) • kkt2 · · · · · · fe (−t1max , t2 , t3max , t4 ) • kkt2 ; fe (t1max , t2 , −t3max , t4 ) • kkt2 ; fe (t1max , t2 , t3max , t4 ) • kkt2 ; · · · · · · fe (−t1max , t2 , t3 , −t4max ) • kkt3 ; fe (−t1max , t2 , t3 , t4max ) • kkt3 ; fe (t1max , t2 , t3 , −t4max ) • kkt3 ; · · · · · · fe (t1max , t2 , t3 , t4max ) • kkt3 ; fe (t1 , −t2max , −t3max , t4 ) • kkt4 ; fe (t1 , −t2max , t3max , t4 ) • kkt4 ; · · · · · · fe (t1 , t2max , −t3max , t4 ) • kkt4 ; fe (t1 , t2max , t3max , t4 ) • kkt4 ; fe (t1 , −t2max , t3 , −t4max ) • kkt5 ; · · · · · · fe (t1 , −t2max , t3 , t4max ) • kkt5 ; fe (t1 , t2max , t3 , −t4max ) • kkt5 ; fe (t1 , t2max , t3 , t4max ) • kkt5 ; · · · · · · fe (t1 , t2 , −t3max , −t4max ) • kkt6 ; fe (t1 , t2 , −t3max , t4max ) • kkt6 ; fe (t1 , t2 , t3max , −t4max ) • kkt6 ; · · · · · · fe (t1 , t2 , t3max , t4max ) • kkt6 ]

(56)

where  1 2 fe (t1 , t2 , t3 , t4 ) = (a1 t1 + a2 t2 + a3 t3 + a4 t4 )2 + (a5 t1 + a6 t2 + a7 t3 + a8 t4 )2

(57)

 kkt1 = kkt2 = kkt3 = kkt4 = kkt5 =

1 ⇐⇒ (−t3max ≤ t3 ≤ t3max ) ∧ (−t4max ≤ t4 ≤ t4max )

0 other case  1 ⇐⇒ (−t2max ≤ t2 ≤ t2max ) ∧ (−t4max ≤ t4 ≤ t4max ) 0 other case  1 ⇐⇒ (−t2max ≤ t2 ≤ t2max ) ∧ (−t3max ≤ t3 ≤ t3max ) 0 other case  1 ⇐⇒ (−t1max ≤ t1 ≤ t1max ) ∧ (−t4max ≤ t4 ≤ t4max ) 0 other case  1 ⇐⇒ (−t1max ≤ t1 ≤ t1max ) ∧ (−t3max ≤ t3 ≤ t3max ) 0

other

case

(58)

(59)

(60)

(61)

(62)

L. Mejia, et al. / Mechanism and Machine Theory 105 (2016) 58–79

 kkt6 =

1 ⇐⇒ (−t1max ≤ t1 ≤ t1max ) ∧ (−t2max ≤ t2 ≤ t2max )

0

other

case

67

(63)

In this solution, Eq. (56) is represented as the maximum value of 24 terms. Each term in Eq. (56) is obtained as the product of Eq. (57) (evaluated with different values of t1 , t2 , t3 and t4 ) and a penalization term (kktn ) as shown in Eqs. (58) to (63). To evaluate the 24 terms in Eq. (56), the components t1 , t2 , t3 and t4 need to be obtained as a function of the constant values of saturation (tnmax ), the desired moment at the end effector of the manipulator (Miz ) and the desired angle of application of the force (h) using Eqs. (64) to (75). t1 (t3 , t4 , Miz , h) = t2 (t3 , t4 , Miz , h) = t1 (t2 , t4 , Miz , h) = t3 (t2 , t4 , Miz , h) = t1 (t2 , t3 , Miz , h) = t4 (t2 , t3 , Miz , h) = t2 (t1 , t4 , Miz , h) =

(Miz − t3 a11 − t4 a12 ) vb (h) + (t3 vc (h) + t4 vd (h)) a10 a9 vb (h) − a10 va (h)

(Miz − t3 a11 − t4 a12 ) va (h) + (t3 vc (h) + t4 vd (h)) a9 a10 va (h) − a9 vb (h)

(Miz − t2 a10 − t4 a12 ) vc (h) + (t2 vb (h) + t4 vd (h)) a11 a9 vc (h) − a11 va (h) (Miz − t2 a10 − t4 a12 ) va (h) + (t2 vb (h) + t4 vd (h)) a9 a11 va (h) − a9 vc (h)

(Miz − t2 a10 − t3 a11 ) vd (h) + (t2 vb (h) + t3 vc (h)) a12 a9 vd (h) − a12 va (h)

(Miz − t2 a10 − t3 a11 ) va (h) + (t2 vb (h) + t3 vc (h)) a9 a12 va (h) − a9 vd (h) (Miz − t1 a9 − t4 a12 ) vc (h) + (t1 va (h) + t4 vd (h)) a11 a10 vc (h) − a11 vb (h)

(64) (65) (66) (67) (68) (69) (70)

t3 (t1 , t4 , Miz , h) =

(Miz − t1 a9 − t4 a12 ) vb (h) + (t1 va (h) + t4 vd (h)) a10 a11 vb (h) − a10 vc (h)

(71)

t2 (t1 , t3 , Miz , h) =

(Miz − t1 a9 − t3 a11 ) vd (h) + (t1 va (h) + t3 vc (h)) a12 a10 vd (h) − a12 vb (h)

(72)

t4 (t1 , t3 , Miz , h) = t3 (t1 , t2 , Miz , h) = t4 (t1 , t2 , Miz , h) =

(Miz − t1 a9 − t3 a11 ) vb (h) + (t1 va (h) + t3 vc (h)) a10 a12 vb (h) − a10 vd (h)

(Miz − t1 a9 − t2 a10 ) vd (h) + (t1 va (h) + t2 vb (h)) a12 a11 vd (h) − a12 vc (h)

(Miz − t1 a9 − t2 a10 ) vc (h) + (t1 va (h) + t2 vb (h)) a11 a12 vc (h) − a11 vd (h)

(73) (74) (75)

5. Mathematical closed-form solution to obtain the force capability in robotic mechanisms with CN = 5 In a similar way as in Section 4, the mathematical closed-form solution to obtain the force capability in manipulators with CN = 5 can be obtained by using an optimization process. In this section the mathematical closed-form solution to obtain the force capability in manipulators with CN = 5 is presented directly as shown in Eqs. (76) to (112).

Fapp (h, Miz ) = max fe1 (t1 , t2 , t3 , t4 (t1 , t2 , t3 ), t5 (t1 , t2 , t3 )) • kkt1 ; · · · · · · fe2 (t1 , t2 , t3 (t1 , t2 , t4 ), t4 , t5 (t1 , t2 , t4 )) • kkt2 ; fe3 (t1 , t2 (t1 , t3 , t4 ), t3 , t4 , t5 (t1 , t3 , t4 )) • kkt3 ; · · · · · · fe4 (t1 (t2 , t3 , t4 ), t2 , t3 , t4 , t5 (t2 , t3 , t4 )) • kkt4 ; fe5 (t1 , t2 , t3 (t1 , t2 , t5 ), t4 (t1 , t2 , t5 ), t5 ) • kkt5 ; · · · · · · fe6 (t1 , t2 (t1 , t3 , t5 ), t3 , t4 (t1 , t3 , t5 ), t5 ) • kkt6 ; fe7 (t1 (t2 , t3 , t5 ), t2 , t3 , t4 (t2 , t3 , t5 ), t5 ) • kkt7 ; · · · · · · fe8 (t1 , t2 (t1 , t4 , t5 ), t3 (t1 , t4 , t5 ), t4 , t5 ) • kkt8 ; fe9 (t1 (t2 , t4 , t5 ), t2 , t3 (t2 , t4 , t5 ), t4 , t5 ) • kkt9 ; · · ·  · · · fe10 (t1 (t3 , t4 , t5 ), t2 (t3 , t4 , t5 ), t3 , t4 , t5 ) • kkt10

(76)

where  1 2 fen (t1 , t2 , t3 , t4 , t5 ) = (a1 t1 + a2 t2 + a3 t3 + a4 t4 + a5 t5 )2 + (a6 t1 + a7 t2 + a8 t3 + a9 t4 + a10 t5 )2

(77)

68

L. Mejia, et al. / Mechanism and Machine Theory 105 (2016) 58–79

 kkt1 =

kkt2 =

kkt3 =

kkt4 =

kkt5 =

kkt6 =

kkt7 =

kkt8 =

kkt9 =

1 ⇐⇒ (−t4max ≤ t4 ≤ t4max ) ∧ (−t5max ≤ t5 ≤ t5max )

(78)

0 other case  1 ⇐⇒ (−t3max ≤ t3 ≤ t3max ) ∧ (−t5max ≤ t5 ≤ t5max )

(79)

0 other case  1 ⇐⇒ (−t2max ≤ t2 ≤ t2max ) ∧ (−t5max ≤ t5 ≤ t5max )

(80)

0 other case  1 ⇐⇒ (−t1max ≤ t1 ≤ t1max ) ∧ (−t5max ≤ t5 ≤ t5max )

(81)

0 other case  1 ⇐⇒ (−t3max ≤ t3 ≤ t3max ) ∧ (−t4max ≤ t4 ≤ t4max )

(82)

0 other case  1 ⇐⇒ (−t2max ≤ t2 ≤ t2max ) ∧ (−t4max ≤ t4 ≤ t4max )

(83)

0 other case  1 ⇐⇒ (−t1max ≤ t1 ≤ t1max ) ∧ (−t4max ≤ t4 ≤ t4max )

(84)

0 other case  1 ⇐⇒ (−t2max ≤ t2 ≤ t2max ) ∧ (−t3max ≤ t3 ≤ t3max )

(85)

0 other case  1 ⇐⇒ (−t1max ≤ t1 ≤ t1max ) ∧ (−t3max ≤ t3 ≤ t3max )

kkt10 =

(86)

0 other case  1 ⇐⇒ (−t1max ≤ t1 ≤ t1max ) ∧ (−t2max ≤ t2 ≤ t2max ) 0

other

(87)

case

In this solution, Eq. (76) is represented as the maximum value of 10 terms. Each term in Eq. (76) is obtained as the product of Eq. (77) evaluated with different values of saturation for t1 , t2 , t3 , t4 and t5 and a penalization term (kktn ) as shown in Eqs. (78) to (87). In order to exemplify the chosen different values of saturation for the actuated joints tn , consider the first element in Eq. (76), where fe1 (t1 , t2 , t3 , t4 (t1 , t2 , t3 ), t5 (t1 , t2 , t3 )) is a composed function that needs to be evaluated by all the possible saturation values of t1 , t2 and t3 . The possible values of saturation for t1 , t2 and t3 are shown in Table 1, and the elements t4 and t5 need to be calculated as a function of these saturated elements and the imposed moment (Miz ) and the desired direction (h) as shown in Eqs. (106) and (107). The same process needs to be followed in order to evaluate all the elements in Eq. (76) but saturating the independent values tn for each element Fen in these equations.

t1 (t3 , t4 , t5 , Miz , h) = t2 (t3 , t4 , t5 , Miz , h) =

(Miz − t3 a13 − t4 a14 − t5 a15 ) vb (h) + (t3 vc (h) + t4 vd (h) + t5 ve (h)) a12 a11 vb (h) − a12 va (h)

(Miz − t3 a13 − t4 a14 − t5 a15 ) va (h) + (t3 vc (h) + t4 vd (h) + t5 ve (h)) a11 a12 va (h) − a11 vb (h)

(88) (89)

t1 (t2 , t4 , t5 , Miz , h) =

(Miz − t2 a12 − t4 a14 − t5 a15 ) vc (h) + (t2 vb (h) + t4 vd (h) + t5 ve (h)) a13 a11 vc (h) − a13 va (h)

(90)

t3 (t2 , t4 , t5 , Miz , h) =

(Miz − t2 a12 − t4 a14 − t5 a15 ) va (h) + (t2 vb (h) + t4 vd (h) + t5 ve (h)) a11 a13 va (h) − a11 vc (h)

(91)

Table 1 Saturation values of t1 , t2 and t3 in the first element of Eq. 76. Case 1 2 3 4 5 6 7 8

Value for t1

Value for t2

Value for t3

−t1max −t1max −t1max −t1max t1max t1max t1max t1max

−t2max −t2max t2max t2max −t2max −t2max t2max t2max

−t3max t3max −t3max t3max −t3max t3max −t3max t3max

L. Mejia, et al. / Mechanism and Machine Theory 105 (2016) 58–79

69

t1 (t2 , t3 , t5 , Miz , h) =

(Miz − t2 a12 − t3 a13 − t5 a15 ) vd (h) + (t2 vb (h) + t3 vc (h) + t5 ve (h)) a14 a11 vd (h) − a14 va (h)

(92)

t4 (t2 , t3 , t5 , Miz , h) =

(Miz − t2 a12 − t3 a13 − t5 a15 ) va (h) + (t2 vb (h) + t3 vc (h) + t5 ve (h)) a11 a14 va (h) − a11 vd (h)

(93)

t1 (t2 , t3 , t4 , Miz , h) =

(Miz − t2 a12 − t3 a13 − t4 a14 ) ve (h) + (t2 vb (h) + t3 vc (h) + t4 vd (h)) a15 a11 ve (h) − a15 va (h)

(94)

t5 (t2 , t3 , t4 , Miz , h) =

(Miz − t2 a12 − t3 a13 − t4 a14 ) va (h) + (t2 vb (h) + t3 vc (h) + t4 vd (h)) a11 a15 va (h) − a11 ve (h)

(95)

t2 (t1 , t4 , t5 , Miz , h) =

(Miz − t1 a11 − t4 a14 − t5 a15 ) vc (h) + (t1 va (h) + t4 vd (h) + t5 ve (h)) a13 a12 vc (h) − a13 vb (h)

(96)

t3 (t1 , t4 , t5 , Miz , h) =

(Miz − t1 a11 − t4 a14 − t5 a15 ) vb (h) + (t1 va (h) + t4 vd (h) + t5 ve (h)) a12 a13 vb (h) − a12 vc (h)

(97)

t2 (t1 , t3 , t5 , Miz , h) =

(Miz − t1 a11 − t3 a13 − t5 a15 ) vd (h) + (t1 va (h) + t3 vc (h) + t5 ve (h)) a14 a12 vd (h) − a14 vb (h)

(98)

t4 (t1 , t3 , t5 , Miz , h) =

(Miz − t1 a11 − t3 a13 − t5 a15 ) vb (h) + (t1 va (h) + t3 vc (h) + t5 ve (h)) a12 a14 vb (h) − a12 vd (h)

(99)

t2 (t1 , t3 , t4 , Miz , h) =

(Miz − t1 a11 − t3 a13 − t4 a14 ) ve (h) + (t1 va (h) + t3 vc (h) + t4 vd (h)) a15 a12 ve (h) − a15 vb (h)

(100)

t5 (t1 , t3 , t4 , Miz , h) =

(Miz − t1 a11 − t3 a13 − t4 a14 ) vb (h) + (t1 va (h) + t3 vc (h) + t4 vd (h)) a12 a15 vb (h) − a12 ve (h)

(101)

t3 (t1 , t2 , t5 , Miz , h) =

(Miz − t1 a11 − t2 a12 − t5 a15 ) vd (h) + (t1 va (h) + t2 vb (h) + t5 ve (h)) a14 a13 vd (h) − a14 vc (h)

(102)

t4 (t1 , t2 , t5 , Miz , h) =

(Miz − t1 a11 − t2 a12 − t5 a15 ) vc (h) + (t1 va (h) + t2 vb (h) + t5 ve (h)) a13 a14 vc (h) − a13 vd (h)

(103)

t3 (t1 , t2 , t4 , Miz , h) =

(Miz − t1 a11 − t2 a12 − t4 a14 ) ve (h) + (t1 va (h) + t2 vb (h) + t4 vd (h)) a15 a13 ve (h) − a15 vc (h)

(104)

t5 (t1 , t2 , t4 , Miz , h) =

(Miz − t1 a11 − t2 a12 − t4 a14 ) vc (h) + (t1 va (h) + t2 vb (h) + t4 vd (h)) a13 a15 vc (h) − a13 ve (h)

(105)

t4 (t1 , t2 , t3 , Miz , h) =

(Miz − t1 a11 − t2 a12 − t3 a13 ) ve (h) + (t1 va (h) + t2 vb (h) + t3 vc (h)) a15 a14 ve (h) − a15 vd (h)

(106)

t5 (t1 , t2 , t3 , Miz , h) =

(Miz − t1 a11 − t2 a12 − t3 a13 ) vd (h) + (t1 va (h) + t2 vb (h) + t3 vc (h)) a14 a15 vd (h) − a14 ve (h)

(107)

The mathematical closed-form solution to obtain the force capability in manipulators with CN = 5 presented herein represents the second most important result obtained in this study. It should be noted that the fifteen kinematic variables (a1 , . . . , a15 ) were obtained from the solution of the forward static problem presented in Eq. (5), and the terms va (h), vb (h), vc (h), vd (h) and ve (h) are obtained as shown in Eqs. (108) to (112). At this point it is important to highlight that although variables va (h) to vd (h) appeared previously in Section 3, in this section these variables assume new values as shown in Eqs. (108) to (111) in order to solve the force capability in manipulators with CN = 5. va (h) = a6 − a1 tan(h)

(108)

vb (h) = a7 − a2 tan(h)

(109)

vc (h) = a8 − a3 tan(h)

(110)

vd (h) = a9 − a4 tan(h)

(111)

ve (h) = a10 − a5 tan(h)

(112)

70

L. Mejia, et al. / Mechanism and Machine Theory 105 (2016) 58–79

6. Mathematical closed-form solution to obtain the force capability in robotic mechanisms with CN = 6 Similarly as in Sections 4 and 5, the mathematical closed-form solution to obtain the force capability in manipulators with CN = 6 was obtained by using an optimization procedure. This section shows in a direct way the mathematical closed-form solution to obtain the force capability in manipulators with CN = 6. This mathematical closed-form solution is presented in Eqs. (113) to (165).

Fapp (h, Miz ) = max fe1 (t1 , t2 , t3 , t4 , t5 (t1 , t2 , t3 , t4 ), t6 (t1 , t2 , t3 , t4 )) • kkt1 ; · · · · · · fe2 (t1 , t2 , t3 , t4 (t1 , t2 , t3 , t5 ), t5 , t6 (t1 , t2 , t3 , t5 )) • kkt2 ; fe3 (t1 , t2 , t3 (t1 , t2 , t4 , t5 ), t4 , t5 , t6 (t1 , t2 , t4 , t5 )) • kkt3 ; · · · · · · fe4 (t1 , t2 (t1 , t3 , t4 , t5 ), t3 , t4 , t5 , t6 (t1 , t3 , t4 , t5 )) • kkt4 ; fe5 (t1 (t2 , t3 , t4 , t5 ), t2 , t3 , t4 , t5 , t6 (t2 , t3 , t4 , t5 )) • kkt5 · · · · · · fe6 (t1 , t2 , t3 , t4 (t1 , t2 , t3 , t6 ), t5 (t1 , t2 , t3 , t6 ), t6 ) • kkt6 ; fe7 (t1 , t2 , t3 (t1 , t2 , t4 , t6 ), t4 , t5 (t1 , t2 , t4 , t6 ), t6 ) • kkt7 ; · · · · · · fe8 (t1 , t2 (t1 , t3 , t4 , t6 ), t3 , t4 , t5 (t1 , t3 , t4 , t6 ), t6 ) • kkt8 ; fe9 (t1 (t2 , t3 , t4 , t6 ), t2 , t3 , t4 , t5 (t2 , t3 , t4 , t6 ), t6 ) • kkt9 ; · · · · · · fe10 (t1 , t2 , t3 (t1 , t2 , t5 , t6 ), t4 (t1 , t2 , t5 , t6 ), t5 , t6 ) • kkt10 ; fe11 (t1 , t2 (t1 , t3 , t5 , t6 ), t3 , t4 (t1 , t3 , t5 , t6 ), t5 , t6 ) • kkt11 ; · · · · · · fe12 (t1 (t2 , t3 , t5 , t6 ), t2 , t3 , t4 (t2 , t3 , t5 , t6 ), t5 , t6 ) • kkt12 ; fe13 (t1 , t2 (t1 , t4 , t5 , t6 ), t3 (t1 , t4 , t5 , t6 ), t4 , t5 , t6 ) • kkt13 ; · · ·  · · · fe14 (t1 (t2 , t4 , t5 , t6 ), t2 , t3 (t2 , t4 , t5 , t6 ), t4 , t5 , t6 ) • kkt14 ; fe15 (t1 (t3 , t4 , t5 , t6 ), t2 (t3 , t4 , t5 , t6 ), t3 , t4 , t5 , t6 ) • kkt15 (113)

where  1 2 fen (t1 , t2 , t3 , t4 , t5 , t6 ) = (a1 t1 +a2 t2 + a3 t3 +a4 t4 + a5 t5 +a6 t6 )2 + (a7 t1 +a8 t2 + a9 t3 +a10 t4 + a11 t5 +a12 t6 )2 (114)  kkt1 =

kkt2 =

kkt3 =

kkt4 =

kkt5 =

kkt6 =

kkt7 =

kkt8 =

kkt9 =

1 ⇐⇒ (−t5max ≤ t5 ≤ t5max ) ∧ (−t6max ≤ t6 ≤ t6max )

0 other case  1 ⇐⇒ (−t4max ≤ t4 ≤ t4max ) ∧ (−t6max ≤ t6 ≤ t6max ) 0 other case  1 ⇐⇒ (−t3max ≤ t3 ≤ t3max ) ∧ (−t6max ≤ t6 ≤ t6max ) 0 other case  1 ⇐⇒ (−t2max ≤ t2 ≤ t2max ) ∧ (−t6max ≤ t6 ≤ t6max ) 0 other case  1 ⇐⇒ (−t1max ≤ t1 ≤ t1max ) ∧ (−t6max ≤ t6 ≤ t6max ) 0 other case  1 ⇐⇒ (−t4max ≤ t4 ≤ t4max ) ∧ (−t5max ≤ t5 ≤ t5max ) 0 other case  1 ⇐⇒ (−t3max ≤ t3 ≤ t3max ) ∧ (−t5max ≤ t5 ≤ t5max ) 0 other case  1 ⇐⇒ (−t2max ≤ t2 ≤ t2max ) ∧ (−t5max ≤ t5 ≤ t5max ) 0 other case  1 ⇐⇒ (−t1max ≤ t1 ≤ t1max ) ∧ (−t5max ≤ t5 ≤ t5max )

kkt10 =

kkt11 =

kkt12 =

0 other case  1 ⇐⇒ (−t3max ≤ t3 ≤ t3max ) ∧ (−t4max ≤ t4 ≤ t4max ) 0 other case  1 ⇐⇒ (−t2max ≤ t2 ≤ t2max ) ∧ (−t4max ≤ t4 ≤ t4max ) 0 other case  1 ⇐⇒ (−t1max ≤ t1 ≤ t1max ) ∧ (−t4max ≤ t4 ≤ t4max ) 0

other

case

(115)

(116)

(117)

(118)

(119)

(120)

(121)

(122)

(123)

(124)

(125)

(126)

L. Mejia, et al. / Mechanism and Machine Theory 105 (2016) 58–79

 kkt13 = 

other

(127)

case

1 ⇐⇒ (−t1max ≤ t1 ≤ t1max ) ∧ (−t3max ≤ t3 ≤ t3max ) 0

 kkt15 =

1 ⇐⇒ (−t2max ≤ t2 ≤ t2max ) ∧ (−t3max ≤ t3 ≤ t3max ) 0

kkt14 =

71

other

(128)

case

1 ⇐⇒ (−t1max ≤ t1 ≤ t1max ) ∧ (−t2max ≤ t2 ≤ t2max ) 0

other

(129)

case

In this new solution, Eq. (113) is represented as the maximum value of fifteen elements where each element in these equations is obtained as the product of Eq. (114) evaluated with different values of saturation for t1 , t2 , t3 , t4 , t5 and t6 and a penalization term (kktn ) as shown in Eqs. (115) to (129). In order to exemplify the chosen different values of saturation for the actuated joints tn , consider the first element in Eq. (113) similarly as was done in Section 5, where fe1 (t1 , t2 , t3 , t4 , t5 (t1 , t2 , t3 , t4 ), t6 (t1 , t2 , t3 , t4 )) is a composed function that needs to be evaluated by all the possible saturation values of t1 , t2 , t3 and t4 . The possible values of saturation for t1 , t2 , t3 and t4 are shown in Table 2, and the elements t5 and t6 need to be calculated as a function of these saturated elements and the imposed moment (Miz ) and the desired direction (h) as shown in Eqs. (158) and (108). The same process needs to be followed in order to evaluate all the elements in Eq. (113) but saturating the independent values tn for each element fen in these equations.

t1 (t3 , t4 , t5 , t6 , Miz , h) =

(Miz − t3 a15 − t4 a16 − t5 a17 − t6 a18 ) vb (h) + (t3 vc (h) + t4 vd (h) + t5 ve (h) + t6 vf (h)) a14 a13 vb (h) − a14 va (h) (130)

t2 (t3 , t4 , t5 , t6 , Miz , h) =

(Miz − t3 a15 − t4 a16 − t5 a17 − t6 a18 ) va (h) + (t3 vc (h) + t4 vd (h) + t5 ve (h) + t6 vf (h)) a13 a14 va (h) − a13 vb (h) (131)

t1 (t2 , t4 , t5 , t6 , Miz , h) =

(Miz − t2 a14 − t4 a16 − t5 a17 − t6 a18 ) vc (h) + (t2 vb (h) + t4 vd (h) + t5 ve (h) + t6 vf (h)) a15 a13 vc (h) − a15 va (h) (132)

t3 (t2 , t4 , t5 , t6 , Miz , h) =

(Miz − t2 a14 − t4 a16 − t5 a17 − t6 a18 ) va (h) + (t2 vb (h) + t4 vd (h) + t5 ve (h) + t6 vf (h)) a13 a15 va (h) − a13 vc (h) (133)

t1 (t2 , t3 , t5 , t6 , Miz , h) =

(Miz − t2 a14 − t3 a15 − t5 a17 − t6 a18 ) vd (h) + (t2 vb (h) + t3 vc (h) + t5 ve (h) + t6 vf (h)) a16 a13 vd (h) − a16 va (h) (134)

Table 2 Saturation values of t1 , t2 , t3 and t4 in the first element of Eq. 113. Case

Value for t1

Value for t2

Value for t3

Value for t4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

−t1max −t1max −t1max −t1max −t1max −t1max −t1max −t1max t1max t1max t1max t1max t1max t1max t1max t1max

−t2max −t2max −t2max −t2max t2max t2max t2max t2max −t2max −t2max −t2max −t2max t2max t2max t2max t2max

−t3max −t3max t3max t3max −t3max −t3max t3max t3max −t3max −t3max t3max t3max −t3max −t3max t3max t3max

−t4max t4max −t4max t4max −t4max t4max −t4max t4max −t4max t4max −t4max t4max −t4max t4max −t4max t4max

72

L. Mejia, et al. / Mechanism and Machine Theory 105 (2016) 58–79

t4 (t2 , t3 , t5 , t6 , Miz , h) =

(Miz − t2 a14 − t3 a15 − t5 a17 − t6 a18 ) va (h) + (t2 vb (h) + t3 vc (h) + t5 ve (h) + t6 vf (h)) a13 a16 va (h) − a13 vd (h) (135)

t1 (t2 , t3 , t4 , t6 , Miz , h) =

(Miz − t2 a14 − t3 a15 − t4 a16 − t6 a18 ) ve (h) + (t2 vb (h) + t3 vc (h) + t4 vd (h) + t6 vf (h)) a17 a13 ve (h) − a17 va (h) (136)

t5 (t2 , t3 , t4 , t6 , Miz , h) =

(Miz − t2 a14 − t3 a15 − t4 a16 − t6 a18 ) va (h) + (t2 vb (h) + t3 vc (h) + t4 vd (h) + t6 vf (h)) a13 a17 va (h) − a13 ve (h) (137)

t1 (t2 , t3 , t4 , t5 , Miz , h) =

(Miz − t2 a14 − t3 a15 − t4 a16 − t5 a17 ) vf (h) + (t2 vb (h) + t3 vc (h) + t4 vd (h) + t5 ve (h)) a18 a13 vf (h) − a18 va (h) (138)

t6 (t2 , t3 , t4 , t5 , Miz , h) =

(Miz − t2 a14 − t3 a15 − t4 a16 − t5 a17 ) va (h) + (t2 vb (h) + t3 vc (h) + t4 vd (h) + t5 ve (h)) a13 a18 va (h) − a13 vf (h) (139)

t2 (t1 , t4 , t5 , t6 , Miz , h) =

(Miz − t1 a13 − t4 a16 − t5 a17 − t6 a18 ) vc (h) + (t1 va (h) + t4 vd (h) + t5 ve (h) + t6 vf (h)) a15 a14 vc (h) − a15 vb (h) (140)

t3 (t1 , t4 , t5 , t6 , Miz , h) =

(Miz − t1 a13 − t4 a16 − t5 a17 − t6 a18 ) vb (h) + (t1 va (h) + t4 vd (h) + t5 ve (h) + t6 vf (h)) a14 a15 vb (h) − a14 vc (h) (141)

t2 (t1 , t3 , t5 , t6 , Miz , h) =

(Miz − t1 a13 − t3 a15 − t5 a17 − t6 a18 ) vd (h) + (t1 va (h) + t3 vc (h) + t5 ve (h) + t6 vf (h)) a16 a14 vd (h) − a16 vb (h) (142)

t4 (t1 , t3 , t5 , t6 , Miz , h) =

(Miz − t1 a13 − t3 a15 − t5 a17 − t6 a18 ) vb (h) + (t1 va (h) + t3 vc (h) + t5 ve (h) + t6 vf (h)) a14 a16 vb (h) − a14 vd (h) (143)

t2 (t1 , t3 , t4 , t6 , Miz , h) =

(Miz − t1 a13 − t3 a15 − t4 a16 − t6 a18 ) ve (h) + (t1 va (h) + t3 vc (h) + t4 vd (h) + t6 vf (h)) a17 a14 ve (h) − a17 vb (h) (144)

t5 (t1 , t3 , t4 , t6 , Miz , h) =

(Miz − t1 a13 − t3 a15 − t4 a16 − t6 a18 ) vb (h) + (t1 va (h) + t3 vc (h) + t4 vd (h) + t6 vf (h)) a14 a17 vb (h) − a14 ve (h) (145)

t2 (t1 , t3 , t4 , t5 , Miz , h) =

(Miz − t1 a13 − t3 a15 − t4 a16 − t5 a17 ) vf (h) + (t1 va (h) + t3 vc (h) + t4 vd (h) + t5 ve (h)) a18 a14 vf (h) − a18 vb (h) (146)

t6 (t1 , t3 , t4 , t5 , Miz , h) =

(Miz − t1 a13 − t3 a15 − t4 a16 − t5 a17 ) vb (h) + (t1 va (h) + t3 vc (h) + t4 vd (h) + t5 ve (h)) a14 a18 vb (h) − a14 vf (h) (147)

t3 (t1 , t2 , t5 , t6 , Miz , h) =

(Miz − t1 a13 − t2 a14 − t5 a17 − t6 a18 ) vd (h) + (t1 va (h) + t2 vb (h) + t5 ve (h) + t6 vf (h)) a16 a15 vd (h) − a16 vc (h) (148)

L. Mejia, et al. / Mechanism and Machine Theory 105 (2016) 58–79

73

t4 (t1 , t2 , t5 , t6 , Miz , h) =

(Miz − t1 a13 − t2 a14 − t5 a17 − t6 a18 ) vc (h) + (t1 va (h) + t2 vb (h) + t5 ve (h) + t6 vf (h)) a15 a16 vc (h) − a15 vd (h) (149)

t3 (t1 , t2 , t4 , t6 , Miz , h) =

(Miz − t1 a13 − t2 a14 − t4 a16 − t6 a18 ) ve (h) + (t1 va (h) + t2 vb (h) + t4 vd (h) + t6 vf (h)) a17 a15 ve (h) − a17 vc (h) (150)

t5 (t1 , t2 , t4 , t6 , Miz , h) =

(Miz − t1 a13 − t2 a14 − t4 a16 − t6 a18 ) vc (h) + (t1 va (h) + t2 vb (h) + t4 vd (h) + t6 vf (h)) a15 a17 vc (h) − a15 ve (h) (151)

t3 (t1 , t2 , t4 , t5 , Miz , h) =

(Miz − t1 a13 − t2 a14 − t4 a16 − t5 a17 ) vf (h) + (t1 va (h) + t2 vb (h) + t4 vd (h) + t5 ve (h)) a18 a15 vf (h) − a18 vc (h) (152)

t6 (t1 , t2 , t4 , t5 , Miz , h) =

(Miz − t1 a13 − t2 a14 − t4 a16 − t5 a17 ) vc (h) + (t1 va (h) + t2 vb (h) + t4 vd (h) + t5 ve (h)) a15 a18 vc (h) − a15 vf (h) (153)

t4 (t1 , t2 , t3 , t6 , Miz , h) =

(Miz − t1 a13 − t2 a14 − t3 a15 − t6 a18 ) ve (h) + (t1 va (h) + t2 vb (h) + t3 vc (h) + t6 vf (h)) a17 a16 ve (h) − a17 vd (h) (154)

t5 (t1 , t2 , t3 , t6 , Miz , h) =

(Miz − t1 a13 − t2 a14 − t3 a15 − t6 a18 ) vd (h) + (t1 va (h) + t2 vb (h) + t3 vc (h) + t6 vf (h)) a16 a17 vd (h) − a16 ve (h) (155)

t4 (t1 , t2 , t3 , t5 , Miz , h) =

(Miz − t1 a13 − t2 a14 − t3 a15 − t5 a17 ) vf (h) + (t1 va (h) + t2 vb (h) + t3 vc (h) + t5 ve (h)) a18 a16 vf (h) − a18 vd (h) (156)

t6 (t1 , t2 , t3 , t5 , Miz , h) =

(Miz − t1 a13 − t2 a14 − t3 a15 − t5 a17 ) vd (h) + (t1 va (h) + t2 vb (h) + t3 vc (h) + t5 ve (h)) a16 a18 vd (h) − a16 vf (h) (157)

t5 (t1 , t2 , t3 , t4 , Miz , h) =

(Miz − t1 a13 − t2 a14 − t3 a15 − t4 a16 ) vf (h) + (t1 va (h) + t2 vb (h) + t3 vc (h) + t4 vd (h)) a18 a17 vf (h) − a18 ve (h) (158)

t6 (t1 , t2 , t3 , t4 , Miz , h) =

(Miz − t1 a13 − t2 a14 − t3 a15 − t4 a16 ) ve (h) + (t1 va (h) + t2 vb (h) + t3 vc (h) + t4 vd (h)) a17 a18 ve (h) − a17 vf (h) (159)

The mathematical closed-form solution to obtain the force capability in manipulators with CN = 6 presented herein represents the third most important result obtained in this study. It should be noted that the eighteen kinematic variables (a1 , . . . , a18 ) were obtained from the solution of the forward static problem presented in Eq. (6), and the terms va (h), vb (h), vc (h), vd (h), ve (h) and vf (h) are obtained now as shown in Eqs. (160) to (165). Similarly as in the previous section, it is important to highlight that although variables va (h) to ve (h) appeared previously, in this section these variables assume new values as shown in Eqs. (160) to (165) in order to solve the force capability in manipulators with CN = 6. va (h) = a7 − a1 tan(h)

(160)

vb (h) = a8 − a2 tan(h)

(161)

74

L. Mejia, et al. / Mechanism and Machine Theory 105 (2016) 58–79

vc (h) = a9 − a3 tan(h)

(162)

vd (h) = a10 − a4 tan(h)

(163)

ve (h) = a11 − a5 tan(h)

(164)

vf (h) = a12 − a6 tan(h)

(165)

7. Applications and results To validate the mathematical closed-form solutions proposed in this paper four cases were studied, the first is a redundantlyactuated 4RRR planar parallel manipulator, the second is a redundantly-actuated 5RRR planar parallel manipulator, the third is a redundantly-actuated 6RRR planar parallel manipulator, and finally, the fourth is a redundantly-actuated 3RRR planar parallel manipulator. The planar parallel manipulators studied herein were shown in Fig. 2(a), (b), (c) and (d) respectively. In these parallel manipulators, the fixed and moving platforms are formed by regular polygons with 3, 4, 5 and 6 sides joined by using the same number of legs. Each leg has three rotational joints whose axes are perpendicular to the (x − y) plane, and some of their joints in each leg are actuated according to its net degree of constraint (CN ). Notice that in the planar parallel manipulators shown in Fig. 2(a), (b), (c) and (d), all of them have a circular envelopment around their moving and fixed platforms. These circular envelopments were constructed in order to normalize the geometry of the studied manipulators. In this study, the fixed and moving platforms of the parallel manipulators are circumscribed about circles with diameters 0f = 1[m] and 0m = 0.3[m] respectively. In all the studied manipulators herein, the legs are formed by two links with lengths l1 and l2 , the end effector of the manipulator is located in the geometrical center of the moving platform (E) and the angle of orientation (a) represents the orientation of the moving platform [11, 12]. Here, the link lengths in each leg were specified as l1 = l2 = 0.6[m], the end effector of the manipulator is located in E = (0, 0)[m], the moving platform is oriented in a = 0◦ and the maximum torque for each actuated joint of the manipulator is imposed as tmax = ±100[Nm]. The closed-form solutions shown in Sections 4, 5 and 6 allow us to know the maximum force (Fapp ) with a prescribed moment (Miz ) that can be applied or sustained in a given direction (h). If all the possible directions of the desired angle h are considered as 0◦ ≤ h ≤ 360◦ , a force capability polygon can be constructed as a polar representation of the maximum force with a prescribed moment at the end effector of the manipulator. Considering the imposed moment Miz = 0, the force capability polygon for the 4RRR redundant planar parallel manipulator (RPPM) is obtained as shown in Fig. 4 (a), the force capability polygon for the 5RRR RPPM is obtained as shown in Fig. 4(b), the force capability polygon for the 6RRR RPPM is obtained as shown in Fig. 4(c) and finally the force capability polygon for the 3RRR RPPM is obtained as shown in Fig. 4(d). Following the same strategy that we used previously, it is possible to obtain different wrench capability polygons for different values of the imposed moment at the end effector of the manipulator. If a three-dimensional representation of several wrench capability polygons is plotted, a complete mapping of the wrench capability at the end effector of the manipulator can be obtained. This kind of graphic representation is called the wrench capability polytope . The wrench capability polytope of the first studied manipulator (the redundantly-actuated 4RRR planar parallel manipulator) is shown in Fig. 5 where it can be noted that the force capability polytope is obtained as a set of individual force capability polygons. In order to illustrate this, it can be observed that the force capability polygon shown as a red-dotted line in Fig. 5, where the prescribed moment is Miz = 0, is the same as the force capability polygon shown in Fig. 4(a). Following the same idea, wrench capability polytopes for the 5RRR, 6RRR and 3RRR redundantly-actuated planar parallel manipulators can be constructed as shown in Figs. 6, 7 and 8. As previously discussed, the force capability polytopes shown in Figs. 6, 7 and 8 are obtained as a set of individual force capability polygons where the force capability polygons shown as red-dotted lines are the same as the force capability polygons shown in Fig. 4(b), (c) and (d). 8. Discussion about the proposed closed-form solutions Some important issues regarding the closed-form solutions obtained are related to its simplicity, generality, versatility and computing time. We comment these aspects below. The closed-form solutions shown in Sections 4, 5 and 6 are mathematical expressions that can be evaluated in a finite number of operations, containing constants, variables, well known operations (e.g., +, −, ×, ÷) and functions (e.g., square root, exponent, maximum) and thus the evaluation is simpler compared with other methodologies found in the literature. The evaluation of the closed-form solutions is based on a knowledge of the kinematic variables (a1,1 , . . . , a3,CN ) obtained from the solution of the forward static problem (presented in Eqs. (4), (5) and (6)), the desired moment and direction of the application of the force at the end effector of the manipulator, and the maximum torque in the actuated joints. Since these characteristics are a common factor in all manipulators with CN = 4, CN = 5 and CN = 6 the proposed solutions represent general solutions for any manipulator with these net degree of constraint.

L. Mejia, et al. / Mechanism and Machine Theory 105 (2016) 58–79

75

Fig. 4. Force capability polygons for: (a) 4RRR PPM, (b) 5RRR RPPM, (c) 6RRR RPPM and (d) 3RRR planar parallel manipulator.

Given that the force capabilities as a function of the desired moment and the fixed force direction can be established relatively easily, it is possible to solve several processes which require easy redefinition of the task. This allows us to affirm that the proposed closed-form solutions offer a high degree of versatility. Finally, as the effort necessary to evaluate the mathematical closed-form solutions is reduced to the time needed to evaluate some finite operations, the computing time is less than that associated with other methodologies using heuristic or meta-heuristic approaches described in the literature. It should be emphasized that the proposed mathematical closed-form solutions obtained in Sections 4, 5 and 6 work only when the manipulator evaluated has a net degree of constraint value equal to four, five or six (CN = 4, CN = 5 or CN = 6). Manipulators with a net degree of constraint values which are not four, five or six (CN = 4, CN = 5 or CN = 6) cannot be solved using the mathematical closed-form solutions proposed in this paper. The results shown in Figs. 4, 5, 6, 7 and 8 are the same as those obtained by using the method proposed by Mejia et al. [11,12] (based on a DE algorithm) for the same manipulators, using the same dimensions, topological structure, posture and maximum torques at the actuated joints. This allowed us to compare and validate the results obtained using the proposed mathematical closed-form solutions. The comparison showed that the force capabilities obtained were exactly the same as the results reported by Mejia et al. [12], but an important difference lies in the computing time used to obtain these results. Using the proposed generalized mathematical closed-form solutions for the manipulators with CN = 4, CN = 5 and CN = 6, the force capability polygon for one pose is completed in 0.4622 s, 1.1894 s and 3.7868 s respectively when running on a P4 2.4 GHz computer. As a comparison, the same results were obtained using the method proposed by Mejia et al. [12] using a

76

L. Mejia, et al. / Mechanism and Machine Theory 105 (2016) 58–79

Fig. 5. Force capability polytope for the studied 4RRR RPPM. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

differential evolution algorithm (DE), but with a very slow response of more than 7560 s for each case when running on the same P4 2.4 GHz computer. The differential evolution algorithm (DE) used to validate the results in this paper has a sensitive performance to the mutation strategy and respective control parameters such as the population size (NP), crossover rate (CR) and the mutation factor (MF). The best settings for the control parameters can be different for different optimization problems and the same functions with different requirements for consumption time and accuracy. The classical DE starts with a population of random solutions generated randomly with uniform distribution. At each generation, a trial population is generated via mutation and crossover operations; and then a selection operation is applied to select the better vectors into the next generation. It evolutes generation by generation until the termination conditions have been met. The aim of this work is not to find the best control parameters nor to test the efficiency and feasibility of the DE approach [3]. Classical values of NP, CR and MF suggested in [32] were used in this work achieving algorithm convergence. The number of objective function evaluation (population size × number of generations) was fixed to 6000. Another important issue with regard to the computing time of the proposed generalized mathematical closed-form solution is that when the force capability of manipulators with CN = 4, CN = 5 and CN = 6 is evaluated in a fixed direction, the time used in that operation is only 0.001284 s, 0.003304 s and 0.010519 s respectively (again, when running on a P4 2.4 GHz computer). This response is very fast and allows us to contemplate applications that require a real-time response in terms of the manipulation of the force, such as grasping, polishing, and milling. A comparison of the computing times required to solve the force capability problem is shown in Table 3.

Fig. 6. Force capability polytope for the studied 5RRR RPPM. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

L. Mejia, et al. / Mechanism and Machine Theory 105 (2016) 58–79

77

Fig. 7. Force capability polytope for the studied 6RRR RPPM. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

9. Conclusions Mathematical closed-form solutions to obtain the force capability in manipulators with a net degree of constraint equal to four, five and six are proposed in this paper. The proposed models are dependent on the torque in the actuated joints of the manipulator and the prescribed moment at the manipulator end effector. The proposed mathematical closed-form solutions represent a simpler, faster and more direct approach compared with other solutions found in literature for the same problem. Also, their implementation process can be easily adapted to a high number of manipulators. Furthermore, the mathematical closed-form solutions proposed herein can be applied in several processes and projects that require a fast response in terms of the manipulation of the force, such as milling, polishing, and grasping.

Fig. 8. Force capability polytope for the studied 3RRR RPPM. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

78

L. Mejia, et al. / Mechanism and Machine Theory 105 (2016) 58–79

Table 3 Comparison of the computing time required for different strategies in order to solve the force capability problem. Study case RPPM with CN RPPM with CN RPPM with CN RPPM with CN RPPM with CN RPPM with CN

=4 =5 =6 =4 =5 =6

Method used

Time needed for one pose

Time needed for one direction

Closed-form solution

0.4622 s 1.1894 s 3.7868 s > 7560 s > 9000 s > 10,800 s

0.001284 s 0.003304 s 0.010519 s > 21 s > 25 s > 30 s

(DE) algorithm

The force capability polytope is composed of the superposition of several force capability polygons and is dependent on several parameters, such as their kinematic position, orientation and working mode. Four examples were shown herein in order to illustrate these force capability polytopes. Using the proposed mathematical closed-form solutions, it is possible to obtain the force capability polytope and the force capability polygon of manipulators for each position within the manipulator’s workspace and although only four examples were shown, the proposed mathematical closed-form solution can be applied to any manipulator with a net degree of constraint equal to four, five or six which satisfy the prescribed conditions detailed in Section 2. This study can be extended in several ways, to be able to use for manipulators with different degrees of freedom and kinematic chains and dynamic behavior could be included. The outcome of a generalized mathematical closed-form solution to obtain the maximum force with a prescribed moment, allows us to reduce the computing time and effort required to evaluate this characteristic in planar mechanisms with a net degree of constraint equal to four, five or six. It opens the possibility of real-time applications in processes such as manipulation and grasping. Acknowledgments The authors would like to thank the Department of Mechanical Engineering of the Federal University of Santa Catarina and the National Council for Scientific and Technological Development (CNPq) which have made the present work possible.

References [1] F. Firmani, A. Zibil, R.P. Podhorodeski, S.B. Nokleby, Wrench Capabilities of Planar Parallel Manipulators and Their Effects Under Redundancy, INTECH Open Access Publisher. 2008. [2] S.B. Nokleby, R. Fisher, R.P. Podhorodeski, F. Firmani, Force capabilities of redundantly-actuated parallel manipulators, Mech. Mach. Theory 40 (2004) 578–599. [3] L. Weihmann, D. Martins, L.S. Coelho, Force Capabilities of Kinematically Redundant Planar Parallel Manipulators, 13th World Congress in Mechanism and Machine Science, 2011, 483-483. [4] F. Firmani, S.B. Nokleby, R.P. Podhorodeski, A. Zibil, Wrench Capabilities of Planar Parallel Manipulators and Their Effects Under Redundancy, Parallel Manipulators, towards New Applications, 2008, 109–132. [5] T.H. Davies, Mechanical networks III: wrenches on circuit screws, Mech. Mach. Theory (1983) 107–112. [6] V. Garg, S.B. Nokleby, J.A. Carretero, Force-Moment Capabilities of Redundantly-Actuated Spatial Parallel Manipulators Using Two Methods, Proceedings of the 2007 CCToMM Symposium on Mechanisms, Machines, and Mechatronics, 2007, pp. 12. [7] T. Yoshikawa, Manipulability and Redundancy Control of Robotic Mechanisms, in Robotics Research: The Second International Symposium, 1985, 1004–1009. [8] M.A. Nahon, J. Angeles, Real-time force optimization in parallel kinematic chains under inequality constraints, IEEE Trans. Robot. Autom. 8 (4) (1992) 439–450. [9] J. Wu, J. Wang, L. Wang, T. Li, Dynamics and control of a planar 3-DOF parallel manipulator with actuation redundancy, Mech. Mach. Theory 44 (4) (2009) 835–849. [10] J.M. Tao, J.Y.S. Luh, Coordination of two redundant manipulators, Proceedings of the 1989 IEEE International Conference on Robotics and Automation, 1989, pp. 425–430. [11] L. Mejia, H. Simas, D. Martins, Force Capability Polytope of a 3RRR Planar Parallel Manipulator, Proceedings of Romansy 2014 XX CISM-IFToMM Symposium on Theory and Practice of Robots and Manipulators, 2014, 537–545. [12] L. Mejia, H. Simas, D. Martins, Force Capability Polytope of a 4RRR Redundant Planar Parallel Manipulator, Advances in Robot Kinematics, 2014, 87–94. [13] L. Mejia, J. Frantz, H. Simas, D. Martins, Influence of the assembly mode on the force capability in parallel manipulators, Proceeding in 23rd ABCM International Congress of Mechanical Engineering (COBEM 2015), 2015. [14] L. Mejia, J. Frantz, H. Simas, D. Martins, Wrench capability polytopes in redundant parallel manipulators, Proceeding in 23rd ABCM International Congress of Mechanical Engineering (COBEM 2015), 2015. [15] J. Wu, X. Chen, L. Wang, X. Liu, Dynamic load-carrying capacity of a novel redundantly actuated parallel conveyor, Nonlinear Dynamics 78 (1) (2014) 241–250. [16] A. Zibil, F. Firmani, S.B. Nokleby, R.P. Podhorodeski, An explicit method for determining the force moment capabilities of redundantly actuated planar parallel manipulators, Mech. Design (2007) 1046–1056. [17] S.B. Nokleby, F. Firmani, A. Zibil, R.P. Podhorodeski, Force moment capabilities of redundantly actuated planar parallel architectures, 12th IFToMM 2007 World Congress (2007) 17–21. [18] L. Mejia, J. Frantz, H. Simas, D. Martins, Modified scaling factor method for the obtention of the wrench capabilities in cooperative planar manipulators, Proceeding in 14th IFToMM World Congress (IFToMM 2015), 2015. [19] J. Frantz, L. Mejia, H. Simas, D. Martins, Analysis of wrench capability for cooperative robotic systems, Proceeding in 23rd ABCM International Congress of Mechanical Engineering (COBEM 2015) (2015). [20] L. Mejia, H. Simas, D. Martins, Force capability in general 3DoF planar mechanisms, Mech. Mach. Theory 91 (0) (2015) 120–134. [21] F. Firmani, A. Zibil, S.B. Nokleby, R.P. Podhorodeski, Wrench capabilities of planar parallel manipulators. Part I: wrench polytopes and performance indices, Robotica 26 (2008) 791–802.

L. Mejia, et al. / Mechanism and Machine Theory 105 (2016) 58–79

79

[22] F. Firmani, A. Zibil, S.B. Nokleby, R.P. Podhorodeski, Wrench capabilities of planar parallel manipulators. Part II: redundancy and wrench workspace analysis, Robotica 26 (2008) 803–815. [23] R. Finotello, T. Grasso, G. Rossi, A. Terribile, Computation of Kinetostatic Performances of Robot Manipulators with Polytopes, IEEE International Conference on Robotics and Automation, 1998, pp. 3241–3246. [24] B. Siciliano, O. Khatib, Springer Handbook of Robotics, Springer, 2008. [25] T.Y. Chow, What is a closed-form number? Am. Math. Mon. 106 (5) (1999) 440–448. [26] T.H. Davies, Mechanical networks I: passivity and redundancy, Mech. Mach. Theory (1983) 95–101. [27] T.H. Davies, Mechanical networks II: formulae for the degrees of mobility and redundancy, Mech. Mach. Theory (1983) 102–106. [28] L. Weihmann, D. Martins, L. dos Santos Coelho, Modified differential evolution approach for optimization of planar parallel manipulators force capabilities, Expert Syst. Appl. 39. [29] L. Weihmann, Modelagem e otimização de forças e torques aplicados por robôs com redundância cinemática e de atuação em contato com o meio, (Ph.D. thesis). Universidade Federal de Santa Catarina, Brazil, 2011. [30] H.R. Cazangi, Aplicação do método de Davies para Análise Cinemática e Estática de Mecanismos de Múltiplos Graus de Liberdade, (Ph.D. thesis). Universidade Federal de Santa Catarina, Brazil, 2008. [31] P. Chiacchio, Force polytope and force ellipsoid for redundant manipulators, J. Robot. Syst. 14 (1997) 613–620. [32] R. Storn, K. Price, Differential evolution: a simple and efficient adaptive scheme for global optimization over continuous spaces, Technical Report TR-95-012, International Computer Science Institute.