Dynamic analysis of planar parallel robots considering singularities and different payloads

Robotics and Computer--Integrated Manufacturing 46 (2017) 114–121

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Dynamic analysis of planar parallel robots considering singularities and different payloads

MARK

Mustafa Özdemir Department of Mechanical Engineering, Faculty of Engineering, Marmara University, Göztepe Campus, Kadıköy, 34722 Istanbul, Turkey

A R T I C L E I N F O

A BS T RAC T

Keywords: Parallel robot Dynamics Singularity Payload handling Motion planning

This paper studies the effects of payload inertial parameters on its consistent locations for a planar threedegree-of-freedom parallel manipulator in the presence of singularities. It is analytically shown that for a given motion, the consistent locations of different payloads form in general concentric circles on the moving platform plane. A motion planning method is also introduced for planar three-degree-of-freedom parallel manipulators passing through a singular configuration with different payloads. Within these scopes, eleven propositions are stated and proved. This work is believed to be highly useful for a wide range of industrial purposes, especially by providing great flexibility and ease for pick-and-place operations.

1. Introduction Parallel manipulators, with numerous advantages such as high payload-to-weight ratio, good positioning accuracy and high speed capacity [1–3], have a wide usage area, including but not limited to machine tools [4], aerospace [1,5] and automotive [6] technologies, and medical robotics [7,8]. However, they have an unacceptably small workspace in practice, basically due to type II singularities encountered within it [9]. These singularities are characterized with a gain of one or more degrees of freedom of the output link, and unbounded growth of the necessary actuator forces, which will unavoidably lead to the loss of the motion control in their neighborhood [9–11]. There are several solutions proposed to deal with this problem. These are the use of actuation [12,13] or kinematic [14,15] redundancy, the “active masses” [16], the “double parallel manipulator” [17,18], the “joint-coupling” [19], and mechanisms having variable structure [20]. As another cost-effective alternative, a motion planning approach based on the consistency of the dynamic model at the singular position has also been extensively studied in the literature for nonredundant parallel robots [21–25]. If the motion is inconsistent at the singular configuration, the dynamic equations will be contradictory, and hence the motion cannot be realized by the manipulator. Although some promising results were reported, different motion plans are needed when different payloads are carried. This will add complexity to the execution of payload handling tasks by using parallel robots in the presence of type II singularities. Besides, the present author [26] has recently derived the additional conditions (other than the consistency of the dynamic model at the singularity) that should be satisfied

for bounded inverse dynamics solutions as the type II singularity is approached. Some recent studies regarding payload handling of parallel manipulators concentrated on dealing with uncertainties [27], and on maximization of the dynamic load-carrying capacity [28]. In another recent work, the present author [29] studied consistent payload positions for parallel robots in the presence of type II singularities. For planar three-degree-of-freedom parallel manipulators, this study [29] showed that given a prescribed trajectory, the locus of such positions of a given payload on the moving platform plane can be a circle, a unique point, a line or the entire moving platform plane itself, or does not exist at all. Focusing on planar parallel robots with three degrees of freedom, the present paper extends the author's previous work [29] through a novel analysis of the effects of inertial parameters of the payload on its consistent locations in the presence of type II singularities. Based on these new insights, another contribution of the paper is to propose a motion planning method for planar three-degree-of-freedom parallel manipulators passing through a singular configuration with different payloads. Thus, a great flexibility is obtained for pick-and-place operations. For detailed analyses of singularity loci of planar parallel manipulators, readers are invited to see [30–34]. It may be also useful to recall that type II singularities are also known in the literature as direct kinematic [16] or drive [21,22] singularities.

E-mail address: [email protected]. http://dx.doi.org/10.1016/j.rcim.2017.01.005 Received 7 August 2016; Received in revised form 19 December 2016; Accepted 19 January 2017 0736-5845/ © 2017 Elsevier Ltd. All rights reserved.

Robotics and Computer--Integrated Manufacturing 46 (2017) 114–121

M. Özdemir

wh = qh − rh for h = 0, 1, 2

(14)

Denote the vectors of the joint variables of the actuated and u u T a a a T unactuated joints by ηa = [ η1 η2 η3 ] and ηu = [ η1 ⋯ ηk ] , respectively. Here, k is the number of the loop closure constraints, i.e. the loop closure constraint equations can be expressed at position level as follows:

ϕi(ηa , ηu) = 0,

i = 1, …, k

(15)

The dynamic equations of the robot-payload system can be written as

Fig. 1. A planar three-degree-of-freedom 2-PRR parallel manipulator.

2. Dynamic analysis

τj =

Consider a nonredundant, planar parallel manipulator with three degrees of freedom. The notations used in the formulations are as follows:

• • •

k

∑ λn n =1

∂ϕn ∂ηiu

=

k

∑ λn n =1

∂ϕn ∂ηja

d ⎛ ∂L ⎞ ∂L ⎟⎟ − ⎜⎜ , dt ⎝ ∂η˙iu ⎠ ∂ηiu

,

j = 1, 2, 3 (16)

i = 1, …, k (17)

where L is the Lagrangian of the robot-payload system, τ = [ τ1 τ2 τ3 ]T the vector of the actuator forces/torques and λ = [ λ1 ⋯ λk ]T the vector of the Lagrange multipliers. In order to study type II singularities of the manipulator, it is more convenient to rewrite Eq. (17) in the following matrix form:

A payload having a mass mp and a centroidal moment of inertia Ip is rigidly attached to the moving platform, its center of mass being fixed at point C on the platform. Axy is the fixed global coordinate system of the manipulator while Bx′y′ is a moving frame attached to the platform. Let θ denote the angle between the x′ and x axes, measured in the counterclockwise direction from the x -axis (i.e., the angle θ defines the orientation of the moving platform with respect to the base).

(18)

Φλ = β where

⎡ ∂ϕ1 ⎢ ∂η1u … ⎢ Φ=⎢ ⋮ ⋱ ⎢ ∂ϕu1 … ⎣ ∂ηk

An example of the usage of these notations can be seen in Fig. 1 of Section 3. The position of C in the fixed frame can be expressed as

⎡ xC ⎤ ⎡ xB ⎤ ⎡ cos θ − sin θ ⎤⎡ x′C ⎤ ⎢y ⎥ = ⎢y ⎥ + ⎢ ⎥⎢ ⎥ ⎣ C ⎦ ⎣ B ⎦ ⎣ sin θ cos θ ⎦⎣ y′C ⎦

⎞ ⎛ d ⎜ ∂L ⎟ ∂L − a⎟ − ⎜ dt ⎝ ∂η˙j ⎠ ∂ηja

∂ϕk ⎤

∂η1u ⎥

⋮ ⎥⎥ ⎥ ∂ηku ⎦ ∂ϕk

(19)

(1)

⎡ ⎛ ⎞ ⎢ d ⎜ ∂Lu ⎟ − ⎢ dt ⎝ ∂η1̇ ⎠ β=⎢ ⋮ ⎢ ⎢ d ⎛ ∂L ⎞ − ⎢⎣ dt ⎜⎝ ∂ηk̇u ⎟⎠

Kinetic and potential energies of the payload (Tp and Vp respectively) are given by

1 1 2 Tp = mp(xĊ 2 + yĊ 2 ) + Ipθ ̇ 2 2

(2)

Vp = mpgyC

(3)

⎤

∂L ⎥ ∂η1u

⎥ ⎥ ⎥ ∂L ⎥ ∂ηku ⎥ ⎦

(20)

At a type II singular configuration, Φ becomes rank deficient [22,35]. Assuming a rank deficiency of one (this is not a restrictive assumption from a practical point of view [22,23]), the following relations hold at the singular position:

where g is the gravitational acceleration acting in the negative y -direction. Using Eq. (1), and its first time derivative whenever required, one can rewrite Eqs. (2) and (3) as

k

Tp = mp[q0 + q1x′C + q2y′C + w3(x′C 2 + y′C 2 )] + Ipw3

∑ αnΦni = 0,

(4)

n =1

Vp = mp(r0 + r1x′C + r2y′C )

(5)

1 2 (xḂ + yḂ 2 ) 2

(21)

where αn are scalars such that at least one of them is nonzero. Thus, by the use of the Rouché-Frobenius theorem [36], the “consistency condition” (as called in [21,22]) that should be satisfied at the singular configuration can be written as

where

q0 =

i = 1, …, k

(6)

k

q1 = − θ (̇ xḂ sin θ − yḂ cos θ )

(7)

q2 = − θ (̇ xḂ cos θ + yḂ sin θ )

(8)

1 2 w3 = θ ̇ 2

(9)

∑ αnβn = 0 n =1

r0 = gyB

(10)

r1 = g sin θ

(11)

r2 = g cos θ

(12)

If this condition is not satisfied, the rank of the augmented matrix will not be equal to that of the coefficient matrix, and hence Eq. (18) will be inconsistent at the singularity. Notice that L = Lo + Lp where Lo is the Lagrangian of the unloaded robot. This suggests the following rearrangement of Eq. (22): k

∑ αnβnp = χ n =1

2

Lp = Tp − Vp = mpw0 + mp[w1x′C + w2y′C + w3(x′C + y′C )] + Ipw3

(23)

where

Then, the Lagrangian of the payload can be expressed as 2

(22)

k

(13)

χ=−

where

∑ αnβno n =1

115

(24)

Robotics and Computer--Integrated Manufacturing 46 (2017) 114–121

M. Özdemir

βio =

∂Lo d ⎛ ∂Lo ⎞ ⎟⎟ − ⎜⎜ , dt ⎝ ∂η˙iu ⎠ ∂ηiu

i = 1, …, k

βi p =

∂Lp d ⎛ ∂Lp ⎞ ⎟⎟ − ⎜⎜ , ∂ηiu dt ⎝ ∂η˙iu ⎠

i = 1, …, k

have a real nonzero radius. Q.E.D. (25)

Proposition 2. There is a unique consistent payload location if

rp 2 − (26)

δ 2 − 4δ0δ3 + δ 2 2 1 χ = 1 mp δ 3 4δ32

Substituting Eq. (13) into Eq. (23) and rearranging the resulting expression yields

Proof. The proof simply follows from the fact that a point is a circle of zero radius. Q.E.D.

mpδ0 + mpδ1x′C + mpδ 2y′C + mpδ3(x′C 2 + y′C 2 ) + Ipδ3 = χ

Proposition 3. There is no consistent payload location if

(27)

where

rp 2 −

k

δm =

∑ αnΔmn ,

m = 0, 1, 2, 3 (28)

n =1

Δmi

d ⎛ ∂w ⎞ ∂w = ⎜⎜ mu ⎟⎟ − mu , m = 0, 1, 2, 3, i = 1, …, k dt ⎝ ∂η˙i ⎠ ∂ηi

Proof. When the inequality given in the statement of the proposition is satisfied, the radicand in Eq. (34) will be negative. Q.E.D. Proposition 4. For any rp , there is no consistent payload location if

(29)

δ 2 − 4δ0δ3 + δ 2 2 1 χ + 1 ≤0 mp δ 3 4δ32

Eq. (27) gives the consistency condition that should be satisfied by the robot-payload system at the singular position. Recalling that the matrix Φ is assumed to become rank deficient by one at the singular configuration, satisfaction of this condition guarantees bounded inverse dynamics solutions near the singularity, provided the first time derivative of the determinant of Φ is nonzero at the singularity time [26]. As long as δ3 ≠ 0 , Eq. (27) can be put into a more concise form as

x′C 2 + y′C 2 + 2μ1x′C + 2μ2 y′C + μ3 = 0

Proof. The proof arises from noticing that the (−rp 2 ) term appearing in Eq. (34) is always negative. Q.E.D. Proposition 5. The center Q is located at the same position for any payload with different inertial parameters.

(30)

Proof. The conclusion of the proposition can be reached by simply examining the definitions of μ1 and μ2 , and noticing that they are not functions of any of the inertial parameters of the payload. Q.E.D.

where

μ1 =

μ2 =

δ1 2δ3

and such that 6. For any rp mp δ12 − 4δ0δ3 + δ 2 2 1 χ < rp − , the consistent locations of different mp δ 3 4δ32 1 χ payloads with different rp 2 − m δ values represent concentric circles Proposition

(31)

2

δ2 2δ3

μ3 = rp 2 −

(32)

p 3

δ 1 χ + 0 mp δ 3 δ3

whereas the consistent locations of different payloads with the same 1 χ rp 2 − m δ value represent the same circle.

(33)

p 3

Here, rp = Ip / mp is the centroidal radius of gyration of the payload. Eq. (30) is in the form of the general equation of a circle in the x′y′ plane. Its center Q is located at x′Q = − μ1, y′Q = − μ2 , and its radius is

R=

δ 2 − 4δ0δ3 + δ 2 2 1 χ − rp 2 + 1 mp δ 3 4δ 3 2

μ12 + μ2 2 − μ3 =

Proof. An examination of Eq. (34) reveals that the inertial parameters ⎛ 1 χ⎞ of the payload appear in the form of −⎜rp 2 − m δ ⎟. The rest of the proof p 3⎠ ⎝ follows from Proposition 5. Q.E.D. Proposition 7. If χ = 0 , then, provided rp 2 ≤

(34)

function of only rp .

Seven propositions are stated and proved below to present the original contributions of this new analysis. But before, the following remark is worth noting.

mpδ0 + mpδ1x′C + mpδ 2y′C = χ

1 χ < mp δ 3

− 4δ0δ3 + δ 2 4δ3

4δ32

, R is a

(35)

Readers are referred to [29] for the explanation of how the circle of consistent payload locations degenerates to a line depending on the angular acceleration of the moving platform. Eq. (35) can be further rearranged as,

δ1x′C + δ 2y′C + ψ = 0

(36)

where

Proposition 1. The consistent payload locations lie on a circle if

δ12

δ12 − 4δ0δ3 + δ 22

Proof. The proof directly follows from setting χ = 0 in Eq. (34). The inequality given in the statement of the proposition is for guaranteeing the existence of at least one consistent payload location. Q.E.D. When δ3 = 0 for a prescribed motion, Eq. (27) reduces to the following line equation, provided that at least one of δ1 and δ 2 is nonzero:

Remark. Consistency of the dynamic model of the unloaded manipulator at a type II singularity is equivalent to χ being zero at this singular configuration. Consider a planar three-degree-of-freedom parallel manipulator. Let Φ be rank deficient by one at a type II singularity. Given a prescribed motion, calculate χ and δm , m = 0, 1, 2, 3 using the values of the position, velocity and acceleration variables at the singularity. Let further δ3 ≠ 0 . Then the following seven propositions hold for the payload locations on the moving platform that will render the dynamic model of the robot consistent at this singular configuration (such payload locations are simply called the consistent payload locations in the propositions below):

rp 2 −

δ 2 − 4δ0δ3 + δ 2 2 1 χ > 1 mp δ 3 4δ32

ψ = δ0 −

2

2

χ mp

(37)

The following three propositions explain for the first time the effects of the payload inertial parameters in such a case. Consider a planar three-degree-of-freedom parallel manipulator. Let Φ be rank deficient by one at a type II singularity. Given a

Proof. The proof directly follows from the fact that the radicand in Eq. (34) should be positive for the circle of consistent payload locations to 116

Robotics and Computer--Integrated Manufacturing 46 (2017) 114–121

M. Özdemir

forces corresponding to s1 and s2 , respectively, and T is the actuator torque corresponding to ζ1. The rest of the model parameters are defined where they first appear. Now some preliminaries are to be given. The loop-closure equations can be written as

prescribed motion, calculate χ and δm , m = 0, 1, 2, 3 using the values of the position, velocity and acceleration variables at the singularity. Let further δ3 be zero and at least one of δ1 and δ 2 be nonzero. Then the following three propositions hold for the consistent payload locations: Proposition 8. rp does not have any effect on the consistent payload locations.

ϕ1 = s1 cos γ1 + b2 cos ζ1 + b5 cos θ − b0 − s2 cos γ2 − b4 cos ζ2 = 0

(38)

Proof. The proof follows from a direct examination of Eq. (35). Q.E.D.

ϕ2 = s1 sin γ1 + b2 sin ζ1 + b5 sin θ − s2 sin γ2 − b4 sin ζ2 = 0

(39)

Proposition 9. If χ = 0 , mp also does not have any effect on the consistent payload locations, i.e., the locus of consistent locations of any payload represents the same unique line for a prescribed motion.

⎡ ∂ϕ1 ⎢ ∂ζ Φ=⎢ 2 ∂ϕ ⎢⎣ ∂θ1

where b0 = AE and b5 = BK . Thus, Φ is

Proof. The proof follows from setting χ = 0 in Eq. (37) and substituting the resulting expression into Eq. (36). Q.E.D.

∂ϕ2 ⎤ ∂ζ2

⎡ b sin ζ2 −b4 cos ζ2 ⎤ =⎢ 4 ⎥ ⎣−b5 sin θ b5 cos θ ⎦ ⎦⎥ ⎥

∂ϕ2 ⎥ ∂θ

(40)

Type II singular configurations are encountered when [38]

detΦ = b4b5 sin(ζ2 − θ ) = 0

Proposition 10. Let χ be nonzero. Then the consistent locations of different payloads with different masses represent parallel lines whereas the consistent locations of different payloads with the same mass represent the same line.

(41) b

At such a configuration, Eq. (21) holds with α1 = 1 and α2 = σ b4 5 where

⎧1 forζ2 = θ σ=⎨ ⎩−1 forζ2 = θ ± π

Proof. It is because δ1 and δ 2 appearing in the equation of the line are not functions of any of the inertial parameters of the payload, and hence the angle between the line and the x′-axis does not change for different payloads. Only the intercepts of the line are changed for different payload masses, as should be clear from Eqs. (36) and (37). Q.E.D. Finally, the following proposition establishes the necessary conditions for motion of a planar three-degree-of-freedom parallel manipulator passing through a type II singularity with different payloads.

(42)

Below is given a step-by-step derivation of the necessary conditions for motion of this robot to pass through a type II singular configuration with different payloads. Step 1. Determine w0 , w1 and w2 using Eqs. (14):

w0 =

1 2 2 (s1̇ + b2 2ζ1̇ ) + b2s1̇ ζ1̇ sin(γ1 − ζ1) − g(s1 sin γ1 + b2 sin ζ1) 2

(43)

Proposition 11. Consider a planar three-degree-of-freedom parallel manipulator. Let Φ be rank deficient by one at a type II singularity. If the motion is planned such that the following five conditions are all satisfied at the singularity, then for any payload and for any payload location on the moving platform, the dynamic equations are consistent at this singular configuration:

w1 = [s1̇ sin(γ1 − θ ) + b2ζ1̇ cos(ζ1 − θ )]θ ̇ − g sin θ

(44)

w2 = [−s1̇ cos(γ1 − θ ) + b2ζ1̇ sin(ζ1 − θ )]θ ̇ − g cos θ

(45)

(i) (ii) (iii) (iv) (v)

Δ01 = Δ02 = Δ11 = Δ21 = Δ31 = 0

(46)

2 Δ12 = s1̈ sin(γ1 − θ ) + b2ζ1̈ cos(ζ1 − θ ) − b2ζ1̇ sin(ζ1 − θ ) + g cos θ

(47)

Notice that w3 is already given by Eq. (9). Step 2. Determine Δmi , m = 0, 1, 2, 3, i = 1, 2 using Eqs. (29):

χ=0 δ0 = 0 δ1 = 0 δ2 = 0 δ3 = 0

2 Δ22 = − s1̈ cos(γ1 − θ ) + b2ζ1̈ sin(ζ1 − θ ) + b2ζ1̇ cos(ζ1 − θ ) − g sin θ

(48)

Proof. It is because by doing so, Eq. (27) is satisfied for any mp , Ip , x′C and y′C . Q.E.D.

Δ32 = θ ̈

3. Numerical examples

(49)

Step 3. Determine δ0 , δ1, δ 2 and δ3 using Eqs. (28):

In this section, based on Proposition 11, first a motion is planned for a planar three-degree-of-freedom parallel manipulator to pass through a type II singularity with any payload at any location on the moving platform. Then, the inverse dynamic problem is solved for three different loading conditions in order to show the effectiveness of the proposed motion planning method. Consider a planar three-degree-of-freedom 2-PRR parallel manipulator, as shown in Fig. 1. This manipulator was previously studied for its dexterity in [37] and for a demonstration of the application of the “singularity robust balancing” method in [38]. Links 1–5 have masses m1 to m5, respectively. Links 2 and 4 are homogenous uniform slender bars with lengths of b2 and b4 , respectively. Link 5 has a centroidal moment of inertia of I5, and the location of its center of mass is given by c5 = BG5 and γ5 = ∠G5BK . The gravity acts in the negative y -direction. The joint variable vector is η = [ s1 ζ1 s2 ζ2 θ ]T where s1 = AD and s2 = EJ . The joint variables s1, ζ1 and s2 are the actuated joint variables, hence ηa = [ s1 ζ1 s2 ]T and ηu = [ ζ2 θ ]T . Note that k = 2 . The vector of the actuator forces is τ = [ F1 T F2 ]T where F1 and F2 are the actuator

δ0 = 0

(50)

δ1 = α2Δ12

(51)

δ 2 = α2Δ22

(52)

δ3 = α2Δ32

(53)

Step 4. Determine β1o and β2o using Eqs. (25) (see Appendix for the Lagrangian of the unloaded robot):

β1o = m4

b4 b2 b s2̈ sin(γ2 − ζ2 ) + m4 4 ζ2̈ + m4 4 g cos ζ2 2 3 2

(54)

β2o = m5c5s1̈ sin(γ1 − θ − γ5) + m5b2c5ζ1̈ cos(ζ1 − θ − γ5) + (m5c52 + I5)θ ̈ 2 − m5b2ζ1̇ c5 sin(ζ1 − θ − γ5) + m5c5g cos(θ + γ5)

Step 5. Determine χ using Eq. (24): 117

(55)

Robotics and Computer--Integrated Manufacturing 46 (2017) 114–121

M. Özdemir

χ = − β1o − α2β2o

selected such that Eqs. (61)-(64) are satisfied at the singular configuration, then the dynamic equations are guaranteed to be consistent at this singularity for any payload that is arbitrarily fixed on the moving platform. A suitable selection for this purpose is as follows:

(56)

Step 6. Plan the motion such that the following equations are satisfied at the singularity:

δ1 = 0

(57)

δ2 = 0

(58)

xṖ (t = 0.6s) = 0.75m/s

(79)

yṖ (t = 0.6s) = − 3.2787m/s

(80)

2

δ3 = 0

(59)

xP̈ (t = 0.6s) = 0m/s

(81)

χ=0

(60)

yP̈ (t = 0.6s) = − 9.81m/s2

(82)

Notice from Eq. (50) that δ0 = 0 is already satisfied for this example. Since α2 is nonzero, the conditions given by Eqs. (57)-(59) can equivalently be written as

Δ12 = 0

(61)

Δ22 = 0

(62)

Δ32 = 0

(63)

Eight-order polynomials can be used to generate the trajectory functions xP(t ) and yP (t ) that fulfill the task requirements given by Eqs. (65)–(82):

xP(t ) = 0.6 − 11.04518t 3 + 48.8553t 4 − 75.73422t 5 + 51.09726t 6 − 13.50568t 7 + 0.51127t 8

yP (t ) = 1.25 + 10.67664t 3 + 67.26982t 4 − 395.38172t 5 + 647.31501t 6 − 436.38454t 7 + 106.22982t 8

Similarly, dividing both sides of Eq. (60) by −1 yields

β1o + α2β2o = 0

The following values are assumed for the robot parameters: b0 = 2 m, b2 = 1 m, b4 = 1 m, b5 = 1.5 m, γ1 = 45°, γ2 = 45°, m1 = 0.5 kg, m 2 = 1 kg, m 3 = 0.5 kg, m4 = 1 kg, m5 = 1.5 kg, c5 = 1.25 m, γ5 = 30°, I5 = 0.03 kg m2, d5 = BP = 1 m, and ε5 = ∠PBK = 60°. Now, consider the following task: The end point P will start its motion from (x, y ) = (0.6, 1.25)m to reach its final destination at (x, y ) = (0.75, 1.35)m . During this motion, the orientation θ of the moving platform will be kept constant and equal to 10°. The task will be completed at time t = 1.2s. The initial and final velocities and accelerations are desired to be zero. These task requirements can be expressed as (65)

xṖ (t = 0) = 0

(66)

xP̈ (t = 0) = 0

(67)

xP(t = 1.2) = 0.75

(68)

xṖ (t = 1.2) = 0

(69)

xP̈ (t = 1.2) = 0

(70)

yP (t = 0) = 1.25

(71)

yṖ (t = 0) = 0

(72)

yP̈ (t = 0) = 0

(73)

yP (t = 1.2) = 1.35

(74)

yṖ (t = 1.2) = 0

(75)

yP̈ (t = 1.2) = 0

(76)

4. Conclusions By focusing on planar three-degree-of-freedom parallel manipulators, the present paper first derives the conditions on the mass and centroidal radius of gyration of a payload for the existence of its consistent locations at a type II singular configuration.

Moreover, while executing the task, the end point is desired to pass through an intermediate position (x, y ) = (0.6717, 1.2972)m at time t = 0.6s, i.e.

xP(t = 0.6) = 0.6717

(77)

yP (t = 0.6) = 1.2972

(78)

(84)

The variation of the determinant of the Φ matrix during the prescribed task can be seen in Fig. 2. Note that the first time derivative of this determinant is nonzero at the singularity time. Table 1 summarizes the three different scenarios considered. Among these, the first scenario represents the unloaded case with the numerical values assumed above for the robot parameters. The values of the inertial parameters of the moving platform are arbitrarily changed in the second and third scenarios to represent two different loading conditions (The used values in the last two scenarios can be interpreted as the resultant values for the corresponding parameters of the system composed of the payload and the moving platform itself). Figs. 3–8 show the required actuator forces and torques to realize the desired task in each of these scenarios. In all the scenarios, the actuator efforts remain bounded near the singularity. A second example of such a planned motion for a planar 2-RRR parallel manipulator can be found in a recent conference paper of the present author [39]. However, this previous work addressed neither the generalization of these motion-planning conditions for all planar threedegree-of-freedom parallel manipulators nor the analysis of the effects of payload inertial parameters on its consistent locations.

(64)

xP(t = 0) = 0.6

(83)

However, a type II singularity will occur at this intermediate point with ζ2 being equal to θ + π . The values of the other joint variables at this singularity are as follows: s1 = 1.4857m , ζ1 = 3.9073rad , and s2 = 1.1196m . If the velocity and acceleration profiles of the end-effector are

Fig. 2. Determinant of the matrix Φ versus time.

118

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M. Özdemir

Table 1 A summary of the considered scenarios. Parameter

Scenario 1 (Unloaded case)

Scenario 2

Scenario 3

m5 c5 γ5

1.5 kg 1.25 m 30°

4.5 kg 2.25 m 45°

8 kg 0.25 m 10°

I5

0.03 kg m2

0.08 kg m2

0.15 kg m2

Fig. 6. Necessary actuator torque T in Scenario 2.

Fig. 3. Necessary actuator forces F1 and F2 in Scenario 1.

Fig. 7. Necessary actuator forces F1 and F2 in Scenario 3.

Fig. 4. Necessary actuator torque T in Scenario 1.

Fig. 8. Necessary actuator torque T in Scenario 3.

Considering the cases wherein the consistent payload locations exist in the form of a circle, the following original conclusions are then drawn for a given prescribed motion:

• • •

Fig. 5. Necessary actuator forces F1 and F2 in Scenario 2.

119

The center of this circle is fixed for any payload. Consistent locations of different payloads with different rp 2 −

1 χ mp δ 3

values will represent concentric circles whereas they will represent 1 χ the same circle for different payloads with the same rp 2 − m δ value. p 3

If the motion is planned such that the dynamic model of the unloaded robot is consistent at the singular position, then the radius

Robotics and Computer--Integrated Manufacturing 46 (2017) 114–121

M. Özdemir

Finally, based on these new insights, a motion planning method is proposed for planar three-degree-of-freedom parallel manipulators passing through a type II singular configuration with different payloads. Five conditions are presented within this context. This method is believed to be highly useful, especially for pick-and-place operations, because it provides great flexibility and ease for handling different payloads. It is worth to mention that in practice, there may be considerable modeling uncertainties due to the manufacturing tolerances of the robot. These modeling errors should be dealt with using a proper control system [23,25] which is out of the scope of the present paper.

of the circle of interest is only a function of the centroidal moment of inertia of the payload. Considering the cases wherein the consistent payload locations are in the form of a line, the following original conclusions are also obtained for a given prescribed motion:

• • •

The centroidal radius of gyration of the payload does not have any effect on these locations. Consistent locations of different payloads with different masses will represent parallel lines whereas they will represent the same line for different payloads with the same mass. The locus of consistent locations of any payload for a prescribed motion describes the same unique line if the motion is such that the dynamic model of the unloaded manipulator is consistent at the singularity.

Funding This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Appendix A. Lagrangian of the robot considered in the numerical examples The kinetic energy of the planar 2-PRR parallel robot can be written for the unloaded case as

To =

1 1 1 1 1 1 1 1 2 2 m1s1̇ 2 + m 2s1̇ 2 + m 2s1̇ b2ζ1̇ sin(γ1 − ζ1) + m 2b2 2ζ1̇ + m 3s2̇ 2 + m4s2̇ 2 + m4s2̇ b4ζ2̇ sin(γ2 − ζ2 ) + m4b4 2ζ2̇ 2 2 2 6 2 2 2 6 1 1 2 2 2 + m5[(s1̇ cos γ1 − b2ζ1̇ sin ζ1 − c5θ ̇ sin(θ + γ5)) +(s1̇ sin γ1 + b2ζ1̇ cos ζ1 + c5θ ̇ cos(θ + γ5)) ] + I5θ ̇ 2 2

(A.1)

Its potential energy, on the other hand, can be expressed as

⎞ ⎛ ⎞ ⎛ 1 1 Vo = m1g(s1 − c1)sin γ1 + m 2g⎜s1 sin γ1 + b2 sin ζ1⎟ + m 3g(s2 − c3)sin γ2 + m4g⎜s2 sin γ2 + b4 sin ζ2⎟ + m5g[s1 sin γ1 + b2 sin ζ1 + c5 sin(θ + γ5)] ⎠ ⎝ ⎠ ⎝ 2 2

(A.2)

where c1 = DG1 and c3 = JG3. Then, the Lagrangian of the unloaded robot can be given as

⎞ ⎞ ⎛1 1 1⎛1 1 1 1 2 2 (m1 + m 2 + m5)s1̇ 2 + ⎜ m 2 + m5⎟s1̇ b2ζ1̇ sin(γ1 − ζ1) + ⎜ m 2 + m5⎟b2 2ζ1̇ + (m 3 + m4 )s2̇ 2 + m4s2̇ b4ζ2̇ sin(γ2 − ζ2 ) + m4b4 2ζ2̇ ⎠ ⎠ ⎝2 2 2⎝3 2 2 6 ⎞ ⎛ 1 1 2 + m5c5θ[̇ s1̇ sin(γ1 − γ5 − θ ) + b2ζ1̇ cos(ζ1 − θ − γ5)] + (I5 + m5c52 )θ ̇ − m1g(s1 − c1)sin γ1 − m 2g⎜s1 sin γ1 + b2 sin ζ1⎟ − m 3g(s2 − c3)sin γ2 ⎠ ⎝ 2 2 ⎞ ⎛ 1 − m4g⎜s2 sin γ2 + b4 sin ζ2⎟ − m5g[s1 sin γ1 + b2 sin ζ1 + c5 sin(θ + γ5)] ⎠ ⎝ 2

Lo = To − Vo =

(A.3)

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