Dynamic decoupling analysis and experiment based on a class of modified Gough-Stewart parallel manipulators with line orthogonality

Mechanism and Machine Theory 143 (2020) 103636

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Research paper

Dynamic decoupling analysis and experiment based on a class of modified Gough-Stewart parallel manipulators with line orthogonality Zhizhong Tong a,∗, Clément Gosselin b, Hongzhou Jiang a a

Department of Mechatronics Engineering, Harbin Institute of Technology, No.2, Yi-kuang Street, Nan-Gang District, Harbin, 150080, China b Department of Mechanical Engineering, Université Laval, Québec, Canada

a r t i c l e

i n f o

Article history: Received 2 July 2019 Revised 18 September 2019 Accepted 29 September 2019

Keywords: Stiffness analysis Dynamic decoupling Cross-coupling Modified Gough-Stewart parallel manipulators Line orthogonality

a b s t r a c t In this paper, a class of modified Gough-Stewart parallel manipulators (MGSPMs) with line orthogonality is presented in order to fulfill the dynamic decoupling along the Z-axis. Relations between kinematic orthogonality, static orthogonality, and dynamic orthogonality are analytically formulated using double hyperboloids. Based on the derived formulas, it is proven that a class of MGSPMs with line orthogonality does exist, which expands the dynamic orthogonality from a point to a line. The evaluating factors are defined to carry out thorough cross coupling investigations. With the aid of numerical examples, a class of MGSPMs with line dynamic orthogonality is shown to exhibit large workspace with low coupling and high precision. A physical prototype is established and the experimental results confirm the high precision and approximately decoupled characteristics. The presented MGSPMs with line dynamic orthogonality possess better mechanical feasibility than conventional GSPMs. Moreover, two major constraints, namely that all six struts must have the same stiffness and a compliant center - which are difficult to satisfy - are removed or relaxed, thus greatly expanding the range of possible applications. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction The Gough-Stewart parallel manipulators (GSPMs) have found wide applications including flight simulation, machining, vibration isolation, and precision pointing. Designing a parallel manipulator at a desired point is often considered a design objective [1], including the local design [2] and the point design [3]. At the desired configuration, it is proven that decoupling of stiffness and dynamic isotropy is possible [4-10]. Nevertheless, the inescapable fact is that synthesizing the geometry to meet desired specifications is still difficult [11]. One of the most restrictive constraints is that the point design leads to good performances in a rather small workspace, and it limits the applications significantly, for example, flight simulation. It is difficult to break down the shackles of small workspace with high precision, but it is of great significance. In the past years, extensive studies have been conducted for GSPMs covering design and control, including geometry orthogonality [2,11-14], kinematic isotropy [15-18], static isotropy [4,19-21] and dynamic isotropy [5-10,17-19]. Compared to the method based on structure, the method based on a desired Jacobian matrix is more effective because it can find ∗

Corresponding author. E-mail address: [email protected] (Z. Tong).

https://doi.org/10.1016/j.mechmachtheory.2019.103636 0094-114X/© 2019 Elsevier Ltd. All rights reserved.

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Z. Tong, C. Gosselin and H. Jiang / Mechanism and Machine Theory 143 (2020) 103636

a class of parallel manipulators described by the same Jacobian matrix. For example, a class of orthogonal Gough-Stewart parallel manipulators was generated by Jafari and Yi et al. [2,12-14]. They also found an asymmetric configuration and a redundant configuration with fault tolerance. However, a numerical search approach was employed so it is hard to analyze through conventional structure parameters. Another method paved the way to deriving analytical formulas in order to obtain a closed form design such as in [5-10,17-19]. Indeed, the goals that all design methods are expected to achieve are to generate parallel manipulators meeting the required specifications. It is essential that the geometric design has a necessary close connection with the control design. Theoretically, stiffness constructs relations from pure geometry to statics and dynamics. It specifies precision poses and conditions a manipulator’s bandwidth for its control. The stiffness performance must be fully considered and evaluated at the design stage, because it has a direct relationship with precision and load capacity [22]. Jafari presented [12] that the design of precision GSPMs operating about a single configuration is very similar to the design of a spatial stiffness matrix. McInroy proposed two decoupling algorithms by combining static input–output transformations with hexapod geometric design [23]. Simaan and Shoham proposed a method that synthesizes the geometry of a 6 DOF variable geometry parallel robot in order to achieve a certain stiffness matrix [24]. However, prior decoupling control methods imposed severe constraints on the allowable geometry and payload [25], and it is important to loosen and remove some constraints limiting strictly a parallel manipulator’s applications. In previous research work, the concept that constructs architectures only at the desired point is dominant. The fact is that point orthogonality or point isotropy is lost when the platform moves away from the desired point, leading to indispensable performance evaluations in its global workspace. Orthogonality facilitates design of force and motion ranges and decoupling control. The motivation in this paper is to break down point design in order to make a parallel manipulator possess continuously orthogonal configurations along a line direction. This approach differs from general methods. To the best of our knowledge, none of the related research investigated such an approach in depth. We will propose a class of MGSPMs with line orthogonality. This paper provides insight into the problem of stiffness orthogonality and dynamic decoupling analysis. Finally, the presented work is investigated with the aid of numerical examples and experimental validation. 2. Problem definition Extensive studies have been discussed in order to design a parallel manipulator at the desired point, especially point orthogonality. In the following, the line orthogonality for a modified Gough-Stewart parallel manipulator will be conducted analytically. 2.1. Jacobian matrix described by line geometry A GSPM consists of a fixed base and a movable platform with six linear struts supporting it. Theoretically, six struts can be arranged arbitrarily except for several classic singular configurations. A modified Gough-Stewart parallel manipulator (MGSPM) was studied in [2,5-8,13-14,26]. For this particular architecture, an approach that is geometrically intuitive, concise and analytical can be derived, so this paper focuses on orthogonality analysis for a MGSPM. As shown in Fig. 1(a), Ai denotes the ith movable platform attachment point, and Bi represents the ith base attachment point in the fixed base frame. In Fig. 1(a), the distribution of point Ai and Bi should be rotational symmetry and repeat every 120°, respectively. All attachment points are partitioned into two groups: the first group with points 1, 3, 5 and the second group with points 2, 4, 6. The attached points of each strut are uniformly spaced on the circumference of two circles on the movable platform and the fixed base respectively.

Fig. 1. The description using double circular hyperboloids for a MGSPM.

Z. Tong, C. Gosselin and H. Jiang / Mechanism and Machine Theory 143 (2020) 103636

3

It has been proven that a class of MGSPMs can be described by double hyperboloids [5]. As shown in Fig. 1(b), one family of straight lines along the directions of the struts 1, 3 and 5 lie on a regulus, and the other family of straight lines along the directions of the struts 2, 4 and 6 on the complementary regulus. Two surface equations S1 and S2 , describing the double hyperboloids, can be given by

S1 :

x2 + y2 (z − az1 ) − =1 ra21 c12

(1)

S2 :

x2 + y2 (z − az2 ) − =1 2 ra2 c22

(2)

2

2

subject to : ra1 > 0, ra2 > 0, c1 > 0, c2 > 0. The throat radii of the double hyperboloids are ra1 and ra2 , respectively. The distances from the centers of the throat circles to the frame origin are az1 and az2 , respectively. It should be noted that the origin of the frame, referred to as the compliant center, should coincide with the payload’s center of mass [25]. In conventional methods, the constraint regarding the compliant center seriously limits the location of the end effector, thus eliminating numerous important applications, such as those requiring cooperating hexapods [25]. In this paper, the double hyperboloids can be cut using two planes parallel to each other and perpendicular to the XOY plane. The distance between the upper intersection and the origin can be chosen with respect to the payload’s center of mass, and the low intersection is determined by the required workspace. As a consequence, the strict constraint is relaxed by a geometric design. Let points a1 and a2 be attached on their own throat circles of the double hyperboloids, and they exhibit mirror symmetry with respect to the X-axis. α denotes the angle between a1 and the X-axis. Then the coordinates of the points a1 and a2 can be written as a1 = [ra1 cos α ra1 sinα az1 ]T ,a2 = [ra2 cos α −ra2 sinα −az2 ]T .The Plücker coordinates of the straight lines 1 and 2 can be expressed as



a1 × Ln1

p1 = Ln1



a2 × Ln2

p2 = Ln2

T

(3)

T

(4)

where Lni is the 3 × 1 spatial unit vector of the ith strut with i = 1, 2 · · · 6. Using the parameters of the double hyperboloids, Ln1 and Ln2 can be written as



Ln1 = −la1 sin α Ln2 = [−la2 sin α

la1 cos α la2 cos α

lc1



(5)

lc2 ]

(6)

Substituting the above two equations into Eqs. (3) and (4) respectively, then yield



p1 = −la1 sin α



p2 = −la2 sinα where la1 = 

ra1 ra21 +c12

la1 cos α

lc1

−az1 la1 cos α + ra1 lc1 sin α

−az1 la1 sin α − ra1 lc1 cos α

−la2 cosα

lc2

−az2 la2 cos α − ra2 lc2 sinα

az2 la2 sin α − ra2 lc2 cosα

, lc1 = 

c1 ra21 +c12

, la2 = 

ra2 ra22 +c22

, lc2 = 

c2 ra22 +c22

ra1 la1 −ra2 la2

T

T

(7) (8)

.

Rotating the straight lines 1 and 2 with the angle of ± 2π /3 along the Z-axis respectively, six Plücker coordinates of all struts can be generated to construct the Jacobian matrix as follows



TR (2π /3 )p2

Jlx = p1

TR (2π /3 )p1

TR (−2π /3 )p2

TR (−2π /3 )p1

p2

T

(9)

where TR (θ ) is the rotation matrix. 2.2. Stiffness modeling In the following, the mathematical formulas of the orthogonal conditions at a desired point will be derived respectively. The most straight forward choice of the desired point is at half stroke of all the struts’ actuators, referred as to the neutral position. Assume that the MGSPM shown in Fig. 1 consists of rigid links, whereas the actuators of all struts are the only nonrigid elements. The stiffness matrix is the ratio of a load to the resulting set of displacement, and hence, exhibits the static characteristics. It should be stressed that the lengths of all struts may be different, thereby we assume that the struts belonging to the same group have an equal stiffness, that is, KL1 = KL3 = KL5 and KL2 = KL4 = KL6 , which is easily satisfied by selecting struts with physically identical constructions in practice. It relaxes the major constraint that all struts must have the same stiffness and length, which is imposed by most decoupling algorithms. The stiffness matrix of six struts can be written in joint space as



Kj = diag kL1

KL2

KL1

KL2

KL1

KL2



(10)

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Z. Tong, C. Gosselin and H. Jiang / Mechanism and Machine Theory 143 (2020) 103636

Theoretically, the stiffness matrix of the GGSPM in task space should consider the variations of the Jacobian matrix and can be given as [22]

 −1

KM = JTlx Kj Jlx + H  JTlx

F

(11)

where the sign ‘’ denotes the tensor product of the matrix, H indicates the total differential of matrix JTlx , and F is an external wrench. The second term of Eq. (11) is usually far smaller than the first term, so Eq. (11) is usually reduced to

KM = JTlx Kj Jlx

(12)

Substituting Eq. (10) into Eq. (12), then yields



KM = where

GF F GMF

GF M GMM



(13)

   3 diag KL1 la21 + KL2 la22 , KL1 la21 + KL2 la22 , 2 KL1 la21 + KL2 la22 2  3 2 2 2 2 2 2 2 2 = diag KL1, ra1 lc1 + KL1 a2z1 la21 + KL2 ra2 lc2 + KL2 a2z2 la22 , KL1, ra1 lc1 + KL1 a2z1 la21 + KL2 ra2 lc2 + KL2 a2z2 la22 , 2   2 2 2 2 2 KL1 ra1 la1 + KL2 ra2 la2

GF F = GMM



GMF

3 = 2

−KL1 ra1 la1 lc1 + KL2 ra2 la2 lc2 KL1 la21 az1 − KL2 la22 az2

−KL1 la21 az1 + KL2 la22 az2 −KL1 ra1 la1 lc1 + KL2 ra2 la2 lc2





2 KL1 ra1 la1 lc1 − KL2 ra2 la2 lc2



GMF = GTMF To meet stiffness orthogonality, then GMF = GTMF = 03×3 . In the sequel, the orthogonal conditions are

KL1 ra1 la1 lc1 = KL2 ra2 la2 lc2

(14)

KL1 la21 az1 = KL2 la22 az2

(15)

The stiffness matrix is the stiffness weighted kinematic transmission matrix, and only if KL1 = KL3 = KL5 = KL2 = KL4 = KL6 , then stiffness orthogonality is equivalent to kinematic orthogonality. Eqs. (14) and (15) address an intrinsic relation between structure geometry and statics. 2.3. Dynamic orthogonality In the field of applications of precision pointing and trajectory tracking, we generally concentrate on the dynamic response of MGSPMs in the task space. The dynamic response is determined by the natural frequency matrix in task space, and thus the natural frequency matrix can be defined as a performance index employed by

GNFM = M−1 t KM

(16)

where Mt denotes the payload’s mass matrix. In practice, it is possible to choose the coordinate system such that it coincides with the orientation of the principal axes of the payload, and then the payload’s mass matrix can be given as



mI3×3 Mt = 03×3

03×3 Ic



(17)

where I3 × 3 is an identity matrix, m is the payload mass, and Ic denotes the inertia matrix with respect to the center of mass, Ic = diag( Ixx , Iyy , Izz ), here Ixx , Iyy and Izz are the moments of inertia. Substituting Eq. (12) into Eq. (16), then rearranging the equation yields



GNFM = where

GNT GNTR

GNRT GNR



(18)

  2  3 2 diag KL1 la21 + KL2 la22 , KL1 la21 + KL2 la22 , 2 KL1 lc1 + KL2 lc2 2m  2 2 2 2 KL1 ra21 lc1 + KL1 a2z1 la21 + KL2 ra22 lc2 + KL2 a2z2 la22 KL1 ra21 lc1 + KL1 a2z1 la21 + KL2 ra22 lc2 + KL2 a2z2 la22 3 = diag , , 2 Ixx Iyy

GNT = GNR



2 KL1 ra21 la21 + KL2 ra22 la22 Izz



Z. Tong, C. Gosselin and H. Jiang / Mechanism and Machine Theory 143 (2020) 103636

GNTR

3 = 2m

⎡ 3⎢ ⎣ 2

GNRT =

−KL1 ra1 lc1 la1 + KL2 ra2 lc2 la2 KL1 az1 la21 − KL2 az2 la22 0

−KL1 az1 la21 + KL2 az2 la22 −KL1 ra1 lc1 la1 + KL2 ra2 lc2 la2 0

−KL1 ra1 lc1 la1 +KL2 ra2 lc2 la2 Ixx KL1 az1 la21 −KL2 az2 la22 Ixx

−KL1 az1 la21 +KL2 az2 la22 Iyy −KL1 ra1 lc1 la1 +KL2 ra2 lc2 la2 Iyy

0

0

5



0 0 2(KL1 ra1 lc1 la1 − KL2 ra2 lc2 la2 )

0 0 2(KL1 ra1 lc1 la1 −KL2 ra2 lc2 la2 ) Izz

⎤ ⎥ ⎦

Obviously, the decoupled conditions for dynamic orthogonality are the same as Eqs. (14) and (15). The detailed derivation is referred in given in [5]. When KL1 = KL3 = KL5 = KL2 = KL4 = KL6 , the conditions of kinematic orthogonality, static orthogonality and dynamic orthogonality have the same formulas as Eqs. (11) and (12). It is a special equivalence and hence, the geometric design is available and can be employed in the field of engineering applications. In fact, if ra1 = ra2 , c1 = c2 and az1 = az2 , then a GGSPM turns to be a standard Gough-Stewart parallel manipulator. The related geometric conditions have been conducted in reference [4] and can be derived from Eqs. (14) and (15). The above derived formulas are pertaining to dynamic orthogonality at a point, and our motivation is to break down point orthogonality in order to make a parallel manipulator possess continuously orthogonal configurations along a line direction. In the following sections, line orthogonality will be conducted. 3. Line orthogonality 3.1. Principle for line orthogonality A GGSPM exhibits geometric symmetry, and this property can be maintained only if the movable platform is translating solely along the Z-axis. Let z denote a translation at time t, then the coordinates of points a1 and a2 are written as a1,t = [ra1,t cos α ra1,t sinα az1,t + z]T , a2,t = [ra2,t cos α −ra2,t sinα −az2,t + z]T . The Plücker coordinates of the straight lines 1 and 2 at time t can be represented in the form



a1,t × Ln1,t

p1,t = Ln1,t



a2,t × Ln2,t

p2,t = Ln2,t

T

(19)

T

(20)

where Ln1,t and Ln2,t are written as



Ln1,t = −la1,t sin α



Ln2,t = −la2,t sin α

la1,t cos α

lc 1,t

la2,t cos α

lc 2,t



(21)



(22)

Accordingly, the natural frequency matrix of the GGSPM at time t is



GNFM,t where

GNT,t = GNTR,t

GNRT,t GNR,t



(23)

  3 diag KL1 la21,t + KL2 la22,t , KL1 la21,t + KL2 la22,t , 2(KL1 lc21,t + KL2 lc22,t ) 2m   3 2 (KL1 ra21,t la21,t +KL2 ra22,t la22,t ) G0 = diag GI 0 , , I I xx yy zz 2

GNT,t = GNR,t

G0 = KL1 ra21,t lc21,t + KL1 (az1,t + z )2 la21,t + KL2 ra22,t lc22,t + KL2 (−az2,t + z )2 la22,t



GNTR,t

G1 3 G2 = 2m 0

−G2 G1 0



0 0 2(KL1 ra1,t lc1,t la1,t − KL2 ra2,t lc2,t la2,t )

G1 = − KL1 ra1,t lc1,t la1,t + KL2 ra2,t lc2,t la2,t G2 = KL1 (az1,t + z )la21,t − KL2 (az2,t + z )la22,t

⎡ G1 I

GNRT,t

3 Gxx = ⎣ Ixx2 2 0

− GIyy2

0

0

2(KL1 ra1,t lc1,t la1,t −KL2 ra2,t lc2,t la2,t ) Izz

G1 Iyy

0

⎤ ⎦

Letting G1 = G2 = 0, then orthogonal conditions at time t are given as

KL1 ra1,t la1,t lc1,t = KL2 ra2,t la2,t lc2,t

(24)

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Z. Tong, C. Gosselin and H. Jiang / Mechanism and Machine Theory 143 (2020) 103636

KL1 la21,t (az1,t + z ) = KL2 la22,t (az2,t + z )

(25)

If Eqs. (24) and (25) can be satisfied at every moment when the GGSPM is translating along the Z-axis, the corresponding natural frequency matrix will be changing but guaranteed to be orthogonal. Such a property is referred to as line dynamic orthogonality. Compared to the parallel manipulator with point orthogonality, this kind of MGSPMs no doubt enlarges the workspace with high precision since orthogonality implies low cross couplings. 3.2. Proof of line orthogonality To carry out dynamic orthogonality along a line, two matchable reguli should be reconfigured in order to meet Eqs. (24) and (25). The most direct solution is that the positions of three or all of the six lower gimbals can be adjusted for reconfiguration, and it is inevitable in this case to introduce kinematic redundancy, leading to mechanical complexity. A question then arises as to whether a class of MGSPMs with line orthogonality exist? Proposition 1. Let GNFM, 0 = diag[λ1 There exists

λ2

λ3

λ4

λ5

λ6 ] at initial time where the GGSPM is at its neutral position.

(I) KL1 la21,t = KL2 la22,t and az1,t = −az2,t are sufficient and necessary conditions such that GNFM, t is a diagonal matrix where

qt = [0 0 z 0 0 0] T . (II) a class of MGSPMs with line orthogonality if and only if Eqs. (24) and (25) are satisfied.

Proof. I) Recalling the orthogonal condition addressed by Eq. (25), a sufficient condition to guarantee line dynamic orthogonality is that Eq. (25) should have be independent from z. It is fulfilled if and only if one of the following equivalent statements is ensured 1. KL1 la21,t = KL2 la22,t = 0 2 2 then la21,t = la22,t = 0 since KL1 = 0 and KL2 = 0. Concerning la21,t + lc1 = 1 and la22,t + lc2 = 1, then lc21,t = lc22,t = 1. It implies ,t ,t that six struts of the MGSPM are parallel to each other and are all perpendicular to the XOY plane. The movable platform and the base are identical, leading to a singular Jacobian matrix. Thereby KL1 la21,t = KL2 la22,t = 0 cannot be satisfied.

2. KL1 la21,t = KL2 la22,t If GNFM, 0 = diag[λ1 λ2 λ3 λ4 λ5 λ6 ] at initial time where the MGSPM is at its neutral position, there exists that two surface equations of double hyperboloids describe a class of MGSPMs. Then we construct a MGSPM by cutting two reguli using two planes, and the upper gimbals of the movable platform lies on two circles on the intersection of the upper plane and the two reguli. The radius of the circle where the odd numbered struts lie is denoted by rA1 , and rA2 represents the radius of the circle where the even numbered struts lie. According to the geometric characteristics of a cylindrical regulus, at time t, rA1 = Since c1 =

lc 1,t r , la1,t a1 ,t

then az1,t =

r a1,t r a2,t cos α , rA2 = cos(−α ) , az1,t = −c1 tan α and az2,t = −c2 tan α . l l − lc1,t ra1,t tan α = − lc1,t rA1 sin α . Similarly, we can obtain a1,t

a1,t

sides of Eq. (24) be divided by KL1 la21,t = KL2 la22,t , one then obtains

r a1,t lc 1,t la1,t

=

a2,t

r a2,t lc 2,t

lc1,t rA1 cos α

(26)

la2,t

Substituting rA1 =

la1,t

l

az2,t = − lc2,t rA2 sin α . Let both

=

r a1,t cos α

and rA2 =

ra2,t lc2,t rA2 cos α la2,t

r a2,t

cos(−α )

⇒

into Eq. (26) provides

lc1,t rA1 sin α la1,t

=

ra2,t lc2,t rA2 sin α

(27)

la2,t

Such that az1,t = az2,t at every moment, GNFM, t is a diagonal matrix where qt = [0 0 z 0 0 0]T . Conversely, if GNFM, t is a diagonal matrix, then Eqs. (24) and (25) have to be satisfied. II) By proof I), when Eqs. (24) and (25) are fulfilled such that the MGSPM can possess line orthogonality. ??  Corollary 1. For a MGSPM with line orthogonality, la21,t = la22,t , lc21,t = lc22,t and ra1,t = ra2,t if KL1 = KL2 . Corollary 2. No MGSPM with line isotropy exists even if it exhibits line orthogonality. Proof. If a MGSPM exhibits line orthogonality, by Proposition 1 and Corollary 1, then az1,t = az2,t , KL1 = KL2 , la21,t = la22,t , lc21,t = lc22,t and ra1,t = ra2,t . When the MGSPM achieves translation isotropy, then la21,t = la22,t =

2 3,

lc21,t = lc22,t =

2 3.

Obviously, these

relations can be fulfilled if and only if the MGSPM is at a certain pose. la21,t , la22,t , and lc21,t , lc22,t vary consequently when the MGSPM translates along the Z-axis, thus there cannot be any MGSPM with line isotropy. 

Z. Tong, C. Gosselin and H. Jiang / Mechanism and Machine Theory 143 (2020) 103636

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Corollary 3. If two reguli of double hyperboloids have the same parameters, then a Gough-Stewart parallel manipulator (GSPM) is described as a special case of MGSPMs. A class of GSPMs possesses line orthogonality if and only if all struts of the reguli meet at right angles i.e. Ln1 ⊥ Ln3 ⊥ Ln5 and, as well as Ln2 ⊥ Ln4 ⊥ Ln6 . However, the GSPM belonging to this family never accomplishes dynamic isotropy, worse than MGSPMs, due to the payload inertial constraint of Izz = 4Iyy that it cannot be obtained in the physical world. Compared to point orthogonality or point isotropy, it is worth mentioning that line orthogonality is significant. In physical world, Proposition 1 and Corollary 1 can be fulfilled feasibly and the corresponding structure is similar to a conventional Gough-Stewart parallel manipulator. Remark. Compared to isotropy, orthogonality can be achieved with less restriction and make control and calibration easier. Anisotropy may be considered as a kind of orthogonality, especially eligible for a robot requiring to control stiffness or force in a special direction. Therefore, orthogonality has the potential to be a promising property in robotics and mechanisms. Among kinematic, static and dynamic orthogonality, stiffness orthogonality constructs relations from the pure geometry to statics and dynamics. It specifies precision poses and conditions a manipulator’s bandwidth for its control, and the structure with stiffness orthogonality facilitates design of force and motion ranges, decoupling control, and Cartesian stiffness [12]. As a consequence, we will direct our attention to stiffness orthogonality which leads to dynamic orthogonality, instead of isotropy. 3.3. Orthogonality index If a parallel manipulator is decoupled, it is easy to realize precision pointing, motion planning, control scheme, calibration and compensation. Stiffness and mass specify precision poses and condition a manipulator’s bandwidth for its control. The natural frequency matrix reflects the cross-coupling, which is significant to determine the dynamic behaviors. For example, it is necessary for flight simulator to decouple motions and ensure the bandwidth uniform in order to replicate the given motions and provide the pilots motion cueing. If the natural frequency matrix will not be a diagonal matrix in the overall workspace and cross coupling occurs to weaken positioning precision and dynamic response. Therefore, a cross coupling index should be presented in order to assess the influence. All off-diagonal elements of the natural frequency matrix reflect whether cross coupling effects exist, and those average value can be employed to quantify the extent of cross coupling effect. As a consequence, local norm-coupling factor (LNCF) is defined in the form

CILNCF =

6 6 1  30



i=1 j=1

GNFM,t (i, j ) 

GNFM,t (i, i ) GNFM,t ( j, j )

( i = j )

(28)

Obviously, a LNCF is dimensionless, and 0 means no coupling and 1 denotes complete coupling. In general, a natural length scale during kinematic orthogonality or static orthogonality design is lack due to the fact that the evaluation matrix is not dimensionally homogenous. For dynamics and control a required LNCF is more reasonable. To analyze stiffness mapping, Merlet proposed the concept of an iso-stiffness curve [1]. Similarly, an iso-coupling curve with a constant LNCF value, denoted by the iso-LNCF curve, can be introduced reasonably. The locus covered by the iso-LNCF curves can be used to bound a high-precision working area. Moreover, if the iso-LNCF curves are cascaded, the iso-LNCF polyhedron can be obtained and its volume can be calculated by

 VCILNCF =

LCILNCF d W

(29)

W

The enclosed polyhedron with the same LNCF implies that the parallel manipulator possesses a cross coupling constrained workspace. Within the cross coupling constrained workspace, the dynamic behavior along a specific direction can be generated accurately with the required cross coupling. It is significant for some applications that need high tracking precision, such as cue simulations and flight simulators. Obviously, it can be used to assess how large the workspace with a required high precision is. By recalling Corollary 2, the extent of maximal regularity, instead of isotropy, should be addressed to evaluate an iso-LNCF curve. MRSF1 and MRSF2 are defined as the shape factors. MRSF1 is the ratio of a short shaft La and a long shaft Lb of the iso-LNCF curve, and MRSF2 is the ratio of the nearest point Lc and the furthest point Ld, related to the center of the iso-LNCF curve. The shape factors represent the extent of maximal regularity along two shafts’ directions, thus another shape factor is proposed to depict the extent to isotropy, denoted by SFI. To find the minimum circumscribed circle of the iso-LNCF curve, then SFI is defined as the ratio of the area of the locus covered by the iso-LNCF curve and the minimum circumscribed circle. As shown in Fig. 2, MRSF1 is reasonable in Fig. 2(a) but not reasonable in Fig. 2(b) as La is approximately equal to Lb, thus MRSF2 should be adopted in Fig. 2(b). In Fig. 2(b) and (c), MRSF1 and MRSF2 cannot reflect how Fig. 2(c) is nearly isotropic, but SFI is effective. Obviously, the closer the three factors are to being equal to 1, the more regular dynamic response of a MGSPM is.

8

Z. Tong, C. Gosselin and H. Jiang / Mechanism and Machine Theory 143 (2020) 103636

Fig. 2. Maximal regular factors for the iso-LNCF curves.

4. Numerical analysis and evaluation The aforementioned method delivered a clear perspective on the line orthogonality possessing large workspace with high precision. In order to further examine this issue, the numerical analysis is developed and examined. 4.1. Numerical examples The characteristics of low coupling and high precision are essential in a typical application of high precision motion control (for example, for telescopes). If the employed parallel manipulator possesses line orthogonality, it is no doubt vested with a promising property. For example, the required specifications are as follows: maximum half cone angle, which usually measures the rotation capability, is no less than 10°. The rotation velocity is no less than 35°/s, and a gross moving payload, mounted in the plane formed by the payload attachment points, is about 50 kg. The geometry should have workspace allowing equal translations in the X-axis and the Y-axis and equal rotations about these same axes. To meet the requirements, we designed optimally two solutions as shown in Fig. 3: one is the architecture based on a MGSPM with line dynamic orthogonality (Fig. 3(a)), and the other possessing (24) and (25) approximately point orthogonality is the architecture based on a GSPM with 3-3 configuration (Fig. 3(b)). The former is marked as the LDO-MGSPM, and the latter is marked as the PO-GSPM. The surface equations describing the double hyperboloids on which the LDO-MGSPM lies are in the form

S10 :

x2 + y2 (z − 0.34 ) − =1 0.252 (1/3 )2

(30)

S20 :

x2 + y2 (z + 0.34 ) − =1 0.252 (1/3)2

(31)

2

2

The structure illustrated in Fig. 3(a) is obtained by cutting the double hyperboloids with α = 30° and two planes, perpendicular to the Z-axis, at origin point and −0.538 m along the Z-axis. The PO-GSPM employs 3-3 configuration, and the nominal radii of the movable platform and the base are 0.33 m and 0.66 m, respectively. If it is similar to the mutually orthogonal geometry, the workspace cannot satisfy the requirements. In fact, a GSPM with point orthogonality does not have a compact structure if the compliant center is located on the movable platform with a certain distance. The cubic structure’s compliant center must coincide with the center of the cube formed by the orthogonal struts [25]. Hence, the neutral height is adjusted to 0.538 m for the PO-GSPM in this paper, which exhibits approximately point orthogonality. 4.2. Cross-coupling analysis Let the required LNCF be 1% in order to guarantee the precision, the numerical examinations are employed to evaluate cross coupling for the LDO-MGSPM and the PO-GSPM, respectively. 41 intersections, parallel to the XOY plane, are chosen to investigate the extent of cross coupling influence in the translational workspace, the orientation of the movable platform being fixed. The iso-LNCF curves on the sampled six intersections for the LDO-MGSPM are illustrated in Fig. 4. At every intersection, the minimum circumscribed circle of the iso-LNCF curve is also depicted to clarify the maximal regularity. Fig. 4 shows that all shapes of six intersections are identical to each other since the minimum circumscribed circles are cascaded to form one circle in the XOY plane. For the PO-GSPM shown in Fig. 5, there are there are six intersections meeting CILNCF = 1%.

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Fig. 3. Schematic illustration for the architectures.

Fig. 4. The iso-LNCF curves (left) and their top view (right) with CILNCF = 1% for the LDO-MGSPM.

The corresponding diameters of the minimum circumscribed circle are decreasing along the Z-axis, and four concentric circles are projected into the XOY plane. It is evident that the LDO-MGSPM possesses larger workspace. To quantify the extent of maximal regularity, the evaluating factors for the LDO-MGSPM and the PO-GSPM are drawn in Fig. 6, in which 41 intersections are adopted. The MRSF1 of the LDO-MGSPM is approximately invariant over the workspace, while the MRSF1 of the PO-GSPM is closely equal to 1 only at neutral position. By recalling Section 3.3, Fig. 6 shows that the LDO-MGSPM is more regular within the workspace. Overall, the figures lead us to the conclusion that the LDO-MGSPM is more regular, more consistent and close to isotropy.

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Fig. 5. The iso-LNCF curves (left) and their top view (right) with CILNCF = 1% for the PO-GSPM.

Fig. 6. Comparison of the evaluating factors (CILNCF = 1%).

4.3. Cross coupling constrained workspace Cascading all locus covered by the iso-LNCF curves to construct an iso-LNCF polyhedron, it is feasible to assess how large the cross-coupling constrained workspace is. Figs. 7 and 8 depict two iso-LNCF surfaces with CILNCF = 1% of the LDO-MGSPM and the PO-GSPM, respectively. Form the top view on the right hand side figure of Fig. 7, it seems to be a circle as demonstrated in Section 4.2, thereby the iso-LNCF surface of the LDO-MGSPM closely approximates a cylinder. By contrast, the workspace enclosed by the surface of the PO-GSPM is rather small, and thus it is employed for micro precision applications. Computing the integral along closed surface using Eq. (42), the volume for the LDO-MGSPM is found to be about 16.45 times the volume of the PO-GSPM. It becomes more evident that the LDO-MGSPM possesses the ability to enlarge the workspace with low cross coupling that means high-precision. The parallel manipulators similar to the PO-GSPM can only perform very small motions, which limit the applications but are in practice accurate for some industrial applications including vibration isolation and precision manipulation. According to the numerical analysis, the LDO-MGSPM, of course, expands a point with promising performances to a line with promising performances. It is surely significant for some applications which require the highly accurate workspace along and around a special direction. The LDO-MGSPM can be employed in potential applications ranging from micromanipulators to macro manipulators, such as machining or motion simulation. 5. Experimental validation 5.1. Prototype To verify the performance of the LDO-MGSPM, a physical prototype was developed with the structural parameters described in Section 4.1. It is driven by six electromechnical actuators with 210 mm stroke and settle switches, each of which

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Fig. 7. The iso-LNCF surface (left) and its top view (right) with CILNCF = 1% of the LDO-MGSPM.

Fig. 8. The iso-LNCF surface (left) and its top view (right) with CILNCF = 1% of the PO-GSPM. Table 1 The structural and physical parameters for the physical prototype. Parameter

Value

Unit

description

ra1 ra2 rb1 rb1 H m Ixx Iyy Izz

0.357 0.357 0.291 0.704 0.538 106 28 28 37

M M M M M M kg m2 kg m2 kg m2

The The The The The The The The The

outer radii of the circles of the group with 1, 3, 5 outer radii of the circles of the group with 1, 3, 5 outer radii of the circles of the group with 2, 4, 6 outer radii of the circles of the group with 2, 4, 6 height of the manipulator at the neutral position equivalent mass of the payload moments of inertia moments of inertia moments of inertia

of of of of

the the the the

movable platform fixed base movable platform fixed base

employs an AC servo motor (Panasonic A6) and a ball screw to convert rotary torque into linear motion. The excursion, velocity and acceleration performances are specified for the physical prototype in Table 1. In some cases, the inertial forces of the actuators cannot be neglected. Therefore, the light weight motion systems requiring high performance motion this has to be reevaluated. In our physical prototype, the shape of the movable platform is a thin cylinder, so the inertia matrix is closely a diagonal one. The mass and inertia including the movable platform and six upper joints are more than the equivalent mass and inertia that are calculated by taking into account the actuator inertial forces. Additionally, the achievable bandwidth is less than 8 Hz, so the mass/inertia of the links can be neglected in our experiments.

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Fig. 9. The architecture of motion control system.

Three levels of controllers with the emphasis on the interfaces are arranged in the hierarchy shown in Fig. 9. The first controller, the low-level controller, is the AC servo driver (Panasonic A6) that is applied to the speed control of the AC servo motor equipping with a 16-bit encoder. The second controller in the middle-level is the motion real-time controller for the position control, and it runs the control model in real-time with xPC-Target. The third controller, the top-level controller, is used as the host monitor for maintenance tests, tuning and troubleshooting. The second controller can interface with the third controller in response to commands received from the host monitor via Ethernet using the TCP/IP protocol. The adopted control strategy is the basic kinematic decoupling control. 5.2. Trajectory tracking test High accuracy and dynamic response are two of the most important requirements for a precision pointing equipment, and the accuracy is generally evaluated by tracking errors, as for dynamic response by the excitation using random signals. Three components of the desired trajectory along the X, Y and Z axes are entered into the system, in which x = 0.05sin0.02t, y = 0.05cos0.02t and z = 0.02sin0.04t. t denotes the running time and is set in the range of 0 s to 50 s. Obviously, the spatial trajectory is a helix. Fig. 10 shows the trajectory’s profile. The spatial tracking errors are less than 1.5E-4 m and the actual trajectory coincides with the desired one. It is demonstrated that the controlled servo motor has an extremely good tracking performance. 5.3. Dynamic orthogonality evaluation In the prototype, the platform frame is selected to coincide with the orientation of the principal axes of the payload. Although the actual payload mass-inertia matrix is not diagonal, it is still diagonally dominant. Further, it contributes surely to dynamic response but has been evaluated in the design. The closed loop system is validated by six separate experiments in the frequency domain from 0.1 Hz to 20 Hz with the exciting of the six independent random signals, and Fig. 11 depicts the tracking curves of the six directions of motion. Remarkable is the fact that the coherence between the desired and

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Fig. 10. The helix trajectory tracking. Table 2 Specifications for the physical prototype. DOF

Surge x

Sway y

Heave z

Roll ϕ

Pitch θ

Yaw ψ

−3 dB bandwidth (Hz)

7.32

7.48

7.3

7.21

7.03

7.41

the actual trajectory is higher in the exciting direction, and the outputs in other five directions seem to be negligible. It is demonstrated that the dynamic decoupling design is experimentally verified in the time domain. To quantify clearly the dynamic response, the Bode plots for the six directions of motion are measured in the experiments using the random signal excitation. According to the frequency matrix, there are 36 frequency domain measurements, in which the diagonal elements are the characteristics of 6 degrees of freedom and the non-diagonal elements denote the influence of the cross coupling. The Bode plots are illustrated in Fig. 12, in which each figure reflects the frequency resonances of one input and six outputs. To satisfy the precision positioning and pointing requirements, the closed loop system is corrected to reduce the overshoot and build up the stability and there are not resonance peaks along the exciting direction. The curves exhibit similar behaviors in different degree of freedom: • At the neutral position, the −3 dB bandwidth of the closed loop system of each degree of freedom is listed in Table 2. It implies that the six directions of motion have very much the same characteristics. • At the neutral position, the outputs of the cross coupling to other degrees of freedom are less than −20 dB along the three translational directions and are less than −30 dB around the rotational directions. Therefore, the corresponding phase angle becomes meaningless except for the one of the main output. • The given motions are within the workspace enclosed by the iso-LNCF surface with CILNCF = 1%, and the −20 dB magnitude attenuation is equivalent to CILNCF = 1%. As a consequence, the experimental cross coupling results fit well with the analysis presented in Section 4.2. • The results imply that the cross coupling can be neglected and the dynamic decoupling is fulfilled by the presented method. In general, in a conventional GSPM, surge-pitch cross coupling and roll-sway cross coupling are present, and the corresponding decoupled control algorithm has to be carried out to improve performance. However, the surgepitch cross coupling and roll-sway cross coupling are decreased to negligible proportions for the LDO-MGSPM in the experiments. To verify the line orthogonality, several configurations are selected gradually along the Z-axis in the range of −0.08 ≤z ≤ 0.08 m, there are similarly experimental results. In addition, some experiments, including moving away from the neutral position and rotating around the Z-axis from −0.1 rad to 0.1 rad, were carried out. For example, the surge is chosen to be the main input, and the bode plots for the two cases with ψ = −0.1 rad and ψ = 0.1 rad are drawn in Fig. 13, respectively. The results reveal that the LDO-MGSPM exhibits low cross coupling with the workspace with the given CILNCF = 1%. It is verified that the cross coupling constrained workspace exists and can be enlarged by the line orthogonality design. The presented design has high accuracy in practice and reaches the performance boundaries, and it is validated experimentally that the modal-space control strategy has practically no contributions to improve performance. The experimental results reveal that the design method for line dynamic orthogonality decouples effectively the system into six independent systems, and also the very high mechanical and controlled stiffness of the electromechanical actuators contribute considerably to dynamic orthogonality.

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Fig. 11. Excitation using of random signals.

The presented decoupling design is rooted in the MGSPMs and the line orthogonality, and it is the synthetical method involving the geometry, payload, stiffness and basic control. In general, if an orthogonal structure that does not possess the line orthogonality is used to meet more highly accurate requirements with a large workspace, a complex controller and sophisticated noise estimation are inevitably required, such as parameter identification and accurate dynamic model. The method presented in this paper performs compared to some previous methods more straightforwardly and favorably.

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Fig. 12. The Bode plots of the main output and the cross coupling outputs.

15

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Fig. 12. Continued

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Fig. 13. The Bode plots when the surge is the main input with ψ = −0.1 rad and ψ = 0.1 rad.

Although it cannot completely replace the decoupled control algorithm, it provides a very solid foundation for the decoupled control algorithm.

6. Conclusions Orthogonality facilitates the design of force and motion ranges, decoupling control, and Cartesian stiffness. This paper presents a class of modified Gough-Stewart parallel manipulators with line orthogonality, and a dynamic decoupling design is fulfilled by geometry. The evaluation matrices for kinematic, static and dynamic orthogonality are formulated by using double hyperboloids, and the relations among them are demonstrated. Focusing on dynamic orthogonality, the concept of line dynamic orthogonality is proposed and it is shown that a class of MGSPMs with line orthogonality does exist. It possesses good mechanical feasibility similar to conventional GSPMs, providing an alternative to a direct solution. Further, the major constraint that all six struts must have the same stiffness has been removed. Another major constraint regarding the compliant center is also alleviated using double hyperboloids. Therefore, the dynamic decoupling puts far less constraints, greatly expanding the applications.

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To carry out a thorough cross-coupling investigation, the orthogonality-evaluating factors, including iso-LNCF curves, isoLNCF volume, MRSF1 , MRSF2 and SFI, are defined. The cross coupling is also investigated with the aid of numerical examples. The numerical analysis illustrates that the LDO-MGSPM exhibits the ability to enlarge the high-precision workspace, superior to GSPMs. A physical prototype is established in order to validate the proposed method. The experimental results indicate that the proposed decoupling design is practical and yields superior performances. We believe that the LDO-MGSPM has the potential to be a solution for high-precision applications including precision pointing, machining, telescope, and interaction robots in contact with surroundings. Declaration of Competing Interest None. Funding This work was supported by the National Natural Science Foundation of China (Grant number 51575121). 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