Journal of Mechanical Science and Technology 31 (3) (2017) 1427~1436 www.springerlink.com/content/1738-494x(Print)/1976-3824(Online)

DOI 10.1007/s12206-017-0243-9

Design and kinemics/dynamics analysis of a novel climbing robot with tri-planar limbs for remanufacturing† Yi Lu1,2,*, Keke Zhou3 and Nijia Ye1 1

College of Mechanical Engineering, Yanshan University, Qinhuangdao, Hebei, 066004, China Advanced Metal Forming Key Laboratory of Education Ministry & Parallel Robot Key Laboratory of Hebei, China 3 SNBC Zenqe Robt Co., LTD., China

2

(Manuscript Received December 15, 2015; Revised August 9, 2016; Accepted November 1, 2016) ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Abstract A novel climbing robot with tri-planar limbs is proposed for remanufacturing, and its kinematics and dynamics are studied systematically. First, a 3D prototype of the climbing robot with tri-planar limbs was designed and constructed; its principle of climbing wall and machining in fixed wall location are explained. Second, the formulae for solving the displacement, the velocity, acceleration, and the dynamic active forces of this climbing robot were derived. Third, a workspace during its operation in fixed location was constructed. Finally, a numerical example is given, and analytical solutions are verified by the simulation solutions. Keywords: Climbing robot; Parallel manipulator; Kinematics; Dynamics ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

1. Introduction Wall-climbing robots have extensive applications for the maintenance and inspection of surface defects of large equipment, clearing waste on surfaces of ship hulls, scanning external defects of surfaces of gas or oil tanks, and inspecting and maintaining a wind power tower. Several different robots can navigate on buildings, or other human-made structures. Zhang et al. presented a compact permanent-magnetic wheeled wallclimbing micro robot [1]. Schmidt and Berns studied different locomotion and adhesion methods for climbing robots [2]. Henrey et al. presented a hexapod legged robot [3]. Hu et al. designed a miniature wall climbing robot with biomechanical suction cups [4]. Kim et al. proposed a wall-climbing robot for climbing a vertical plane [5]. Qian et al. designed a wallclimbing robot for moving on smooth glass surfaces using dual vacuum suction cups [6]. Choi et al. developed a selfcontained wall climbing robot for scanning external surfaces of gas or oil tanks to find defects [7]. Yue et al. established a dynamic model and presented a path planning method for the City-Climber robot [8]. Vidoni and Gasparetto studied force distribution and leg posture for a bio-inspired spider robot [9]. Meng et al. developed a Gecko-like robot that simulates a gecko's ability to climb walls and ceilings [10]. Provancher presented a bio-inspired dynamic climbing robot [11]. Gao et *

Corresponding author. Tel.: +86 15243477660, Fax.: +86 3358057031 E-mail address: [email protected] † Recommended by Associate Editor Kyoungchul Kong © KSME & Springer 2017

al. developed a climbing robot with wheel for inspecting and maintaining a wind power tower [12]. Xu et al. proposed a suction method based on a mechanism utilizing hook-like claws and presented the design of a robot system for inspecting rough concrete walls [13]. Wang et al. built a kinematic model of a wall-climbing caterpillar robot to reveal the validity and the benefits of the closed-chain kinematics of the fourlinkage mechanism to a crawling gait [14]. Liu et al. designed a docking wall-climbing robot for barrier-crossing [15]. Koo et al. presented a wall climbing robot system with driving wheels [16]. Hirai et al. proposed an arm-equipped reconfigurable multi-modules wall-climbing robot [17]. Liang et al. designed a climbing robot for inspection of glass curtain walls [18]. Lu et al. studied a motion control approach to transfer locomotion types of a multi-locomotion robot from walking to brachiating for maneuver performance [19]. Above wall climbing robots are applied for visual inspection of surface defects, clearing waste on surface. However, up to now, what has not been solved is how to remanufacture or to repair the surface defects of various large heavy equipment such as ironsteel making furnace hulls, large containers of oil/gas in highaltitude work-fields or ship hulls in deep water using wall climbing robots. For this reason, a novel climbing robot with tri-planar limbs was designed to climb on the surface of various large heavy equipment and repair surface defects by laser melting coating, drilling and grounding. Since all the operation workloads of the climbing robot and the inertial wrenches of the moving

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(a)

(b) Fig. 1. A 3D prototype of climbing robot with tri-planar limbs (a); composite spherical joint (b).

links must be transmitted from tri-planar limbs to the magnetic feet and borne by the magnetic feet, it is a significant and challenge issue to establish its kinematics and dynamics model to ensure it operates safely during wall climbing. Hence, we focused on the kinematics, dynamics and workspace of the climbing robot with tri-planar limbs.

2. Structure characteristics of climbing robot with tri-planar limbs A 3D prototype of the novel climbing robot with tri-planar limbs is designed and approved for a Chinese invent patent (ZL201310651656.5) [20], see Fig. 1. It includes a moving platform m, three planar limbs Lpi (i = 1, 2, 3), 3 SPS-type linear elastic limbs Lei, six magnetic feet and a return mechanism. Here, m is a circle plate with three connection points bi (i = 1, 2, 3) and three connection points ai. bi and ai (i = 1, 2, 3) are disitrubuted uniformly on m at the same circumference. Each Lpj includes an upper beam gi, two SPR-type linear active legs rij (i = 1, 2, 3; j = 1, 2), two revolute joints Rij, a univeral joint Ui, a lower beam Gi, magnetic feet and a small spring. gi is connected with m by a

revolute joint Ri at bi. Each rij includes a linear actuator, the upper end of rij is connected with gi by Rij at the one end point bij of gi; the lower end of rij is connected with Gi at the point Bj. Gi is connected with magnetic feet by univeral joint Ui at Bi. Let ||, ^, | be the parallel, perpendicular, coincident constraints. In structure, Ri1||Ri2 and Ri^Rij are satisfied. Thus, rij and gi are located in the same plane. Each Lei (i = 1, 2, 3) includes an SPS-type linear actuator, two spherical joints S, a magnetic feet, a drag spring, a return spring and a small spring. The upper end of Lei is connected with m by S joint at ai. The lower end of Lei is connected with the magnetic disc by S joint. When Lei is stretched out, the magnetic feet are sucked onto B. When Lei draws back, a pulling force is applied onto m by the drag spring as the magnetic feet are sucked onto B. When the magnetic feet are away from B, Lei and the magnetic feet are returned to their original postions by the return spring and the small spring. The return mechanism includes a pressing-plate and a translational actuator. The pressing-plate can be reciprocated by the translational actuator. The three Lpi can be returned to their original positions by pressing-plate and three upper beam gi. The machining processes are explained as follows: (1) Six magnetic discs are always sucked onto the surface of the equipment B; (2) three Lei are drawed back properly, three pulling forces are applied onto m by the drag spring; (3) three Lpi are stretched out properly, a parallel minipulator with triplanar limbs is formed. The processes of climbing wall are explained as follows: (1) Three Lpi stretched out, the magnetic feet of Lei are away from B and Lei and its magnetic feet are returned to their original postions; (2) drive three Lpi, m and three Lei are translated to new position in any direction and the magnetic feet of Lei are sucked onto B at new position; (3) three Lpi are drawn back, the three magnetic feet of Lpi are away from B, Lei and its magnetic feet are returned to their original postions by the return mechanism and small spring; (4) repeat steps (1) to (3) to continue climbing wall. The climbing robot has merits as follows: (1) Can move in translation to any direction during climbing wall. (2) The tri-planar limbs with revolute joint R and prismatic joint P are simple in structure and easy in manufacturing; (3) R joint has a larger capability of pulling force bearing than that of spherical joint S; (4) The workspace of the proposed manipulator can be increased because the rotation range of R joint is larger than that of S joint before interference. When an inspector and various tools (such as the laser welding gun, the milling, drilling, grinding spindle) are attached on the moving platform, the climbing robot can climb on the surface of various large heavy equipment and repair their surface defects in the high-altitude work-field by laser melting coating, milling, drilling and grinding during

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Em is the distance from bi to o. Lm is the distance between two bi. ± is “+” as j = 1; ± is “-” as j = 2. This condition is also suitable for Eqs. (4)-(6) with ±. Let Rm→B be a rotational matrix from {m} to {B} in order of ZYZ. Let (Xo, Yo, Zo) be position components of m at its central point o. Let (xl, yl, zl, xm, ym, zm, xn, yn, zn) be the nine orientation parameters of m in {B}. Rm→B and o in {B} are represented as follows: æ xl ç Rm®B = ç xm çx è n

Fig. 2. Kinematics model and dynamic model of PM with tri-planar limbs.

3. DoF and displacement of PM with tri-planar limbs When three magnetic feet of the tri-planar limbs are fixed (sucked) onto the surface of the equipment, the climbing robot with tri-planar limbs is transformed into a Parallel manipulator (PM) with tri-planar limbs. The PM with tri-planar limbs includes n = 20 links for (1 m, 1 B, three upper beams, six piston rods, xix cylinders, three lower beams; g = 24 joints for (15 R, 6 P, 3 U); Σfi = 27 for (15R, 6P, 3U), redundant constraint is υ = 3υp = 9 because the redundant constraint of each of three planar limbs is υp = 3; passive DoF is ζ = 0. Thus, the DoF of PM with tri-planar limbs is calculated based on revised Grübler-Kutzbach formula [21, 22] as below: g

(1)

i =1

A kinematic model of the PM with tri-planar limbs is shown in Fig. 2. Let {m} be the coordinate system of m, {B} be coordinate system of B. Let α, β, γ be three Euler angles of m in {B}. Let φ be one of {θ, 2θ, (i-1)θ, α, β, γ; (i = 1, 2, 3)}. Set sφ = sinφ, cφ = cosφ, tφ = tanφ. The position vectors Bi of the connection points Bi of B in {B}, the position vectors mbi of the connection points bi of m in {m} are represented [23] as follows: æ -s( i-1)q ö æ -s( i-1)q ö ç ÷ m ç ÷ E = 3L / 3 Bi =E ç c( i-1)q ÷ , bi =Em ç c( i-1)q ÷ , ç 0 ÷ ç 0 ÷ Em = 3 Lm / 3 è ø è ø æ ±e ö æ - Em sq ± ecq ö æ - Em s2q ± ec2q ö ç ÷ ç ÷ ç ÷ m b1 j = ç Em ÷ , mb2 j = ç ± esq + Em cq ÷ , mb3 j = ç ± es2q + Em c2q ÷ . ç 0 ÷ ç ÷ ç ÷ 0 0 è ø è ø è ø

zl ö æ Xo ö ÷ ç ÷ zm ÷ , o = ç Yo ÷ , çZ ÷ zn ÷ø è oø

æ - sa cb sg + ca cg ç Rm®B = ç ca cb sg + sa cg ç sb sg è

- sa cb cg - ca sg ca cb sg - sa sg sb cg

(3) sa sb ö ÷ -ca sb ÷ . cb ÷ø

The position vectors bij of bij of beam Gi in {B} and bi are derived as follows:

maintenance and inspection of large vertical structures.

M =6(n-g -1) + å f i + u =6 ´ (20-24-1) + 27 + 9 = 6 .

yl ym yn

(2)

Here, e is the distance from bi to bij (i = 1, 2, 3; j = 1, 2); E is the distance from Bi to O. L is the distance between two Bi.

bij = Rm®B mbij + o, (i = 1, 2,3; j = 1, 2)

(4)

æ ± exl + Em yl + X o ö ç ÷ b1 j = ç ± exm + Em ym + Yo ÷ , ç ± ex + E y + Z ÷ n m n o ø è æ (- Em sq ± ecq ) xl + ( Em cq ± esq ) yl + X o ö ç ÷ b2 j = ç (- Em cq ± ecq ) xm + ( Em cq ± esq ) ym + Yo ÷ , ç (- E c ± ec ) x + ( E c ± es ) y + Z ÷ q q m q n m q n o ø è æ (- Em s2q ± ec2q ) xl +( Em c2q ± es2q ) yl +X o ö ç ÷ b3 j = ç (- Em s2q ± ec2q ) xm +( Em c2q ± es2q ) ym +Yo ÷ , ç ÷ è (- Em s2q ± ec2q ) xn +( Em c2q ± es2q ) yn +Z o ø æ - Em s( i-1)q xl +Em c( i-1)q yl +X o ö ç ÷ bi = ç - Em s( i-1)q xm +Em c( i-1)q ym +Yo ÷ . ç -E s ÷ è m ( i-1)q xn +Em c( i-1)q yn +Z o ø

The inverse displacement of the PM with tri-planar limbs is derived from Eqs. (4) and (2) as follows: rij = bij - Bij , (i =1, 2, 3; j = 1, 2) æ ± exl + Em yl + X o ö ç ÷ r1 j = ç ± exm + Em ym - E + Yo ÷ , ç ± ex + E y + Z ÷ n m n o è ø

(5)

æ (-Em sq ± ecq )xl + (Em cq ± esq )yl + Esq + X o ö ç ÷ r2 j = ç (-Em cq ± ecq )xm + (Em cq ± esq )ym -Ecq + Yo ÷ , ç ( - E c ± ec )x + (E c ± es )y + Z ÷ m q n m q n 0 q q è ø æ (-Em s2q ± ec2q )xl + (Em c2q ± es2q )yl + Es2q + X o ö ç ÷ r3 j = ç (-Em s2q ± ec2q )xm + (Em c2q ± es2q )ym -Ec2q + Yo ÷ . ç (-E s ± ec )x + (E c ± es )y + Z ÷ m 2q 2q n m 2q 2q n o è ø

During a wall climb, the PM with tri-planar limbs is moved in translation. In this case, α = β = γ = 0º are satisfied. Thus,

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the inverse displacement of the PM with tri-planar limbs is derived from Eq. (4) as follows: æ ±e + X o ö æ Dsq ± ecq + X o ö ç ÷ ç ÷ r1 j = ç - D + Yo ÷ , r2 j = ç ±esq - Dcq + Yo ÷ , ç Z ÷ ç ÷ Zo o è ø è ø æ Ds2q ± ec2q + X o ö D = E - Em , ç ÷ r3 j = ç ±esq - Dc2q + Yo ÷ , a = b = g = 0°, ç ÷ Zo è ø 2 1j

2

2 o

Let ζ be a vector, zˆ be skew-symmetric matrix, E be a 3×3 unit matrix. They satisfy [23], z ´ = zˆ , zˆ T = -zˆ , - zˆ 2 = E - zz T .

(6)

(9)

Here ζ may be one of (ei, eij, δi, δij, Ri1, Ri2). vrij is derived from Eqs. (9) and (10) as below, vrij =vbij δij =[v + ω ´ (ei + eij )]δij =J vrijV ,

(

J vrij = d ijT

2 o

ìr =( X o ± e) + Y + Z , D = E - Em , ïï 2 2 2 2 ír2 j =( Dsq ± ecq + X o ) +(± esq - Dcq + Yo ) +Z o ï 2 2 2 2 ïîr3 j =( D2q ± ec2q + X o ) +(± es2q - Dc2q + Yo ) +Z o

(10)

-d ijT (eˆi + eˆij ) )1´6 .

Let frij (i = 1, 2, 3; j = 1, 2) be the active force applied on and along rij, (F, T) be the workload wrench applied on m at o. The general input velocity Vr, the general forward velocity V and the statics are derived based on the principle of the virtual work [23] and Eq. (10) as follows:

ìr112 - r122 =4eX o , ï 2 2 ír21 - r22 =4( Dsq + X o )ecq + 4( Dcq + Yo )esq , ï 2 2 îr32 - r31 =4( Ds2q + X o )ec2q + 4( Dc2q + Yo )es2q .

Vr =JV , V =J -1Vr ,

The forwards displacement is derived from Eq. (6) as, Xo =

r112 - r122 r2 - r2 r2 - r2 , Yo = 21 22 - 11 12 , 4e 4esq 4esq cq

Fr TV + ( F T

(7)

2

a =b =g =0°, Z o = r112 - ( e + X o ) - Yo 2 .

4. Velocity/acceleration of PM with tri-planar limbs

(11)

æF ö T T )V =0, Fr =-(J T ) -1 ç ÷ , èT ø

Vr = ( vr11 vr12

T

vr 31 vr 32 ) ,

vr 21 vr 22

Fr = ( f r11

f r12

f r 21

J = ( J vr11

J vr12

J vr 21

f r 22 J vr 22

T

f r 32 ) ,

f r 31 J vr 31

T

J vr 32 ) .

Here, J is a 6×6 Jacobian matrix.

4.1 Velocity of 6-DoF PM with tri-planar limbs The velocity analysis provides a theoretical foundation for the derivation of statics and acceleration of the 6-DoF PM with tri-planar limbs. Let (V, v, ω, A, a, ε) be the general forward velocity, the translational velocity, the angular velocity, the general forward acceleration, the translational acceleration, and the angular acceleration of m at o, respectively, see Fig. 2. Let (^, ||, |) be a perpendicular, a parallel constraint, a collinear constraint, respectively. Let ei be the vector from o to bi; eij be the vector from bi to bij, Rij be the unit vector from bi to bij; δi be the unit vector from Bi to bi, δij be the unit vector from Bi to bij. Let vbi be the velocity of m at bi in {B}, vbij be the velocity of the upper beam gi at bij in {B}, ωgi be the angular velocity of gi in {B}; ωri be the angular velocity of ri in {B}, ωrij be the angular velocity of rij in {B}, ωij3 be the scalar angular velocity of gi about rij at bij, vri be the scalar velocity along ri, vrij be the input scalar velocity along rij. Let ωi1 be the scalar angular velocity of gi about Ri at bi. Let Ri1 be the unit vector of Ri. Since Ri1|eij is satisfied, there is Ri1×eij = 0. Thus, vbi and vbij are derived as follows:

vbij = vbi + ωgi ´ eij = vbi + (ω - wi1 Ri1 ) ´ eij = v + ω ´ ei + ω ´ eij .

Let ωi2 be the scalar angular velocity of gi about ri at bi. Let Ri2 be the unit vector of ωi2. Let ωi3 be the scalar angular velocity of gi about rij at bij. Let Ri3 be the unit vector of ωi3. Ri1⊥Ri2, Ri2⊥ri and Ri3||Ri2 are satisfied. Thus, the relation between the angular velocity ωgi of gi, the angular velocity ωri of ri, and the angular velocity ωrij of rij can be represented from Eq. (9) as below: ωgi =ω-wi1 Ri1 =ωri -wi 2 Ri 2 =ωrij -wi 3 Ri 2 .

(12)

ωri and ωrij can be represented from Eq. (12) as follows: (13a) (13b)

ωri = ω - wi1 Ri1 + wi 2 Ri 2

ωrij = ω - wi1 Ri1 +wi 3 Ri 2 .

Cross multiply both sides of Eq. (13a) by ri, from Eq. (8): ω ´ ri - wi1 Ri1 ´ ri +wi 2 Ri 2 ´ ri = vbi - vri δi = (d i × d i )νbi - (νbi × d i )d i =-d ´ (d ´ ν )=-dˆ 2ν =dˆ 2 (-ν + eˆ ω) .

vbi = v + ω ´ ei = vrid i + ωri ´ ri , ωgi +wi1 Ri1 = ω,

4.2 Angular velocity of upper beam and active limbs

i

i

bi

i

bi

i

(14a)

i

(8) Similarly, cross multiply the both sides of Eq. (13b) by rij, from Eq. (9):

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ω ´ rij - wi1 Ri1 ´ rij +wi 3 Ri 2 ´ rij

(14b)

= vbij - vrij δij = δˆ (-v + eˆi ω + eˆij ω) . 2 ij

Dot multiply both sides of Eq. (14a) by Ri1 and Ri2, respectively:

i

(15)

(ω ´ ri ) Ri 2 - (wi1 Ri1 ´ ri ) Ri 2 +(wi 2 Ri 2 ´ ri ) Ri 2 = RT δˆ 2 (-v + eˆ ω). i2 i

J wri

D2 δˆi (ri ω - δˆi v + δˆi eˆi ω) = J wriV , di

æ D δˆ 2 =ç 2 i ç di è

(21)

D δˆ (r + δˆi eˆi ) ö E- 2 i i ÷. ÷ di ø

The angular velocity ωrij of rij is derived from Eqs. (13b) and (20) as follows:

(ω ´ ri ) Ri1 -(wi1 Ri1 ´ ri ) Ri1 +(wi 2 Ri 2 ´ ri ) Ri1 = RT δˆ 2 (-v + eˆ ω), i1 i

ωri = ω -

ωrij =ω-D2 δˆij (rij ω-δˆij v + δˆij eˆi ω + δˆij eˆij ω) / dij

(22)

=J wrijV ,

i

Jwrij =

ωi1 and ωi2 can be derived from Eq. (15) as follows:

1 D2 δˆij2 d ij

(

)

d ij E -D2 δˆij (rij + δˆij eˆi + δˆij eˆij ) .

The angular velocity ωgi of gi is derived from Eq. (12) as follows:

(ω ´ ri ) Ri 2 + RiT2 δˆi2 (v - eˆi ω) wi1 = , ( Ri1 ´ ri ) Ri 2 (ω ´ ri ) Ri1 + RiT1 δˆi2 (v - eˆi ω) -wi 2 = . ( Ri 2 ´ ri ) Ri1

(16)

J w gi = 3´6

Dot multiply the both sides of Eq. (14b) by Ri1 and Ri2, respectively:

(23)

ωgi = ω - wi1 Ri1 = J w giV , 1 Ri1 RiT2 δˆij2 di

(

)

d i E + Ri1 (ri ´ Ri 2 )T -Ri1 RiT2 δˆi2eˆi .

4.3 Acceleration of 6-DoF PM with tri-planar limbs Differentiating Eq. (11), a scalar acceleration arij along rij is derived from Eqs. (10) and (11) as follows:

(ω ´ rij ) Ri1 - (wi1 Ri1 ´ rij ) Ri1 +(wi 3 Ri 2 ´ rij ) Ri1 = Ri1T δˆij2 (-v + eˆi ω + eˆij ω),

(17)

(ω ´ rij ) Ri 2 - (wi1 Ri1 ´ rij ) Ri 2 +(wi 3 Ri 2 ´ rij ) Ri 2

arij ={a + ε ´ (ei + eij ) + ω ´ [ω ´ (ei + eij )]}δij + (v + ω ´ ei + ω ´ eij )δij¢

= Ri 2T δˆij2 (-v + eˆi ω + eˆij ω).

=δij a + [(ei + eij ) ´ δij ]ε + ω ´ [ω ´ (ei + eij )]δij + [(v + ω ´ ei + ω ´ eij ) 2 - vrij2 ] / rij

ωi3 and ωi1 can be derived from Eq. (17) as follows:

=δij a + [(ei + eij ) ´ δij ]ε + [ω ´ (ei + eij )](δij ´ ω) -wi 3 =

wi 1 =

(ω ´ rij ) Ri1 -RiT1 δˆij2 (-v + eˆi ω + eˆij ω) ( Ri 2 ´ rij ) Ri1

(ω ´ rij ) Ri 2 -R δˆ (-v + eˆi ω + eˆij ω) T 2 i 2 ij

( Ri1 ´ rij ) Ri 2

- ( J vrijV ) 2 + {[ E3´3 ,

(18)

= J vrij A +

1 E rij

Let D1 = Ri1×Ri2, D2 = Ri1RTi2-Ri2RTi1, di = riD1, dij = rijD1. From Eq. (16): D2 δˆi (ri ω - δˆi v + δˆi eˆi ω) . di

(19)

From Eq. (18),

wi1 Ri1 -wi 3 Ri 2 =

D2 δˆij (rij ω-δˆij v + δˆij eˆi ω + δˆij eˆij ω) d ij

-(eˆi + eˆij )V

T

) (E

-(eˆi + eˆij )V

)

T + ωT (eˆi + eˆij )δˆij ω - (V T J vrij J vrijV ) / rij

.

arij =J vrij A +

wi1 Ri1 - wi 2 Ri 2 =

(

-eˆi - eˆij ]V }2 / rij

-(eˆi + eˆij ) ö 1 Tæ E V çç ÷V ˆi + eˆij ) -(eˆi + eˆij ) 2 ÷ ( e rij è ø

0 æ0 ö ÷V +V T ç ç 0 (eˆi + eˆij )δˆij ÷ è ø T æ ö d ij d ij -d ij d ijT (eˆi + eˆij ) 1 - VT ç V. T ç (eˆi + eˆij )d ij d ij - (eˆi + eˆij )d ij d ijT (eˆi + eˆij ) ÷÷ rij è ø

(24)

The scalar acceleration arij and the general input acceleration Ar of the 6-DoF PM with tri-planar limbs are derived from Eqs. (10), (11) and (24) as follows: .

(20)

The angular velocity ωri of ri is derived from Eqs. (13a) and (19) as follows:

arij =J vrij A + V T hijV , i =1, 2, 3; j =1, 2, æ - δˆij2 hij = ç ç -(eˆi + eˆij )δˆij2 6´6 è

δˆij2 (eˆi + eˆij ) ö ÷, ÷ h ø

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h=rij (eˆi + eˆij )δˆij + (eˆi + eˆij )δˆij2 (eˆi + eˆij ), Ar =JA + V T HV , A = J -1 ( Ar - V T HV ), Ar = ( ar11

ar12

H = ( h11

h12

ar 21 h21

ar 22 h22

ar 31 h31

T

ar 32 ) ,

(25)

T

h32 ) .

Here, H is a 6×6×6 Hessian matrix of the 6-DoF PM with tri-planar limbs; hij is a 6×6 sub-Hessian matrix for the linear active leg rij.

the angular velocity, the general velocity, the translation acceleration, the angular acceleration of moving part of rij at its mass center in {B}, respectively. When g is p and q, the moving part of leg rij in the 6-DoF PM with tri-planar limbs is the piston rod and the cylinder, respectively. The kinematic parameters (vpij, ωpij, Vpij, apij, εpij) of the piston rod are derived as follows: v pij =ωrij ´ l pij δij = - l pij δˆij ωrij = J vpijV ,

(

)

ω pij =ωrij , ( J vpij )3´6 = J vpija

J vpijb ,

4.4 Angular accelerations of ri, rij and gi

J vpija =(-l pij δˆij D δˆ )/d ij , i =1, 2, 3; j =1, 2

Several relative formulae for solving angular accelerations are derived as follows:

J vpijb =-l pij δˆij + l pij δˆij D2 δˆij (rij + δˆij ei + δˆij eij ) / dij ,

D1¢=( Ri1 ´ Ri 2 )¢ =Ri¢1 ´ Ri 2 + Ri1 ´ Ri¢2 ,

2 2 ij

Ri¢1 =( ei1 |ei1|)¢ = -eˆi1ω |ei1| - ei1 eˆi1ω |ei1|2 ,

Ri¢2 = ( Ri1 ´ δi |Ri1 ´ δi |)¢ δˆi Ri¢1 + Rˆ i1δˆi ωri δˆi Ri1| - δˆi Ri¢1 - Rˆ i1δˆi ωri | =+ . |δˆi Ri1| |δˆi Ri1|2

vqij =vrij δij + ωrij ´ (rij - lqij )δij =[v + ω ´ (ei + eij )]δij δij + (lqij -rij )δˆij ωrij ,

Differentiating Eq. (22) with respect to time, the angular acceleration of rij is derived as below,

(

vqij =J vqijV , J vqij = J vqija

(27)

J vqijb

)

3´6

,

J vqija =δ δ + (lqij - rij )δˆij D δˆ / dij , T ij ij

2 2 ij

J vqijb =E -δij δijT (ei + eij ) +

εrij =ε - {D2¢ [rij δˆij ω + v - (eˆi + eˆij )ω - vrij δij ]

T

æ v pij ö æ J vpij ö V pij = çç ÷÷ =J pijV , J pij = çç ÷÷ . w 6´6 è rij ø è J wrij ø

The kinematic parameters (vqij, ωqij, Vqij, aqij, εqij) of the cylinder are derived as follows:

δij¢ =( rij rij )¢ =(v + ω ´ ei + ω ´ eij - δij vrij )/rij ,

-D2 [(vrij δij -rij δij¢ ) ´ ω + rij δˆij ε + a -(eˆi + eˆij )ε ]

(29)

=l pij (εrij ´ δij ) + l pij ωrij ´ (ωrij ´ δij ), e pij =e rij ,

(26)

δi¢= (ri ri )¢ =(v + ω ´ ei - δi vri )/ri ,

-D2 [ω ´ (-(eˆi + eˆij )ω)-arij δij + vrij δˆij ]

a pij =(ωrij ´ l pij δij )¢

D2 δˆij (rij + δˆij ei + δˆij eij ) d ij

,

ωqij =ωrij , e qij =e rij , i = 1, 2, 3; j = 1, 2; aqij =[vrij δij + ωrij ´ (rij - lqij )δij ]¢

T ij

aqij =arij δij + 2vrij (ωrij ´ δij ) + (rij -lqij )(εrij ´ δij )

+ (ω - ωrij )[(vrij δij - rij δij¢ ) D1 + r D1¢]} / d ij .

+ (rij - lqij )[ωrij ´ (ωrij ´ δij )],

Differentiating Eq. (23) with respect to time, the angular acceleration of gi is derived as below,

æ vqij ö æ J vqij ö Vqij = çç ÷÷ =J qijV , J qij = çç ÷÷ . w 6´6 è rij ø è J wrij ø

(30)

ε gi = ε + {Ri¢1 (ri δˆi Ri 2 )T ω + Ri1 (rri δˆi Ri 2 )T ε

5.2 Velocity/acceleration of upper beam

+ Ri1 (vri δˆi Ri 2 + ri δi¢ ´ Ri 2 + ri δˆi Ri¢2 )T ω T i2

T

- Ri¢1 R (v - eˆi ω - vri δi ) + Ri1 Ri¢2 (v - eˆi ω - vri δi )

(28)

T i2

- Ri1 R [a + (eˆi ω) ´ ω - eˆi ε - ari δi + vri δi¢] + (ω - ωgi )(vri δiT D1 - ri δi¢D1 + riT D1¢)} / d i .

5. Dynamics of 6-DoF PM with tri-planar limbs 5.1 Velocity/acceleration of piston rod and cylinder A dynamics model of 6-DoF PM with tri-planar limbs is shown in Fig. 2. Each rij includes the piston rod and the cylinder. Let pij, qij be mass center of the piston rod and the cylinder, respectively. Let lpij be the distance from pij to Bi. lqij be the distance from qij to bij. Let (vgij, ωgij, Vgij, agij, εgij) (i = 1, 2, 3; j = 1, 2; g = p, q) be the vector position, the translation velocity,

Let (vbi, ωbi, Vbi, abi, εbi Abi,; i = 1, 2, 3) be the translation velocity, the angular velocity, the general velocity, the translation acceleration, the angular acceleration, the general acceleration of upper beam gi at it mass center bi in {B}, respectively. They can be solved as follows: vbi = v + ω ´ ei = vrid i + ωri ´ ri = J vbiV , J vbi = ( E

-eˆiT ) , i = 1, 2, 3

(31a)

3´6

w bi = w gi =J w giV , e bi = e gi , abi = a + ε ´ ei + ω ´ ( ω ´ ei ), æ J vbi ö æv ö Vbi = ç bi ÷ =J biV , J bi = çç ÷÷ . w 6 ´ 6 è bi ø è J w gi ø

(31b)

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Table 1. Given geometric parameters of proposed PM and 6SPS PM and solved workspace volume.

5.3 Dynamics of 6-DoF PM with tri-planar limbs Let Fr be the general input force of the input active legs of the 6-DoF PM with tri-planar limbs. Let τ be the moving link, here τ may be one of (moving platform m, the upper beam gi, the piston rod and the cylinder in rij). Let (aτ, ετ, w, Mτ, Iτ, Gτ) be the translation acceleration, the angular acceleration, the gravity acceleration, the mass, the inertial moment, and gravity of τ. Let (fτ, tτ) be the inertial force and torque of τ. Let (fs, ts) be the static external concentrated force and torque applied on m. (Fr, fs, ts, Gτ) represented as follows:

Given geometric parameters of proposed PM Explanation

Symbol

Value

Vm, m3 0.0167

Length of upper beam

lg, m

0.16

Extension of active leg

rij, m

0.3→0.4

Distance from o to bi

ei, m

0.2

Increment of rij

Δrij, m

0.02

Distance from O to Bi

ei, m

0.2

ft = - M t at , Gt =M t w ,

(32)

tt = - It εt - ωt ´ ( It ωt ),

t =m, gi , pij , qij , (i =1, 2, 3; j =1, 2).

A power equation of PM with tri-planar limbs is derived based on the principle of virtual work [21, 22] and Eqs. (11) and (32) as below: T

T

3 æ f +G ö æ f + f s + Gm ö gi gi FrTVr + ç m ÷V ÷ V + å çç t gi ÷ø bi i =1 è è tm + ts ø T

T

(a)

(33)

æ f pij + G pij ö æ f qij + Gqij ö + åå (çç ÷÷ V pij + çç ÷÷ Vqij ) = 0 . t i =1 j =1 è pij ø è t qij ø 3

2

Here, the first item is the power generated by the active forces of the PM; the second item is the power generated by the inertial wrench of m at o and the workload applied on m; the third item is the power generated by the inertial wrenches of the three balancing beam gi; the final item is the power generated by inertial wrenches of the piston rod and the cylinder in rij (i = 1, 2, 3; j = 1, 2). A formula for solving the general input force of applied input active legs of the PM is derived from Eqs. (11), (23), (2933) as follows: æ f pij + G pij ö æ f +f +G ö 3 2 Fr =(J T ) -1{ ç m s m ÷ +åå J Tpij çç ÷÷ t + t m s è ø i=1 j =1 è t pij ø æ f qij + Gqij ö 3 T æ f gi + Ggi ö + åå J çç ÷+ J bi çç ÷÷ } . f qij ÷ø å i =1 j =1 i =1 è è t gi ø 3

(b)

(34)

2

T qij

(c) Fig. 3. Isometric view (a); tope view (b); front view (c) of reachable workspace of the PM with tri-planar limbs.

6. Workspace of PM with tri-planar limbs The reachable workspace W is a critical index for evaluating the characteristics of PM. It is all the positions that can be reached by the center of m in the limited extension of active legs. Let Vm be the volume of W. The parameters of PM are given in Table 1. Generally, W is formed by a family of similar spatial surfaces, which are cascaded from a lower boundary surface to an upper boundary surface. Each of the similar spatial surfaces is formed by a family of similar spatial curves, see Fig. 3. The construction procedures of the workspace volumes are explained as follows:

Step 1. Set the extension range of six active legs rij (i = 1, 2, 3; j = 1, 2) of the 6-DoF PM with tri-planar limbs in rijmin = 0.3 m → rijmax = 0.4 m; set gi = 0.16 m, ei = 0.2 m. Step 2. Set four of six rij as rijmin or rijmax using permutation and combination, and increase other two rij from rijmin to rijmax by 0.2 m in each step. Step 3. Solve the central position of m, and generate a group of spatial cures. Step 4. Transfer all the solved position data into the advanced CAD software, generate a family of similar spatial

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Y. Lu et al. / Journal of Mechanical Science and Technology 31 (3) (2017) 1427~1436

Table 2. Given dimensions, workload, inertia moment tensor matrices and mass of 6-DoF PM with tri-planar limbs. Symbol, unit

Value

Symbol, unit

Value

e, mm

23

α, β, γ, °

Initial (0, 0, 0)

54

Xo, Yo, Zo, mm

E, mm 2

2

Initial (0, 0, 186)

ar11, mm/s

3

ar22, mm/s

2.4

ar12, mm/s2

2.6

ar31, mm/s2

2.4

ar21, mm/s2

2.6 T

ar32, mm/s2

2

g, m/s

[0, 0, -9.8]

fs, kN

[0, 0, 1]T

lpij, lqij, mm

90, 90

ts, N·m

[0, 0, 1]T

2

Table 3. Absolute errors between analytic and simulation of kinematics and active forces solutions. Kinematic parameter of m, active forces

The maximum of absolute errors of component

Velocity error, ×10-14 mm/s

Δvx 0.933

Δvy 2.665

Δvz 0

Acceleration error, ×10-15 mm/s2

Δax 0.090

Δay 0.073

Δaz -0.472

Angular velocity error, ×10-14 º/s

Δωx 0.533

Δωy 1.132

Δωz 0

Δεx 0.165

Δεy 0.486

Δεz 0.9021

Mpij, Mqij, kg

2, 2

Io, Igi, kg·m

diag [1 1 1]

Angular acceleration error, ×10-16 º/s2

Mm, Mgi kg

4, 2

Ipij, Iqij, kg·m2

diag [1 1 1]

Active force errors

Δfr11

Δfr21

Δfr21

Δfr22

Δfr13

Δfr23

Value, ×10-3 N

1.2

0.45

0.5

0.6

0.4

1

2

dm, mm

50

FN, kN

1.8

h, mm

20

Ft, kN

1.6

curves, and construct a lower and an upper boundary surface from the family of similar spatial curves using the loft command. Step 5. Generate a family of similar spatial surfaces, which are cascaded from a lower boundary surface to an upper boundary surface using the loft command. Step 6. Generate the workspace volumes of the PM by generating a family of similar spatial surfaces. Step 7. Assemble all the 3D workspaces with the same B, see Fig. 3. The measured volumes of every 3D workspace are listed in Table 1.

(a)

(b)

7. Numerical example of PM with tri-planar limbs Some dimensions of 6-DoF PM with tri-planar limbs, the workload applied on the moving platform at its center, the inertia moment tensor matrices of Iτ, Mτ, (τ = m, gi, pij, qij; i = 1, 2, 3; j = 1, 2) are given in Table 2. The kinematics and dynamics solutions of 6-DoF PM with tri-planar limbs can be solved using relative analytic formulae in Secs. 3, 4, 5. When given aij (i = 1, 2, 3; j = 1, 2) in Table 2, rij (i = 1, 2, 3; j = 1, 2) are solved, see Fig. 4(a); (xo, yo, zo, α, β, γ) of m at o are solved, see Fig. 4(b); the translational velocity/acceleration (vxo, vyo, vzo, axo, ayo, azo) of m are solved, see Fig. 4(c); the angular velocity and acceleration (ωx, ωy, ωz, εx, εy, εz) of m are solved, see Fig. 4(d); the dynamic active forces frij (i = 1, 2, 3; j = 1, 2) are solved, see Fig. 4(e). The absolute errors between analytic and simulation of kinematics and active forces solutions are given in Table 3. It is known that all derived analytical formulae are correct because the maximum of absolute errors of each item are very small. A prototype of the magnetic feet was developed, see Fig. 5. It is connected with the limb by spherical joint. The two permanent magnetic discs are set into the cavity of each magnetic feet. Let dm and h be the diameter and the height of each magnetic disc. Let FN and Ft be the normal sucking force and the friction force of each magnetic foot. The experiment

(c)

(d)

(e) Fig. 4. Kinematics and dynamics solutions of 6-DoF PM with triplanar limbs.

Y. Lu et al. / Journal of Mechanical Science and Technology 31 (3) (2017) 1427~1436

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References

Fig. 5. Prototype of magnetic feet.

results of FN and Ft are given in Table 2. When the permanent magnetic discs are replaced by the eletromagnet discs, FN and Ft may be increased to 4→5 times of the permanent magnetic discs. When the number of permanent magnetic discs is increased in each magnetic feet, FN and Ft can be increased largely.

8. Conclusions (1) A novel climbing robot with tri-planar limbs is composed of a moving platform, tri-planar limbs with magnetic feet, three SPS-type linear elastic limbs with magnetic feet and a return mechanism. When the magnetic feet of the tri-planar limbs are sucked onto the surface of the equipment, the climbing robot with tri-planar limbs is transformed into a 6DoF parallel manipulator with tri-planar limbs. (2) The kinematics and dynamics formulae for solving the velocity and the acceleration of the moving links and dynamic active forces of the 6-DoF parallel manipularor with tri-planar limbs were derived and can be used to solve the kinematics and dynamics of the climbing robot with tri-planar limbs. (3) When given the input kinematics of the parallel manipulator, the velocity and the acceleration of the moving links in the climbing robot with tri-planar limbs can be solved. The inertial wrench of the moving links can be solved. (4) When given the workload applied on the platform, the dynamic active forces of the climbing robot with tri-planar limbs can be solved. (5) A climbing robot with tri-planar limbs has potential applications for climbing on the surface of various large heavy equipment and repairing their surface defects in the highaltitude work-field by laser melting coating, milling, drilling and grinding during maintenance and inspection of large vertical structures. (6) The further study will focus on singularity analysis, the stiffness model, the optimization, and the experimental study of the climbing robot with tri-planar limbs.

Acknowledgments The authors would like to acknowledge (1) Project (E2016203379) supported by Natural Science Foundation of Hebei, (2) Project (51175447) supported by National Natural Science Foundation of China (NSFC).

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Yi Lu got his B.S. and M.S. (Engineering) at Northeast Heavy Machinery Institute in Qiqihar, China, and Dr. Sc. Tech. degree at University of Oulu, Finland, 1997. Dr. Yi Lu has been Professor of Mechanical Engineering since 1998, at Yanshan University in Qinhuangdao, P.R. China. Keke Zhou is a Master’s student of College of Mechanical Engineering, Yanshan University, Qinhuangdao, Hebei, P.R. China.

Nijia Ye is a Ph.D. candidate at College of Mechanical Engineering, Yanshan University, Qinhuangdao, Hebei, P.R. China.