Dynamics analysis of flexible space robot with joint friction

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Dynamics analysis of flexible space robot with joint friction

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Xiaofeng Liu, Haiquan Li, Jingsen Wang, Guoping Cai ∗

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Department of Engineering Mechanics, State Key Laboratory of Ocean Engineering, Shanghai Jiaotong University, Shanghai 200240, China

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a r t i c l e

i n f o

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a b s t r a c t

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Article history: Received 20 March 2015 Received in revised form 31 July 2015 Accepted 27 September 2015 Available online xxxx Keywords: Flexible space robot Jourdain’s velocity variation principle Joint friction Active control

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It is well known that friction is an inevitable phenomenon existing in almost all mechanical systems including robotic systems. Friction can harm dynamic characteristics of mechanical systems as well as the accuracy of manual control. In this paper, we have comprehensively investigated the dynamic process of a flexible space robot with joint friction. Dynamic equation of the system is established based on the Jourdain’s velocity variation principle and the single direction recursive construction method. The Coulomb friction model and LuGre friction model are adopted to describe the joint friction. The calculation method for joint friction is discussed in detail. Moreover, an active controller is designed by the computed torque control method for trajectory tracking control. Simulation results indicate that the proposed model is effective in describing the dynamics response of the system. Joint friction can cause oscillation of the joint response and vibration of the flexible link, as well as affecting the precision of position control of the system. From the results we present, it can also be inferred that the Coulomb friction model is limited in describing the nonlinear features of friction. © 2015 Published by Elsevier Masson SAS.

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1. Introduction

37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59

Flexible space robot is one of the typical unrooted tree dynamic systems with strong nonlinear coupling and time-variant features. This kind of robot has been widely used in space missions due to the advantages such as large load-to-weight ratio and low energy consumption. However, the large load-to-weight ratio will lead to this space robot manipulator to large deformation and considerable amplitude of vibration [1–3]. Although researchers have conducted many investigations on modeling and control of flexible space robot, many of them do not consider the effect of joint friction that cannot be ignored in the space robot system. From the perspective of controller design, joint friction may be regarded as an external disturbance. One can neglect the nonlinear characteristics of joint friction by enhancing the robustness and adaptability, paying the penalty of decreasing control performance. However, the nonlinear characteristics of friction, such as hysteresis, stick–slip and limit cycle are becoming more standing out when it comes to cases where the space robot is moving in low speed, since the coupling characteristics between joint friction and manipulator deformation would be amplified. These nonlinear characteristics could cause complex dynamic behaviors and notable inaccuracy in control result. So the effect of joint friction on the space robot system should not be neglected.

60 61 62 63 64 65 66

*

Corresponding author. E-mail address: [email protected] (G. Cai).

http://dx.doi.org/10.1016/j.ast.2015.09.030 1270-9638/© 2015 Published by Elsevier Masson SAS.

Ambrose and Askew [4] experimentally studied the joint friction of space robot system. Their results showed that static friction of joint was found to increase by a factor of 5 over the ExtraVehicular Activity temperature ranging from 223 K to 373 K, and the kinematic fiction, mapped in a speed–torque curve, was found to vary by 10 or more. The experiment results of ROKVISS, a German space robotics technology experiment, showed that the joint friction of space robot would increase notably in space in comparison with the value on ground [5,6]. Comparing with the ground measurement results, the total frictions increase by 50% and 20% in two different joints of a space robot, respectively. The results of these experiments indicate that joint friction could more negatively affect the dynamics of the system in the severe space environment than that on ground. So the necessity to investigate the effect of joint friction on space robot is obvious. Today some researchers have already carried out studies on this friction problem. For example, Katoh et al. [7] proposed a real time path planning method of a space robot in saving the energy consumed by the mechanical viscous friction forces and the armature’s resistances of motors. But the frictional force adopted is only related with velocity, making the model exercise limited in practice. Breedveld et al. [8] used a new friction model, which is based on a hypothesis that friction tries to stop the system, to simulate the effects of friction on a flexible space manipulator. The friction model simulated the effects of the friction accurately in only little computing time and without numerical instability problems. It can also be used to simulate Stiction and fluid friction, as well as serving as a useful aid to model friction in a discrete simulation model of a position or

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velocity controlled mechanical system. Zavrazhina and Zavrazhina [9] established the dynamic equations of a space manipulator considering the flexibility and friction of hinge joint based on the Euler–Lagrange and Lagrange 2nd kind equations, and analyzed the influence of elastic and friction of hinge joints on the space manipulator. Martins et al. [10] proposed an adaptive neural network control design method for trajectory tracking control of space robot manipulators considering the influence of friction and payload. Xue et al. [11] derived the dynamic model of a space robot, as well as the low speed friction model, and applied two different distributed variable structure control methods for a space robot. The analysis has shown that a proper variable structure control law can not only have better robustness and adaptability, but also reduce the complexity of computation. Zhao and Bai [12] presented a computational methodology for analysis of space robot manipulator system considering the effects of the joint clearances and joint friction, which is modeled by the Coulomb friction model. Krzyzak et al. [13] derived the dynamics of two-link flexible space manipulator based on the Euler–Lagrange formulation, where the nonlinear stiffness and friction components of joints are considered. Liu et al. [14] derived the dynamics of a space robot with a 6-DOF manipulator considering joint friction, and studied the influence of joint friction on the system. From above we can see that the friction in a space robot has gained attention in the field of space robot and some research results have been published. However, there are still some problems worth further exploring. For example, in the above studies for joint friction, the dynamic equations of space robot are often established by the conventional Lagrangian formulation or Newton–Euler method, making the dimensions of the dynamic equations bigger than the degrees of freedom of the system, which increases the difficulty in solving the dynamic equations. Besides the friction model adopted cannot fully describe frictional behaviors, and the detailed calculation of joint friction was not given. The expression of revolute joint in the above studies is overly simplified, or alternatively, planar model of joint is adopted, both leading to the result that the calculated frictional force cannot reflect the true value in actual situation. Considering the flexibility of the space robot manipulator, the effect of joint friction on the space robot system becomes complex. So it is necessary to investigate the problem mentioned above in detail. In this paper, dynamics modeling and active control of a flexible space robot considering joint friction are investigated. The influence of the joint friction on the flexible space manipulator is analyzed. This paper is organized as follows. Section 2 briefly presents the expressions of dynamic model of the system, including the kinematic and dynamic equations of single flexible body, the kinematical recursive relations of the system, the calculation of joint friction and the establishing of dynamic equations of the system. The controller design for trajectory tracking is given in Section 3. Section 4 provides simulation and comparison studies. Finally, a concluding remark is given in Section 5.

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2. Dynamic model of the system

67 68 69 70 71 72 73 74 75 76 77

Fig. 1. Structural model of space robot.

78 79 80 81 82 83 84 85 86 87 88

Fig. 2. Single flexible body.

89 90

2.1. Kinematic and dynamic equations of single flexible body

91 92

Fig. 2 shows a single flexible body B. The lumped mass finite element method is used to divide the body B into l elements. The mass matrix of the k-th node can be expressed as

 k

M =

mk

0

0

Jk



60 61 62 63 64 65 66

95

∈

,

k = 1, . . . , l n

(1)

where mk ∈ 3×3 and J k ∈ 3×3 are the translational mass matrix and matrix of moment of inertia of the k-th node, respectively; ln is the amount of nodes of the body B. As shown in Fig. 2, the point C is the mass center of the body B before the deformation, and the floating frame e is established on it. The relative position vector of the node k is ρk0 before the deformation and ρk after the deformation. The vectors r and r k are the absolute position vectors of the mass center C and the  k and ϕk are the translational node k, respectively. The vectors u and rotational deformation vectors of the node k, respectively. They can be described by the modal coordinates, given by

97 98 99 100 101 102 103 104 105 106 107 108 109 110 111

k

 x k = Φ u

(2)

k ϕ = Ψ x

(3)

k

 k and Ψ k are the translational and rotational modal vector where Φ arrays of the node k, respectively; x is the modal coordinate vector of the body B. Choosing the s modes of the body B, we have k

k

 = [ϕk , . . . , ϕsk ], Ψ = [ψ k , . . . , ψ sk ] and x = [x1 , . . . , xs ]T . The coΦ 1 1  k and Ψ k in the absolute reference frame ordinate matrices of Φ are Φ k and Ψ k , respectively, which can be expressed as

112 113 114 115 116 117 118 119 120 121 122

57 59

94 96

6×6

56 58

93

In this paper, a simplified flexible space robot shown in Fig. 1 is used to study the effect of joint friction on the system, which consists of a spacecraft base and a manipulator. The manipulator is composed of Link 1 and Link 2, where Link 1 is rigid and Link 2 is flexible. A payload is fixed on the tip of Link 2 and it is regarded as a lumped mass in this paper. In this section, the dynamic equations of this system are established by the Jourdain’s velocity variation principle and the single direction recursive construction method [15].

Φk = AΦ  k ,

Ψ k = AΨ  k

(4)

where A is the direction cosine matrix relating the floating frame e and the absolute reference frame; Φ  k and Ψ  k are the coordinate k

k

 and Ψ in the floating frame e, which are both matrices of Φ constant matrices. As shown in Fig. 2, the absolute position vector of the node k can be expressed as

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r k = r + ρk = r + ρk0 + u k

(5)

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The translational velocity and acceleration vectors of the node k can be expressed as k k r˙ = r˙ + ρ˙ = r˙ + ω  × ρk + v kr



r¨ = r¨ + ρ¨ = r¨ + ω ˙ × ρk + akr + 2 × ω  × v kr + ω × ω  × ρ k

k

 k

(6) (7)

˙ are the angular velocity and acceleration vectors of  and ω where ω the floating frame e with respect to the absolute reference frame; v kr and akr are the translational velocity and acceleration vectors of the node k with respect to the floating frame e , respectively, and their matrix forms in the absolute reference frame can be written as v kr

= Φ x˙ , k

= Φ x¨

akr

k

(8)

Considering Eq. (8), the coordinate vectors of Eqs. (5)–(7) in the absolute reference frame can be expressed as

ωk = D k v B

(23)

ω˙ = D v˙ B + τ k

k

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r k = r + ρ k = r + ρ k0 + Φ k x r˙ = r˙ + ρ˙ = r˙ − ρ˜ k

k

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ω + Φ x˙ k

(10)

k ˜ ωρ ˙ + Φ k x¨ + 2ω ˜ Φ k x˙ + ω ˜ k r¨ k = r¨ + ρ¨ k = r¨ − ρ˜ ω

(11)

where and throughout the text the symbol “∼” denotes a skew symmetric matrix, such as

⎡

26 27

k

(9)

−ω3

0

ω˜ = −ω˜ = ⎣ ω3 −ω2 T

0

ω1

⎤

ω2 −ω1 ⎦ ∈ 3×3 , 0

⎡

(12)

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Writing Eqs. (10) and (11) in a matrix from, we have

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(13)

r¨ k = R k v˙ B +  k

(14)

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k

where k



R = I3

−ρ˜

k

Φ

k



∈

3×(6+s)

˜ k ∈ 3×1  k = 2ω˜ Φ k x˙ + ω˜ ωρ

v B = r˙

T

ω

T

 T T

x˙

∈

(6+s)×1

(15)



D = 0

I3

Ψ

ω˙ k = ω + α kr + ω ×ω kr 

 rk



v˙ k = Π k v˙ B +

ω = Ψ x˙ ,

α = Ψ x¨

k

k r

k

(20)

Using Eq. (20), the coordinate vectors of Eqs. (18) and (19) in the absolute reference frame can be expressed as

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ωk = ω + Ψ k x˙

(21)

ω˙ = ω˙ + Ψ x¨ + ω˜ Ψ x˙ k

k

k

Writing Eqs. (21) and (22) in a matrix form, we have

∈

(22)

3×(6+s)

(25)

72 73

(26)

k

 τk

(27)



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Πk =

84

R

ωkT

k

Dk

T

86

∈ 6×1

(29)



δ v kT · M k v˙ k +

(30)

89 90

F

k

T

k



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− δ ε˙ kT σ k = 0

(31)

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where F k and T k are the resultant external force and resultant external torque on the node k, respectively; εk and σ k are the strain and stress of the node k, δ ε˙ kT σ k is the virtual power of node stress of the node k. The sum of virtual power of node stress of the body B can be calculated by

99 100 101 102 103 104 105

δ ε˙ σ = δ x˙ (C x x˙ + K x x) kT

T

k

(32)

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where C x ∈ s×s and K x ∈ s×s are the modal damping and stiffness matrices of the body B. Substituting Eqs. (27), (28) and (32) into Eq. (31) and rearranging the expression, we have

 T

δ v B − M v˙ B − f ω + f o − f

 u

M=

111 112

(33)

114 116 117

Π kT M k Π k ∈ (6+s)×(6+s)

ln

 Π kT M k

k =1 ln



Π kT

k =1



= 0

T

(34)

118 119

k =1

fo =

110

115

ln

fω =

109

113

=0

where

f

87 88

∈ 6×(6+s)

The kinematic relation of single flexible body is deduced in the above. Here we deduce the dynamic equation of a single flexible body. Based on the Jourdain’s velocity variation principle, the dynamic equation of the body B can be written as

u

83 85

v k = r˙ kT

ln

81 82

(28)

where

k =1

 rk

69

80

vk = Πk v B

(17)

(19)

68

71



Equations (23) and (24) are the rotational coordinate velocity and acceleration relation between the node k and the body B. From Eqs. (13), (14), (23) and (24), we can obtain the relation between the configuration velocity of the node k and the configuration velocity of the body B, which can be expressed as

(16)

where ω and α are the angular velocity and acceleration vectors of the node k with respect to the floating frame e , and their coordinate matrices in the absolute frame can be expressed as k r

k

τ k = ω˜ Ψ k x˙ ∈ 3×1

(18)

˙ 

67

70



k

ln

The parameter v B given in Eq. (17) is the configuration velocity of the body B. Equations (13) and (14) show the relation of the translational coordinate vectors between the node k and the body B. Next we deduce the rotational coordinate vectors relation between the node k and the body B. The absolute angular velocity and acceleration vectors of the node k are

ω k =ω +ω rk

(24)

k =1

r˙ = R v B k

50 51

⎤

ω1 ω = ⎣ ω2 ⎦ ω3

k

where

18 19

3

T

0

F

k

T

k

 k

 τk



120

∈ (6+s)×1

(35)

121 122 123 124

∈ (6+s)×1  T T

(C x x˙ + K x x)

(36)

125 126

∈

(6+s)×1

(37)

where M v˙ B and f ω are the acceleration-dependent and velocitydependent generalized inertia forces of the body B, respectively; f o and f u are the generalized external force and generalized elastic force.

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 hT 0 H hT i = Ai H i h

1 2

T = HΩ i

3 4

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Fig. 3. Schematic diagram of a rigid body and a flexible body.

13

2.2. Kinematical recursive relations

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We can see from Fig. 1 that the space robot system consists of three bodies except for the lumped mass, where Base and Link 1 are both rigid and Link 2 is a flexible body. In this section, the kinematical relation of two adjacent bodies is deduced. We briefly deduce the relation of a rigid body and a flexible body. The relation of two adjacent rigid bodies can be obtained by ignoring the flexible items in the relation of a rigid body and a flexible body deduced before. Fig. 3 is the schematic diagram of a rigid body and a flexible body. We firstly introduce the coordinate frames of this paper as following. As shown in Fig. 3, body j is a rigid body and body i is a flexible body. The absolute reference frame O –XYZ is established on the body B 0 and O is the origin. The floating frame ei is established on the point C i that is the mass center of B i before the deformation. The body fixed frame e j is established on the point C j that is the mass center of B j . We assume that the number of bodies and joints are regularly labeled [15–18], or more exactly, the body i is connected to its inboard body j by the joint i, as shown in Fig. 3, where Q i and P i are the definition points of joint i on the body j and body i. The point P i0 is the joint definiP tion point before the deformation. The local coordinate frame e i i0 is built on the point P i0 , which parallels ei . The frames Q e j i

62 63 64 65 66

and

are the local frames of the points P i and Q i . The local frames h

h

of the joint i are represented by ei and ei 0 , which are fixed on bodies i and j, and the origins of the two frames are P i and respectively. In this paper, the joint is considered as a rigid connection −−−→ i = − Q i P i describes tween the two bodies. As shown in Fig. 3, h

Pi ri

the Q i, bethe

ω =

P Ψ i i x˙ i

(43)

relative translation of the joint i, and ω ri will be used to describe the relative revolution of the joint i. As shown in Fig. 3, the absolute position of the mass center of the body i can be expressed as

ri = r j + ρ

Qi j

+ hi − ρ

Pi i

(38)

Taking the first derivative of Eq. (38) and considering Eq. (10), we have



Pi 

r˙ i = r˙ j + −ρ˜ j i − h˜ i + ρ˜ i



Q

Pi 



 ΩT

˜P ω j + H hT i + ρi H i

q˙ i

P + −Φ i i − ρ˜ iP Ψ i x˙ i

where

P Φi i

(39)

are the translation modal vector matrix of the point P

P i in the absolute reference frame, respectively; Ψ i i are the rotational modal vector matrices; x˙ i is the first derivative of modal coordinate vectors of B i , qi is the vector of joint coordinates; H hT i T and H Ω denote the translational and rotational kinematics relai tions of joint i, and those can be expressed as [15–18]

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Considering Eq. (43), Eq. (42) can be rewritten as

86

Pi T ˙ ˙ ωi = ω j + H Ω i q i − Ψ i xi

(44)

In the absolute reference frame, we define the coordinate vector of the configuration velocity of the two bodies as

vi =



r˙ Ti

ω

T x˙ Ti ,

T i

vj =



r˙ Tj

ω

 T T j

(45)

and redefine the generalized coordinate vector of the body i as



y i = qT

x

 T T

where y i ∈ δi +si , δi is the degree of freedom of the joint i, δi ≤ 6, and si is the amount of modal coordinates of the body i. From Eqs. (39) and (44), the recursive relation of configuration velocity between the body i and the body j can be expressed as

v i = T i j v j + U i y˙ i ,

j = L (i ),

⎡

I3

T ij = ⎣ 0

⎢

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i = 1, . . . , N

(47)

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where

⎡

87

94

(46)

i

Pi ⎤

Q −ρ˜ j i − h˜ i + ρ˜ i

I3

0 P

T H hT + ρ˜ i i H Ω i i

Ui = ⎣

104 105

⎦ ∈ (6+si )×6

0 P

P

P

−Ψ i i

0

I si

(48a)

106 107

Pi ⎤

−Φ i i − ρ˜ i i Ψ i

T HΩ i



60 61

P ei i

Pi ri

67

77

(42)

where ωri = i is i is the joint angular velocity [15–18]; ω the angular velocities of the point P i caused by the deformation, which can be expressed as

16

19

with respect to

H Ω T q˙

14

18

is the direction cosine matrix of e

h0

ωi = ω j + ωri − ωriP i

12

17

(41)

the absolute reference frame, H i hT is the matrix whose non-zero columns are composed of the joint translational axis vectors, H i Ω T is the matrix whose non-zero columns are composed of the joint rotational axis vectors. The relation of angular velocities of B i and B j is

6

15

h Ai 0

where

5

(40)

h A i 0 H i Ω T

108 109

⎥ (6+si )×(δi +si ) ⎦∈

110 111

(48b)

112 113 114

Taking the derivative of Eq. (44), we have

115

ω˙ i = ω˙ j + H Ω T q¨ i − Ψ iP i x¨ i + β i2

(49)

117

where

118

˜ j ωri − ω ˜ iω β i2 = ω

Pi ri

+ ηi

(50)

T˙ HΩ i q i [15–18].

where η i = By taking the derivative of Eq. (39) and considering Eq. (11), one can obtain



Pi 

r¨ i = r¨ j + −ρ˜ j i − h˜ i + ρ˜ i

+

116



Q

P −Φ i i

 ˜ iP i Ψ iP i x¨ i

−ρ

˜ P i Ω T q¨ i ω˙ j + H hT i + ρi H i

+ β i1

120 121 122 123 124





119

125 126

(51)

127 128

where

129

Qi j

Pi i

˜ jω ˜ jω ˜ iω ˜ jρ + ω ˜ j hi − ω ˜ iρ β i1 = ω  Pi  Pi ˜ j v ri − ω ˜ i v ri + ρ˜ i β i2 +2 ω

130 131

(52)

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where

5

2.3. Friction model

67

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

68

P

P

v rii = Φ i i x˙ i

From Eqs. (47), (49) and (51), the recursive relation of configuration acceleration between the two bodies can be written as

v˙ i = T i j v˙ j + U i y¨ i + β i ,

23 24 25 26

j = L (i ), i = 1, . . . , N

(54)

where β i is

 T T



β i = β Ti1 β Ti2 0

∈ (6+si )×1

(55)

As deduced above, we obtain the kinematical recursive relations of configuration velocity and configuration acceleration of a rigid body and a flexible body. Similarly, the kinematical recursive relations of two adjacent rigid bodies can be obtained. For this case, y k = qk , k = i , j, the matrices T i j , U i and β i can be expressed as

 T ij =



21 22

(53)

Ui =

I3 0 H hT i



β i = β Ti1

−ρ˜ j i − h˜ i + ρ˜ i Q

Pi

I3 P T + ˜ i i HΩ i ∈ T HΩ i T β Ti2 ∈ 6×1

ρ





∈ 6×6

(56)

6×δi

(57) (58)

where

27 28 29 30 31 32 33 34 35 36 37 38 39

˜ jω ˜ iω ˜ j ρ Qj − ω ˜ i ρ iP + 2ω ˜ j v ri + ρ˜ iP β i2 , β i1 = ω ˜ j ωri + η i β i2 = ω

Below we deduce the kinematical recursive relation of the system. For a multibody system with N bodies, the relation can be expressed as

⎧ ⎪ vi = G ik y˙ k ⎪ ⎪ ⎨ k: B k ≤ B i ⎪ ⎪ v˙ i = G ik y¨ k + g ik , ⎪ ⎩

42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

(i = 1, . . . , N and k = 0)

(60)

k: B k ≤ B i

40 41

(59)

where

⎧ ⎨ T i j G jk when B k < B i G ik = U i when B k = B i , (i , k = 1, . . . , N ) ⎩ 0 when B k =< B i , ⎧ T g when Bk < B i ⎨ i j jk g ik = β i when B k = B i , (i , k = 1, . . . , N ) ⎩ 0 when B k =< B i ,

(61a)

(61b)

In Eq. (61), B k represents the k-th body in the system, “B k < B i ” means that the body k is on the way from the root body to the body i, “B k = B i ” the body k is the body i and “B k =< B i ” the body k is not on the way from the root body to the body i. Writing Eq. (60) in a matrix form, we have



v = G y˙

(62)

v˙ = G y¨ + g Iˆ N

where v = [ v T1 , . . . , v TN ]T is the total vector of configuration veloc-

ity of the system and Iˆ N ∈  N ×1 is a N-dimensional vector with all elements being 1. The parameters G and g in Eq. (62) are

⎡

G 11

⎢ . G = ⎣ ..

G N1

⎤

. . . G 1N .. ⎥ .. . . ⎦, . . . G NN

⎡

g 11

⎢ . g = ⎣ ..

g N1

... .. .

g 1N

...

g NN

⎤

.. ⎥ . ⎦

(63)

Space robot is a multibody system, consisting of several components, which are connected by revolute joint. Friction is inevitable in the joints. Because of the exposure to the high or low temperature in the space, the thermal expansion and contraction phenomenon of the joint materials could aggravate the friction in joints, further affecting the motion of manipulators and the accuracy of positioning. Existed space experiments have shown that the joint friction of space robot may increase by 20%–50%, compared with that on ground [5,6]. So the joint friction of space robot deserves indepth investigation. Previously, in order to describe the joint friction of robots, most researchers adopt classical inconsistent friction models, among which Coulomb friction model is most widely used. It is the first friction model proposed by researchers and is easy to use because of its simple form. Along with the development of technology, laser measurement and high-speed photography make it possible for people to further investigate friction intensively. It is shown in previous research that the friction between two bodies has the nonlinear characteristics of pre-sliding, hysteresis, and stick–slip. But these nonlinear characteristics could not be described by classical Coulomb friction model. So further and indepth studies on friction are carried out by researchers, who propose new friction models, such as Dahl model [19], Bristle model [20], LuGre model [21], Leuven model [22] and Generalized Maxell-slip model [23], based on experiment results. Among those models, LuGre model is a consistent friction model, which could effectively capture those friction characteristics above, and it has been widely used in friction modeling and friction compensation [24–27]. The space robot system in this paper is a hybrid rigid-flexible multibody system. Its dynamics function has strong coupling and nonlinear features. If inconsistent friction model is used to describe the joint friction, the dynamics model will be non-smooth, bringing great difficulty in numerical simulation. LuGre friction model is established by De Wit et al in 1995, based on Dahl model and bristle model [21], which can capture the nonlinear behaviors of friction effectually. It is the expansion of Dahl model, and is inspired by Bristle model at the same time. Bristle model describes the arbitrary behaviors of friction, while LuGre model describes the average behaviors of bristle model. Bristle is a virtual object with stiffness and damping, which is introduced by researchers to describe the friction. Comparing with Dahl model and Bristle model, LuGre model can describe the Stribeck phenomenon. Besides, LuGre model have a simpler form to model friction than other advanced friction model, such as Generalized Maxell-slip model, Leuven model and so on. Since the proposed of the LuGre model, this model is widely adopted to study the joint friction of mechanical system with good results [24–27]. So we adopt this friction model to describe the joint friction of space robot. In LuGre model, the average deflection of bristle is [21]

70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118

τ0 d˙ = v d − | v d |d s( v d )

(64)

where v d is the relative velocity between the two surfaces in contact, τ0 is the stiffness of the bristles, and the function s( v d ) models the Stribeck effect (the decrease of the friction amount as velocity increases within the low-velocity), which can be expressed as

s( v d ) = F f c + ( F f s − F f c )e

69

−( v d / v s )

(65)

where F f c is the Coulomb friction, F f s is the static friction and v s is the Stribeck velocity. F f c and F f s can be calculated by

119 120 121 122 123 124 125 126 127 128 129 130 131

F f c = μd F N ,

F f s = μs F N

(66)

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6

The equivalent pressure of rotational joint by the constraint moments is

1 2



3 4

N3 =

5

8 9 10

N

12

15 16 17 18 19 20 21

where μd and μs are the coefficients of Coulomb friction and static friction, respectively, F N is the normal load. The LuGre friction force can be expressed as [21]

F = τ0 d + τ1 d˙ f

where τ1 is the damping. Substituting Eq. (64) into Eq. (67), we can get

22 23 24

F = τ0 d 1 −

d ss =

29 30 31 32

s( v d )

 | v d | + τ1 v d

(68)

vd

|vd|

s( v d ) = sign( v d )s( v d )/τ0

= τ0 dss = sign( v d )s( v d )

37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52

55 56

The joint of the robot manipulator considered in this paper is a revolute joint that has only one rotational degree of freedom. The geometric model of revolute joint considered in this paper is shown in Fig. 4, where F x , F y , F z , T y and T z are the ideal constraint forces and ideal constraint moments in the three axis directions of the joint coordinate frame O r – X r Y r Z r , respectively. The coordinate vector of ideal force and joint friction force can be expressed as T

= [ F x , F y , F z , 0, T y , T z ] ,

f Fr

T

= [0, 0, 0, T f , 0, 0]

(71)

For the rotational joint considered in this paper, the friction moment may produce when the two connected bodies move relatively. The calculation of this friction moment depends on the acquirement of normal pressure of the joint. The normal pressure is produced by the actions of the constraint forces and constraint moments of the joint, and its algorithm is given below. The equivalent pressure by the axial constraint force is

N1 = | F x|

58

(72)

The equivalent pressure by the transverse constraint force is

57

N2 =



ˆ ˆ

−(θ /θs ) f c )e

T f = τ0 d 1 −

+

N T f s3

(78)

61 62 63

F 2y + F z2

(73)

78 79 80 81 82

(79)



85 86 87 88 89 90 91

τ1 ˆ |θ| + τ1 θˆ

(80)

ˆ s(θ)

92 93



τ1 ˆ ˆ 0, 0 = 0, 0, 0, τ0 d 1 − |θ| + τ1 θ, s(θˆ )

94

T

95

(81)

96 97 98 99

It is well known that the joint friction is a function of ideal constraint force of the joint. From the Lagrange theory, we know that the ideal constraint force can be expressed by the Jacobi matrix and the Lagrange multipliers. So we can use the Lagrange multipliers to express the joint friction. However the introduction of Lagrange multipliers may lead to the increase of the dimension and unknown variables of the system dynamic equations, which will increase the difficulty of calculating. From the Newton–Euler theory, we know that the ideal constraint force can also be expressed by the Cartesian coordinates of system. Considering that the Cartesian coordinates of a system can be expressed by the generalized coordinates of the system, the joint friction force may be written as a function of the generalized coordinates. Below the relationship between the ideal constraint force and the generalized coordinates is firstly investigated. Then the virtual power made by the frictional force is deduced, thus the contribution of frictional force to dynamic equation of the system can be obtained, which will be used in the building of the system dynamic equation of in Section 2.6. Fig. 5 is a space robot system topology map, and the joint n limits the relative movement of the bodies B n−1 and B n that is at the end of the space robot. If B n is a flexible body, the centroid dynamic equation of it based on the Newton–Euler principle and Eq. (33) can be written as

101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125

So the Coulomb frictional moment and the Stiction frictional moment equivalent from the constraint forces are

f oj F nc + F n = F na + F nω − F no + F nu − F n

(82)

f

N T f c1

= μd N 1 R n ,

N T f s1

= μs N 1 R n

(74)

= μd N 2 R p ,

N T f s2

= μs N 2 R p

(75)

65

N T f c2

66

where R n is the friction arm and R p is the pin radius.

64

77

84 2

59 60

75

83

Substituting Eq. (80) into (71), we can have



74

100

53 54

+

N T f s2

2.5. Virtual power of frictional force of joint

2.4. Revolute joint friction

F rc

T fs =

N T f s1

where θˆ is the angular velocity of the joint and θˆs is the Stribeck velocity of the joint friction. From Eqs. (79) and (68), the rotational joint friction can be written as

(70)

35 36

+

N T f c3 ,

s(θˆ ) = T f c + ( T f s − T

f Fr

33 34

(77)

Substituting Eq. (78) into Eq. (65), we can have

(69)

and the friction force is f F ss

+

N T f c2

When v d is a constant, the deflection d approaches the value

26 28

τ1

f

25 27

(67)

T fc =

N T f c1

73

76

N

T f s3 = μs N 3 R p

From Eqs. (74), (75) and (77), the total Coulomb and Stiction frictional moments of the joint can be written as

Fig. 4. Revolute joint model.

13

71 72

T f c3 = μd N 3 R p ,

11

14

70

(76)

where R b is the bending reaction arm. The Coulomb frictional moment and the Stiction frictional moment equivalent from the constraint moments are

7

68 69

T 2y + T z2 Rb

6

67

where F nc and F n are the ideal constraint force and the friction force of the joint n, respectively; F na and F nω denote the acceleration-dependent and velocity-dependent inertia forces of the n-th body, respectively; F nu is the elastic force of the n-th body; oj

F n is the resultant constraint force of outboard joints; and F no is

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Substituting Eq. (90) into Eq. (86), one can obtain

1



2

f

P

4

and F n

fP Fn n

6 8 9

 fPn

 c Pn

10

+ Fn

Fn

11 P

13 14

Fig. 5. The topology structure of the space robot system.

15 16

19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

the external force. If the n-th body is a rigid body, the elastic force F nu will be a zero vector. cP

Define F n n being the ideal constraint force on the point P n , cQ and F n n being the ideal constraint force on the point Q n , both can be expressed as

T

= F ncTP n M ncTP n ∈ 6×1 , T

cQ F n n = F ncTQ n M ncTQ n ∈ 6×1

(83)

 c in the absolute reference frame as shown in Fig. 5, respecM n Pn

tively; F nc Q n and M nc Q n are the coordinate vector from of Fnc Q n and

 c , respectively. The relation between F n M nQn

c Pn

cQn

and F n

40 41

− F nc Q n

(84)

fP

fQ

42 43 44 45 46 47 48 49 50 51 52 53

fQ n

F n n = −F n

(85)

Based on the Jourdain’s velocity variation principle, the relaf

c Pn

tionship of F nc , F n , F n

 T

56

PnT

P nP n = v n F nc + F n = v n

c Pn

Fn

fPn 

+ Fn

ρ

ω

ω

ω

(87)

+ Ψ nP n x˙ n

(88)

P

Pn ˙n n + Φn x

58

In the matrix form, Eqs. (88) and (89) can be written as

ω

63 64 65 66

P T

v nP n = B n n v n

(89)

(90)

cP

v nP n = P Bn n

=

P T rn n



I3 0

P n T T ∈ 6×1 , n  Pn − ˜ n Φ nP n T ∈ (6+sn )×6 I3 Ψ nP n

cP

cP



fP Tn n

80 81

84 85 86

in

fP

fP Tn n

79

83

 c Pn

cP

78

82

(94)

cP

76 77

cP cP c P T F nz n , 0, T ny n , T nz n ,

can be expressed by y¨ , y˙

is a function of y¨ , y˙ , y and dn .

87 88 89 90 91 92

 c Pn 

= Θn ( y¨ , y˙ , y , dn ), = Θn F n T

 fPn F n = 0, 0, 0, Θn ( y¨ , y˙ , y , dn ), 0, 0

93 94

(95)

Similarly to Eq. (93), the dynamic equation of the body B n−1 can be established as  cP

F n−1n− +

95

=



P n − 1 P n − 1 +  a B n− F n −1 1 A n −1

oj  + F nω−1 − F no −1 + F nu−1 − F n−1 (96)

As shown in Fig. 5, the joint n is one and only external joint of oj B n−1 , so F n−1 can expressed as oj v nT−1 F n−1

=

 v nQ−n 1T F nc Q n

+

fQ  Fn n

(91)

(97)

Similarly to Eq. (90), we have



= 

B n−n 1 =

Q

rn−n 1

ω

(98)

101 102 103 104 105 106 107

112

 ∈

Qn n −1

113

6×1

−ρ˜ nQ−n 1 Φ nQ−n 1

0

Ψ nQ−n 1

I3

(99)

T

cQn

∈ (6+δn−1 )×6

(100)

fQ n 

+ Fn

116 117 118 119 120

(101)

Substituting Eq. (101) into Eq. (96), we can obtain the dynamic equation of the body B n−1 about the joint point P n−1 in the frame h en−1 . Similarly to Eq. (95), we have



114 115

Substituting Eq. (98) into Eq. (97), we have



110 111

I3

Q

100

109

Q T

Q v n−n 1

98

108

v n−n 1 = B n−n 1 v n−1 where

97 99

 fP F n−n1−1

 c Pn 

= Θn−1 F n−1 = Θn−1 ( y¨ , y˙ , y , dn−1 ), T

 fPn−1 F n−1 = 0, 0, 0, Θn ( y¨ , y˙ , y , dn−1 ), 0, 0

ω

ρ

cP F ny n ,

and y, it can be known that We can set

fP 1 T n−n− 1

where



cP F nx n ,

F n−1 = B n−n 1 F n

59

62

75

(93)

where F nx n , F ny n , F nz n , T ny n and T nz n are the items of F n

oj

Taking the derivative of r n n , one can obtain

˜ nP n

73 74

= T

fP = 0, 0, 0, T n n , 0, 0

Q

P P rn n = rn + n n Pn Pn n = n + ri

r˙ nP n = r˙ n + ρ˙ nP n = r˙ n + ρ

61

 fPn

Fn



Q

where denotes the sum of virtual power of the ideal constraint force and joint friction of the joint n, v n is the velocity coordinate vector of the mass center C n of the body B n before the P deformation in the absolute reference frame, v n n is the velocity coordinate vector of the point P n in the absolute reference frame. From Fig. 5, we have

57

60

Fn

(86)

P nP n

54 55

fP

and F n n can be expressed as

f

72

96

is

Set F n n being the joint friction force on the point P n , and F n n cP being the joint friction force on the point Q n . The relation of F n n cQn and F n is

39

 c Pn

 c Pn

where F nc P n and M nc P n are the coordinate vector from of Fnc P n and

fP

with

P

Considering Eqs. (80)–(81) and F n

37 38

71

h en

+  a  oj  = B nP n AnP n F n + F nω − F no + F nu − F n

h

c Pn

=

and

three axes of the frame en , and T n n is the joint friction torque.

Fn

cP Fn n

c Pn

where ( B n n A n n )+ is the Moore–Penrose pseudo-inverse matrix of P P B n n An n . Equation (93) is the dynamic equation of the body B n h about the joint point P n in the en frame. Considering the joint is a revolute joint in this paper and using Eq. (71), we have

12

69 70

in frame and is the direction cosine matrix of respect to the absolute reference frame O –XYZ. Substituting Eq. (92) into Eq. (82), one can be obtained

7

68

(92)

are the coordinate matrices of F n

P An n

h en ,

67

 fP   fP  + F n n = B nP n AnP n F n c P n + F n n

 fPn

 c Pn

where F n

5

18

c Pn

F nc + F n = B n n F n

3

17

7

121 122 123 124 125 126 127 128

(102)

Similarly, we can establish the dynamic equations of B n−2 − B 1 in turns as following

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8

1 2 3 4

 fP

 cP

F n−2n−2 ( y¨ , y˙ , y ) + F n−n2−2 ( y¨ , y˙ , y , z )



.. .

6

 c P1

8 9 10 11 12 13 14 15 16

P

= B n−n−22 An−2

5 7

P n−2 + 

oj F na −2 + F nω−2 − F no −2 + F nu−2 − F n−2



 fP1

( y¨ , y˙ , y ) + F 1 ( y¨ , y˙ , y , z )  P 1 P 1 +  a oj  o u = B 1 A1 F1 + Fω 1 − F1 + F1 − F1

F1

(103)

where Q n −1 

oj

F n −2 = B n −2

.. . oj F1

=

fQ n−1 

cQ

F n−n1−1 + F n−1

Q  cQ B1 2 F 2 2

+

fQ  F2 2

(104)

And the joint friction of B n−2 − B 1 can be expressed as

17

where v i is the vector of generalized configuration velocity of the i-th body; M i v˙ i denotes the acceleration-dependent generalized inertia force acting on the i-th body; f ω i is velocity-dependent generalized inertia force acting on the i-th body; f oi is generalized external force of system acting on the i-th body; f ui is the generalized elastic force acting on the i-th body (if the i-th body is a rigid body, f ui is a zero vector); P is the sum of virtual power of the inner forces of system. Expressing P by the generalized variables ey ey ey y of the system, we have P = y T ( f e + f nc ), where f nc is the generalized force corresponding to the non-idealized constraint ey force, and f e is the generalized force corresponding to others the inner forces of system. For the space robot system considered in this paper, the active control torque acted on the joint is the geney eralized force f e and the joint frictional force is the non-idealized ey constraint force f nc . For the joint friction, from Eq. (110) we have ey ey ¨ ˙ f nc = f f ( y , y , y , d). In order to express Eq. (111) conveniently, we set

18 19 20

23

26 27 28 29 30 31 32 33 34 35

 fP F n−n2

T  fP = 0, 0, 0, Θn−2 ( y¨ , y˙ , y , dn−2 ), 0, 0 , . . . , F 1 1 T

= 0, 0, 0, Θ( y¨ , y˙ , y , d1 ), 0, 0

38

B 1 is the base of flexible space robot and it is a free-floating body, and there is no joint friction in the joint 1, so we have fP T1 1

41 42 43 44

 cPi

Fi

 fP i

+ Fi

where

P (Bi i



P

Bi i =

= [0 0 0 0 0 0]

(106)

=



P Bi i

P A i i )−1

− ˜ nP n

ρ

I3 0

I3

P − 1  a Ai i Fi

oj  o + Fω i − Fi − Fi

is the inverse matrix of

T

P Bi i

P Ai i ,

(107) and

P Bi i

∈ 6×6

is

(108)

Below we deduce the sum of virtual power of the joint friction, which be expressed as

n  T fQ fP  Q P v k−1 B k−k 1 F k k + v kT B k k F k k =

(109)

k =2

48

Considering Eq. (62), Eq. (109) may be repressed using the generalized variables y of the system as

51 53

P f = yT

56 57

n

ey f kf ( y¨ ,

y˙ , y , d) = y T f

ey ( y¨ , f

y˙ , y , d)

(110)

k =2

54 55

ey

Q

fQ k

where f kf ( y¨ , y˙ , y ) = G kT−1 B k−k 1 F k [ G k1 , . . . , G kn ].

P

fP

+ G kT B k k F k k ,

Gk =

58 59

2.6. Dynamic equation of space robot

60 61 62 63 64 65 66

70 71 72 73 74 75 76 77 78 79 80 81 82 83

Here we establish the dynamic equation of the system. Based on the Jourdain’s velocity variation principle and Eq. (33), the dynamic equation of the multibody system can be expressed as n i =1

85 86 87 88

v Ti (− M i v˙ i + f i ) + P = 0

(113)

91

In matrix form, Eq. (113) can be written as

92

v T (− M v˙ + f ) + P = 0

(114)

where v = [ v T1 , . . . , v nT ]T is the total vector of configuration velocity T 1, . . . ,

T T N]

of the system; M = diag[ M 1 , . . . , M N ] and f = [ f f are the matrices of generalized mass and generalized external force of the system, respectively. ey ey Using Eq. (62) and considering P = y T [ f e + f f ( y¨ , y˙ , y , d)] given in the above, Eq. (114) can be written as

 ey ey y˙ T − Z y¨ + z + f e + f f ( y¨ , y˙ , y , d) = 0

89 90

i =1

93 94 95 96 97 98 99 100 101

(115)

102

where Z = G MG, z = G ( f − M g Iˆ n ), d = [d1 , . . . , dn ]T is the vector of the average deflection of the bristles, y = [ y T1 , . . . , ynT ]T is

104

T

T

the vector of generalized coordinates of the system. Since all the elements of y are independent, Eq. (115) becomes ey ey − Z y¨ + z + f e + f f ( y¨ , y˙ , y , d) = 0

k =2

47

52

69

103 105 106 107 108

n  Q k T fQ k P T fP  v k −1 F k + v k k F k k P = f

46

50

F1

T

If the i-th body is a rigid body, the dynamic equation of B i h about the joint point P i in the e i frame can be written as

45

49

 fP1

= 0,

39 40

n

(105b)

(112)

So Eq. (111) becomes

(105a)



36 37

o u fi =−fω i + fi − fi

fP

24 25

68

84

fP

2 1 ¨ ˙ ¨ ˙ T n−n− 2 = Θn−2 ( y , y , y , dn−2 ), . . . , T 1 = Θ1 ( y , y , y , d1 )

21 22

67

  o u v Ti − M i v˙ i − f ω i + f i − f i + P = 0

(111)

(116)

110

Because of the introduction of d, the number of unknown variables is more than that of equations. So the additional formulas should be supplemented for solving this equation. Considering Eq. (64), the dynamic equation of the space robot system with joint friction can be written as ey

− Z y¨ + z + f e + f d˙ 1 = q˙ 1 −

τ0 s(q1 )

ey ( y¨ , f

111 112 113 114 115 116

y˙ , y , d) = 0

117 118

|˙q1 |d1

119 120

.. . d˙ n = q˙ n −

109

121

τ0 s(qn )

122

|˙q1 |dn

(117)

Because Eq. (117) is a nonlinear equations, nonlinear iterative matrix solution method, such as Newton–Raphson method, is needed to solve Eq. (117) to get y¨ and d˙ k (k = 1, . . . , n) firstly. And then the classical numerical method for ordinary differential equations could be used to solve the dynamic equation of system. For the space robot shown in Fig. 1, the dimension of the dynamic equations established by the method in this paper is 8 + nm , where nm is the number of the extracted modes of Link 2. The

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1 2 3 4 5 6 7

number of unknown variables is 8 + nm , too. The dimension of the dynamic equations by the conventional Lagrangian formulation or Newton–Euler method is greater or equal to 18 + nm . The number of unknown variables is also greater or equal to 18 + nm . From this we can see that the method given in this paper has the advantage of lower-dimension of dynamic equations and fewer unknown variables, so the computational efficiency can be improved.

9

67 68 69 70 71 72 73

Fig. 6. Mode shape of Link 2.

8 9

74

3. Active control

75

where

10 11

76

3.1. Trajectory planning

h( y˙ , y ) = Z −1 z

(123)

In order to complete the assigned missions, such as point to point assignment and continuous path assignment, the motion of joint need to be planned in joint-space. In this paper, the thirdorder polynomial trajectory planning method is used to design the motion of joint. Set the joint trajectory as

B = Z −1

(124)

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

θi (t ) = ai0 + ai1t + ai2t 2 + ai3t 3 (i = 1, . . . , jn )

where ai0 , ai1 , ai2 and ai3 are the multinomial coefficients, jn is the amount of joints, the initial and final conditions of θi (t ) and θ˙i (t ) are

θi (0) = θi0 ,

θ˙i (0) = 0,

θi (t f ) = θi f ,

θ˙i (t ) = ai1 + 2ai2t + 3ai3t

2

ai0 = θi0 ,

ai1 = 0,

(120)

ai2 =

35

ai3 =

36

u=B

−1



f +B

−1

h( y˙ , y )

(125)

f = y¨ d + K D e˙ + K P e

(126)

where K D and K P are the differential and proportional gain matrices, respectively, which are both positive-definite matrices. Substituting Eq. (125) into Eq. (122), we have

41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61

(θi f − θi0 ) 3

(121)

tf

From Eq. (127) we know that e will tend to zero when K D > 0 and K P > 0, thus y will approach to y d . 4. Numerical simulations

85 86 87 88 89

92 93 94 95 96 97 99 101 102 104 105

Using the method in Section 3.1, we can calculate the joint kinematics for a desired geometric path of a robot. Substitution of the joint kinematics in equations of motion provides the actuator commands. Applying the commands will move the robot on the desired path ideally. However, because of perturbations and nonmodeled phenomena, the robot will not follow the desired path. In order to minimize or remove the difference between the desired path and the actual path, we need to adopt a suitable control method. Space robot is a nonlinear dynamical system, and there is no general method for designing a nonlinear controller to be suitable for every robot in all of missions. In order to get a better control result, some classical control methods, such as PID control method, computed torque control method, model-referenced adaptive control method and transpose Jacobian control method are adopted to design the space robot controller [28]. In this paper, the computed torque control method is used to design the controller of joint-space tracking trajectory control. From Eq. (116), we get

y¨ = h( y˙ , y ) + Bu

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In this section, numerical simulations are carried out to illustrate the effect of the joint friction on the dynamic characteristics of the flexible space robot. The physical parameters of the flexible space robot shown in Fig. 2 are given in Table 1. The first two modes of Link 2 are considered in this paper and the modal functions of cantilever beam are used. The first two natural frequencies of Link 2 are 5.912 Hz and 5.912 Hz, respectively. The mode shapes are shown in Fig. 6, where the solid line represents the first-order mode shape in X 2 O 2 Y 2 plane and the dash line represents the second-order mode shape in X 2 O 2 Z 2 plane. The parameters of the revolute joint are R P = 0.03 m, R n = 0.05 m and R b = 0.05 m. The manipulator given in Fig. 2 has two degrees of freedom, denoted by Axis 1–Axis 2 respectively. The absolute reference frame O –XYZ is fixed on the origin O . The body-fixed frame O –XYZ, which parallels to the absolute reference frame, is fixed on the body Base, where the origin O is the mass center of Base. The fixed frame O 1 – X 1 Y 1 Z 1 and the floating frame O 2 – X 2 Y 2 Z 2 are fixed on Link 1 and Link 2, respectively, and the origins O 1 and O 2 are on the mass centers of the two links.

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Table 1 Physical parameters of the space robot.

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Mass (kg)

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I y y (kg m2 )

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9.822 × 106 25.75 471.82

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Substituting the initial and final conditions into Eqs. (118) and (120) yields

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4.1. Effect of joint friction on the motion of flexible manipulator In order to study the effect of joint friction on the motion of space flexible robot, a series of simulations are carried out. In these simulations, the space robot is controlled to track desired joint trajectories, in which the initial attitude angles of the Base and initial joint angles of the flexible manipulator are [0◦ , 0◦ , 0◦ , 0◦ , 0◦ ], the desired angles are [0◦ , 0◦ , 0◦ , 3◦ , 3◦ ], and the initial and desired angular velocity and acceleration are all zero. The desired trajectories could be designed by with the method in Section 3.1. For these simulations, the Coulomb friction and the LuGre friction are respectively used to describe the friction in all the joints of a space robot. The simulation results are then compared with the results of simulations without joint friction. For the two models above, the coefficients of kinetic friction are all chosen as μd = 0.05. For the LuGre model, the coefficient of static friction and the Stribeck velocity are chosen as μs = 0.1 and θˆs = 5.73◦ /s. For the LuGre model, the√ coefficients τ0 and τ1 are chosen as τ0 = 20 × 105 and τ1 = 20 × 105 , respectively. A 5000 kg payload is fixed at the tip of Link 2. In controller design, K P and K D are chosen as K P = diag[0 0 0 50 50 50 100 100 50 50] and K D = 0.1 × K P . The simulation results are given in Figs. 7, 8 and 9, where Figs. 7 and 8 show the time histories of the Base attitude and the joint angles of the manipulator in the process of trajectory tracking, and Fig. 9 shows the tip response of Link 2 in the floating frame O 2 – X 2 Y 2 Z 2 . It is observed from Figs. 7–9 that, for the simulations without friction, the desired angles of the Base attitude and two joints of the manipulator can be obtained using the controller given in this paper, and the flexible link’s vibration may be suppressed at the same time. But in the presence of joint friction, the attitude of the Base appears with fluctuations, trajectory tracking errors and stick–slip phenomena in the joints motion of the manipulator are obvious, and the vibration of Link 2 cannot be suppressed. Observing simulation results, we find that the obvious phenomena stated above behave in a similar pattern and pace. More specifically, these phenomena have obvious coupling effect. So it can

be indicated that joint friction will not only result in tracking error, but also bring about the vibration of the flexible space robot system. Comparing the simulation results of models, which adopt Coulomb friction and LuGre friction, we find that the result of LuGre model has more obvious stick–slip phenomenon. Thus we could prove that these two models behave differently in describing the stick–slip phenomenon. Considering the fact that stick–slip phenomenon may cause the vibration of the flexible manipulator, which could further affect the dynamic behaviors of the whole system, we need to adopt the advanced models, represented by LuGre model, rather than classical friction models, to describe the effect of joint friction on flexible space robot accurately. 4.2. Effect of joint friction on the operation of flexible space robot

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The results above indicate that joint friction could have significant effect on the dynamic characteristics of flexible space robot. Next we will carry out two sets of simulations to illustrate influences of the parameters τ0 and τ1 in the LuGre model on system dynamics are studied. In the simulations, all the parameters of the system are the same as those in the first set except for τ0 and τ1 . The simulation results are given in Figs. 10–15, where Figs. 10–12 are the results under different bristle stiffness τ0 (2 × 104 , 2 × 105 and 2 × 106 are chosen in this paper; τ1 = 1000 × 20.5 , μd = 0.05 and μs = 0.1), and Figs. 13–15 are the results under different bristle damping τ1 (10 × 20.5 , 100 × 20.5 and 1000 × 20.5 are chosen in this paper; τ0 = 2 × 106 , μd = 0.05 and μs = 0.1). From Figs. 10–12, it is observed that the trajectory errors of the Base attitude and the two manipulator axes and the vibration of Link 2 become larger along with the increase of τ0 , and the Stick–Slip phenomena of joint motions of the manipulator become more obvious too. From Eq. (67) we know that the friction force can be regarded as a spring-damper, where τ0 represents the spring stiffness and τ1 the damper damping. The parameter τ0 determines the value of the pre-sliding displacement. The relation of τ0 with the pre-sliding displacement is that τ0 increases at the cost of

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decrease of the pre-sliding displacement. So we know that the presliding phenomenon could have a significant effect on dynamic characteristics of the system. From Figs. 13–15, we can observe that different value of τ1 may lead to apparent difference of control result. The parameter τ1 can be regarded as the damper damping, different value of τ1 may result in different control result.

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5. Concluding remark

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In this paper, dynamics and control of a flexible space robot with joint friction are investigated. The LuGre friction model is adopted to describe the joint friction. The single direction recursive construction method and the Jourdain’s velocity variation principle are used to establish the dynamic model for the flexible space robot. The derivation of joint friction is presented in detail. A controller for the system trajectory tracking is designed by the computed torque control method. Simulation results indicate that the joint friction has big effect on system response, and it may cause the trajectory error of joint angle and the vibration of flexible link, as well as may affect the control performance.

Conflict of interest statement

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None declared.

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Acknowledgements

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This work is supported by the Natural Science Foundation of China (11132001, 11272202 and 11472171), the Key Scientific Project of Shanghai Municipal Education Commission (14ZZ021) and the Natural Science Foundation of Shanghai (14ZR1421000).

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