A novel mathematical approach for finite element formulation of flexible robot dynamics

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A novel mathematical approach for finite element formulation of flexible robot dynamics

A. M. Chu , C. D. Nguyen , X. B. Duong , A. V. Nguyen , T. A. Nguyen , C. H. Le &

Michael Packianather

To cite this article: A. M. Chu , C. D. Nguyen , X. B. Duong , A. V. Nguyen , T. A. Nguyen , C. H. Le & Michael Packianather (2020): A novel mathematical approach for finite element

formulation of flexible robot dynamics, Mechanics Based Design of Structures and Machines, DOI:

10.1080/15397734.2020.1820874

To link to this article: https://doi.org/10.1080/15397734.2020.1820874

Published online: 18 Sep 2020.

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A novel mathematical approach for finite element formulation of flexible robot dynamics

A. M. Chu^a, C. D. Nguyen^b, X. B. Duong^c, A. V. Nguyen^d, T. A. Nguyen^e , C. H. Le^f, and Michael Packianather^g

aInstitute of Simulation Technology, Le Quy Don Technical University, Hanoi, Vietnam;^bLe Quy Don Technical University, Hanoi, Vietnam;^cCenter of Advanced Technology, Le Quy Don Technical University, Hanoi, Vietnam;^dDepartment of Special Robotics and Mechatronics, Le Quy Don Technical University, Hanoi, Vietnam;^eDepartment of Physics, Le Quy Don Technical University, Hanoi, Vietnam;^fFaculty of Engineering and Science, University of Greenwich, Kent, UK;^gSchool of Engineering, Cardiff University, Cardiff, UK

ABSTRACT

In conventional finite element–Lagrangian methods, the dynamics model of a flexible robot is usually formulated based on a critical assumption that the kinetic energy of an element is approximately calculated with an inte- gral of mass point energy. Since the energy integral is implicit, the formu- lation of the dynamics model is also very complex and implicit. Hence, this article develops a new mathematical approach for the dynamic modeling of a general flexible/rigid robot. The proposed method is more compre- hensive and efficient in comparison with the previous ones because it no longer requires the calculation of the symbolic integrals and the implicit expressions of the elemental and global mass matrices. Besides, the pro- posed approach is applicable for both the flexible robots and the hybrid flexible/rigid robots. To validate the proposed method, numerical simula- tions and experimental results are presented.

ARTICLE HISTORY Received 22 March 2020 Accepted 4 September 2020

KEYWORDS

Flexible robots; flexible/rigid robots; kinematics of machines; dynamics of machines; finite elements method

1. Introduction

In comparison with the conventional rigid robots or manipulators, the flexible ones have out- standing advantages such as lower overall mass, smaller actuators, lower energy consumptions, and a greater payload-to-manipulator-weight ratio. However, the dynamic modeling of the flex- ible robots is much more complex.

The finite elements method (FEM) has been widely used to model and analyze the dynamics of flexible robots (Usoro, Nadira, and Mahil1986; Naganathan and Soni 1987; Jonker 1990; Pan, Scott, and Ulsoy1990; Amirouche and Xie1993; Du, Lim, and Liew 1996; Al-Bedoor and Khulief 1997; Al-Bedoor and Almusallam 2000; Płosa and Wojciech 2000; Zakhariev 2000; Tokhi, Mohamed, and Shaheed 2001; Pan et al.2003; Wang and Mills 2006; Augustynek and Adamiec- Wojcik 2012; Mahto2014; Sayahkarajy, Mohamed, and Mohd Faudzi 2016; Karag€ulle et al.2017;

My et al.2019). As compared with the assumed modes method (Hewit et al.1997; Choi, Lee, and Thompson 1998; Hong and Park 1999; Hermle and Eberhard 2000; Subudhi and Morris 2002;

Mehrez and El-Badawy 2010; Raouf et al. 2017), the main advantage of FEM is that it is ideally suited for the dynamic modeling of the flexible robots which consist of multilinks and different joint types (Zakhariev2000; Pan et al. 2003; Zahariev 2006; My et al. 2019). In addition, FEM is

CONTACT A. M. Chu myca@lqdtu.edu.vn, mychuanh@yahoo.com Institute of Simulation Technology, Le Quy Don Technical University, 236 Hoang Qu^oc Vi^e_:t, Hanoi, Vietnam.

Communicated by Davide Spinello.

ß2020 Taylor & Francis Group, LLC

capable of handling nonlinear conditions and irregularities in the structure and mixed bound- ary conditions.

In the literature, to establish equations of motion for the flexible robots, FEM has been used in conjunction with different analytical approaches, such as (i) the Lagrange’s equations-based approach (Usoro, Nadira, and Mahil 1986; Pan, Scott, and Ulsoy 1990; Al-Bedoor and Khulief 1997; Hewit et al. 1997; Choi, Lee, and Thompson 1998; Hong and Park 1999; Hermle and Eberhard 2000; Tokhi, Mohamed, and Shaheed2001; Subudhi and Morris 2002; Mehrez and El- Badawy 2010; Raouf et al. 2017; My et al. 2019), (ii) the Newton–Euler’s equations-based approach (Naganathan and Soni 1987; Jonker 1990; Augustynek and Adamiec-Wojcik2012), and (iii) Kane’s equations-based approach (Amirouche and Xie1993).

In addition, there have also been some different finite element approaches that were developed for the dynamic modeling of flexible multibody systems, such as the floating frame of reference (FFR) formulation (Shabana 1997a; Shabana and Schwertassek 1998; Berzeri, Campanelli, and Shabana 2001; Shabana and Yakoub 2001; Garcıa-Vallejo, Sugiyama, and Shabana 2005; Nada et al. 2010), the absolute nodal coordinate formulation (ANCF) (Shabana 1996a, 1996b, 1997b), the co-rotational finite element method (CRFEM) (Elkaranshawy and Dokainish1995; Hsiao, Lin, and Lin 1999), and the rigid finite element method (RFEM) (Kermanian, Kamali, and Taghvaeipour 2019; Wittbrodt, Adamiec-Wojcik and Wojciech 2007). In FFR formulations, two sets of coordinate systems are usually employed. One set characterizes the location and orienta- tion of the bodies, while the other one describes the elastic deformation of the bodies (Shabana 1997a, 1997b; Shabana and Schwertassek 1998; Berzeri, Campanelli, and Shabana 2001; Shabana and Yakoub2001; Garcıa-Vallejo, Sugiyama, and Shabana 2005; Nada et al.2010). In ANCF for- mulations (Shabana1996a, 1996b, 1997b), since the position of each point on a flexible body is calculated with a global shape function defined in a fixed reference frame, the global mass matrix of the dynamic equation is a constant matrix. In CRFEM method (Elkaranshawy and Dokainish 1995; Hsiao, Lin, and Lin1999), a moving frame, the so-called the co-rotated frame, is attached to each element of a body in order to define the elastic deformation of the corresponding elem- ent. As for RFEM method (Kermanian, Kamali, and Taghvaeipour 2019; Wittbrodt, Adamiec- Wojcik and Wojciech 2007), each body is divided into rigid finite elements, and all the elements are connected each other via springs and dampers.

Alongside the aforementioned methods, there have also existed other noticeable contributions about the dynamics formulation of a flexible multibody system (Wu, Haug, and Kim 1989;

Wallrapp 1991, 1994; Mbono Samba and Pascal 2001; Shi, McPhee, and Heppler 2001; Shi and McPhee2002; Lugrıs et al.2007; JarzeRbowska, Augustynek, and Urbas2018; Grossi and Shabana 2019; JarzeRbowska, Urbas, and Augustynek 2020). In FEM approach, JarzeRbowska, Urbas, and Augustynek (2020) and JarzeRbowska, Augustynek, and Urbas (2018) formulated and analyzed the dynamics model of a robotic system with some flexible links. In these valuable works, the kine- matic constraints of the system were taken into account. In order to simplify the dynamic model- ing of a flexible multibody system, Shi and McPhee (2002) used Maple software to produce a symbolic programming algorithm. Also, a standardization and a linearization scheme for the dynamic modeling of flexible multibody systems were proposed in the works by Wallrapp (1991, 1994), and the variational approach was employed to analyze the dynamics model of a flexible multibody system (Wu, Haug, and Kim 1989). Focusing on some different numerical methods for flexible multibody dynamics, Shi, McPhee, and Heppler (2001) investigated polynomial shape functions. In particular, when considering some constrained systems, Grossi and Shabana (2019) studied the deformation basis and kinematic singularities of a system.

Although many studies have been devoted to the dynamic modeling of flexible robots, few articles have carried on increasing the accuracy and reducing the computational complexity for the formulation of the dynamics model. Many of the previous FE formulations of the flexible robot dynamics are usually based on an important assumption that each link of a robot is

assumed as an infinitely thin flexible rod. In most of the FE formulations, to formulate elemental mass matrices for constructing the global mass matrix of the dynamic equation, a symbolic inte- gration of the kinetic energy of a mass point along the length of an element is employed. In prac- tice, this computational scheme has some considerable disadvantages. In particular, the construction of the global mass matrix of the dynamic equation from all the elemental mass matrices suffers from a computational burden since several complex and implicit mathematical transformations are required.

In this article, a novel mathematical modeling method is developed and validated for applica- tion in formulating the dynamics model of the flexible/rigid robots. In the proposed approach, the motion of a flexible robot is decomposed into rigid and flexible motions. Both the flexible and rigid motions of each element of a link are taken into account when calculating the total kin- etic energy of an element. By using the Jacobian matrices and a global vector of the generalized velocities, all elemental mass matrices and the global mass matrix for a general flexible/rigid robot are explicitly calculated to determine the mass and all inertia moments of all elements. In this manner, the dynamic equation for a rigid robot can be obtained by eliminating all the elastic dis- placements in the dynamic equation of a flexible robot. Thus, the proposed method is applicable for not only the flexible robots but also the flexible/rigid robots which consist of links of a given geometry. Furthermore, since all the elemental mass matrices can be formulated without any symbolic integrals, the time complexity of the formulation is reduced by O(g), where g is the number of finite elements on all links. It was well demonstrated that the dynamic modeling scheme proposed in this study for a flexible robot is similar to the commonly used schemes of the dynamic modeling for the conventional rigid robots. Thus, the proposed method is very use- ful and helpful to users who have been familiar with the dynamic modeling and control of rigid robots. Therefore, the proposed method is more efficient, advantageous, and useful when com- pared with previous methods. Finally, to present validations of the newly developed dynamic modeling method, the numerical examples and experimental results are presented.

Figure 1. Linki1 and linkiof a flexible robot.

2. Equations of motion

This section presents a new mathematical approach to establish the dynamic equation of a flexible robot. In the proposed approach, the rigid and flexible motions of an element of a link are con- sidered when calculating the linear velocity, the angular velocity, and the kinetic energy of each element of a link. Besides, the global mass matrix of the dynamic equation can be effectively cal- culated and explicitly expressed in terms of Jacobian matrices and the global generalized velocities without any complex assembling procedures.

Let’s consider a planar flexible robot consisting of n links and n joints. Figure 1 presents a generalized schematic of an arbitrary couple of flexible links of the robot, namely link i1 and link i: In this figure, linki1 connects with link i by a jointi which can be a revolute joint, a sliding prismatic joint, or a fixed prismatic joint (Al-Bedoor and Almusallam 2000). Link i of length li is divided intoni elements of equal lengthlie: The vector of flexural displacements and the flexural slopes of an elementj isPij¼½uð Þ2ji 1 uð Þ2ji uð Þ2ji þ1 uð Þ2ji þ2^T:

Suppose that the homogeneous motion of an element j includes the two motions. The first motion is a rigid motion which implies that the motion of the elementjwill take place regardless of the elastic deformation of the link i: The second motion is a flexible motion relative to the rigid motion, which is caused by the elasticity effect of the linki:

Let’s define OiXiYiZi as the local coordinate system attached to the link i, where the origin Oi is fixed to the proximal end of the link i and the Xi points in the direction of the link i: Similarly, Oi1Xi1Yi1Zi1 is defined for the link i1: O0X0Y0Z0 is the reference coordinate system fixed to the base.

LetO^r_ijx^r_ijy^r_ijz^r_ijbe a local frame, of which the origin point O^r_ij is located at the center of mass of the rigid elementj, and the axis x^r_ijpoints in the direction of the rigid link i:Similarly, the frame O^f_ijx^f_ijy^f_ijz^f_ijis defined for the flexible element j: The superscript“r”stands for a rigid element, and

“f”stands for a corresponding flexible element.

In essence, the frame O^f_ijx^f_ijy^f_ijz^f_ij can be obtained by translating and rotating the frame O^r_ijx^r_ijy^r_ijz^r_ijat a flexural displacement wijðxÞalong y^r_ijaxis, and at a flexural slope w⁰_ijð Þx aroundz^r_ij axis, respectively. The transformation matrix can be expressed as follows:

H^fij¼

cosw⁰_ij sinw⁰_ij sinw⁰_ij cosw⁰_ij

0 0

0 wij

0 0

0 0

1 0

0 1

2 66 64

3 77

75¼ A^fij u^fij

0 1

" #

, (1)

where the flexural displacement wijðxÞ is calculated at the center of mass, x¼^lie

2, as wijð Þ ¼x P4

m¼1;mð Þx u_ðiÞð2j2þmÞ, where;m are the shape functions.

In the frameOiXiYiZi attached to a link i, a pointrijðxÞon an elementjof a link i is calcu- lated as follows:

rð Þiji ð Þ ¼x

j1 ð Þlieþx

wijðxÞ 0 1 2 66 4

3 77

5 (2)

In the precede frame O_i1X_i1Y_i1Z_i1, the point rð Þiji ð Þx is denoted asrði1Þijð Þx which can be determined with the following kinematic relationship:

rði1Þijð Þ ¼x Uði1Þirð Þiji ðxÞ (3) where Uði1Þi is the transformation matrix describing the kinematic relationship between a link i1 and a link i for a flexible robot, which was proven in our previous study (Al-Bedoor and Almusallam2000).

In order to determine the position and the orientation of a flexible elementj, the position and the orientation of the rigid element j should be calculated first. In O0X0Y0Z0, the point r^r0ij¼ O^r_ij, and the orientation ofO^r_ijx^r_ijy^r_ijz^r_ijis characterized by the following transformation matrix:

H^r0ij¼Yⁱ

n¼1

Uðn1ÞnH^rij¼ Aij r^r0ij

0 1

(4) where

H^rij¼ 1 0 0 0

0 1 0 0

0 0 1 0

j1 ð Þlieþx

wijðxÞ 0 1 2

66 4

3 77

5 (5)

Consequently, in O0X0Y0Z0, the position vector r^f0ij (the point O^f_0ij) can be determined as fol- lows:

r^f0ij¼r^r0ijþA0iju^fij (6) It is clearly seen that the center of mass of any element of a given link can be determined explicitly withEq. (6).

Let’s consider the dynamic equation of the flexible robot:

M qð Þ€qþC q,ð q_Þq_ þKqþG qð Þ ¼F (7) Equation (7) is written with respect to the global vector of the generalized coordinates, q¼ p1 j^T₁ p2 j^T₁ ::: ::: pn j^T_n

T

, where pi is the joint variable i, and ji¼ u_{ð Þ1i} u_{ð Þ2i} ::: u_{ð Þð2ni iþ1Þ} u_{ð Þð2ni iþ2Þ}

T

is the vector of the elastic displacements of all ele- ments of a link i: The applied forces/torques imposing on n joints of the robot are represented asF¼

s₁ 0 ::: 0

|fflfflffl{zfflfflffl}

2n1þ2

s_n 0 ::: 0

|fflfflffl{zfflfflffl}

2nnþ2

T

:

2.1.Formulation of the global mass matrixMðqÞ

Taking time derivative both sides ofEq. (6)yields the following:

_

r^f0ij¼r_^r0ijþA_0iju^fij þA0iju_^f0ij (8) Note thatu^fij¼A^T0iju^f0ij:Therefore,

v^f0ij¼v^r0ijþA_0ijA^T0iju^f0ijþA0iju_^fij

v^f0ij¼v^r0ijþx~^r0iju^f0ijþA0iju_^fij

v^f0ij¼v^r0iju~^f0ijx^r0ijþA0iju_^fij

(9)

wherev^f0ij is the velocity of the center of massO^f_ij of an element j:x^r0ij is the angular velocity of an elementjwith regardless to the elastic deformation of link i:

It is noticeable that

v^f0ij¼@r^r0ij

@q q,_ x^f0ij¼@x^r0ij

@q_ q, and_ u_^fij¼@u^fij

@q q,_

Hence, the velocity v^f0ij can be expressed in terms of Jacobian matrices and the generalized velocities as follows:

v^f0ij¼ ðJ^rij Tð Þ u^f0ijJ^rij Rð ÞþA0ijJ^fijÞ q_ (10) where the Jacobian matrices are determined as follows:

J^r_{ij Tð Þ}¼@r^r0ij

@q , J^rij Rð Þ¼@x^r0ij

@q_ and J^fij¼@u^fij

@q (11)

The matricesJ^r_{ij Tð Þ} andJ^r_{ij Rð Þ} are the translational and rotational Jacobian matrices of an elem- entjregardless of the elastic displacements of the elementj:Meanwhile, the Jacobian matrix J^fijis calculated with respect to the relative flexural displacement of the elementj:

Let’s denote

J^r_{ij Tð Þ}¼ u^f0ijJ^r_{ij Rð Þ}þA0ijJ^fij (12) J^r_{ij Tð Þ} plays a role of a translational Jacobian matrix with regard to the elastic deformation of the elementjof the linki:The absolute linear velocityv^f0ij can be rewritten as follows:

v^f0ij¼ ðJ^r_{ij Tð Þ}þJ^f_{ij Tð Þ}Þ_q (13) The absolute angular velocity x^f0ijof a flexible element jof a flexible link ican be determined as follows:

x^f0ij¼x^r0ijþx^fij (14) where

~

x^f0ij¼A_^fijðA^fijÞ^T (15) In terms of the Jacobian matrices and the generalized velocities q, the angular velocity_ x^f0ij can be rewritten as follows:

x^f0ij¼ ðJ^r_{ij Rð Þ}þJ^f_{ij Rð Þ}Þ q_ (16) where

J^f_{ij Rð Þ} ¼@x^fij

@q_ (17)

Assuming that the size of an element j is small, and link i is dived into a finite number of ele- ments, the kinetic energy of an elementjof flexible linkican be approximated as follows:

Tijffi1

2milie v^f0ij

T

v^f0ijþ1 2 x^f0ij

T

Iijx^f0ij (18)

SubstitutingEqs. (14)and(17)into Eq. (19)yield Tij¼1

2milie J^rijðTÞþJ^f_ijðTÞ _

h qiT

J^rijðTÞþJ^f_ijðTÞ

q_

h i

þ1

2hðJ^r_ijðRÞþJ^f_ijðRÞÞq_iT

Iij J^r_ijðRÞþJ^f_ijðRÞ _

h qi (19)

Let’s denote

JijðTÞ¼J^r_{ij Tð Þ}þJ^f_{ij Tð Þ}, (20) and

JijðRÞ¼J^rij Rð ÞþJ^f_{ij Rð Þ} (21) RewritingTij in terms of the elemental mass matrix gives

Tij¼1

2q_^TMijð Þq q_ (22)

Therefore, the elemental mass matrixMijcan be expressed as follows:

Mij¼milieJ^T_{ij Tð Þ}JijðTÞþJ^T_{ij Rð Þ}IijJijðRÞ (23)

The total kinetic energy ofnlinks of the robot is calculated as follows:

T¼Xn i¼1

X^nj

j¼1

Tij¼1

2q_^TM qð Þq_ (24)

Figure 2. The algorithm for symbolic formulation of the global mass matrix.

Finally, the global mass matrix is obtained as follows:

MðqÞ ¼Xⁿ

i¼1

X^nj

j¼1

MijðqÞ

¼Xⁿ

i¼1

X^nj

j¼1

milieJ^TijðTÞJijðTÞþJ^TijðRÞIijJijðRÞ

(25)

2.2. Remarks

It is clearly seen that the form of the global mass matrix of the dynamic equation of a flexible robot is similar to the form of the global mass matrix of the dynamic equation of a rigid robot.

The dynamic equation for the flexible robots is more generalized than the dynamic equation for the rigid robots. The dynamic equation of a rigid robot can be obtained by eliminating all the components J^f_{ij Tð Þ} and J^f_{ij Rð Þ} in Eqs. (20) and (21) that characterize the elasticity effects in the dynamic equation of a flexible robot. Therefore, Eq. (25) is a generic formulation of the global mass matrix of the dynamic equation which can be used for both the flexible robot and the rigid robot. Furthermore, this formulation is also applicable for the case when a part of some rigid links of a robot is flexible.

A flowchart of the algorithm for the symbolic formulation of the global mass M qð Þ is pre- sented inFigure 2.

For every element j of link i, the vector u^fij and the matrix A^fij are calculated with Eq. (1), and the position vectorr^r0ijðxÞand the rotation matrixA0ij are calculated withEq. (4). The angu- lar velocity x^r0ij is then determined via x^r0ij¼A_0ijA^T0ij: The Jacobian matrices J^r_{ij Tð Þ}, J^r_{ij Rð Þ}, and J^fij are calculated with Eq. (11), and the matrix J^f_ijðTÞ is calculated with Eq. (12). The relative angular velocity x^fij is calculated with Eq. (15), and the Jacobian J^f_{ij Rð Þ} is determined with Eq.

(17). The elemental mass matrixMij is calculated withEq. (23). For all elements on all links, the above steps are recursively looped to archive all Mij: Finally, the global matrix M is obtained withEq. (25).

Note that, to complete the formulation of the dynamic equation Eq. (7), the global stiffness matrix K and the Coriolis and centrifugal matrix C are formulated by using the method pre- sented in our previous work (My et al.2019).

It is also noticeable that the dynamic modeling method of flexible robots proposed in this sec- tion is different from the FFR formulations (Shabana 1997a; Shabana and Schwertassek 1998;

Berzeri, Campanelli, and Shabana2001; Shabana and Yakoub2001; Garcıa-Vallejo, Sugiyama, and Shabana2005; Nada et al. 2010). In FFR formulations, two sets of coordinates are usually used to describe the configuration of the deformable bodies (Shabana 1997a); the first set describes the position of the bodies, while the second set describes the deformation of the bodies with respect to their coordinate systems. As discussed in Shabana (1997a), one critical issue that arises when using FFR formulations is the definition of the sets of the coordinate systems on the bodies, since the shape of deformation of a body is defined in its coordinate system. Therefore, in the dynamic modeling approach proposed in this article, some following differences are made, in comparison with the FFR formulations (Shabana1997a; Shabana and Schwertassek 1998; Berzeri, Campanelli, and Shabana 2001; Shabana and Yakoub 2001; Garcıa-Vallejo, Sugiyama, and Shabana 2005;

Nada et al.2010):

i. One more set of element coordinate systems O^r_ijx^r_ijy^r_ijz^r_ij is defined. Moreover, all the link frames and element frames are defined in a way that the novel transformation matrix that

was proven in our previous work (My et al. 2019) can be used effectively to simplify the kinematic formulation of a multilink flexible robot with different joint types.

ii. Jacobian matrices are taken into account for the dynamic modeling of a flexible robot. In this manner, the form of the dynamic equation of a flexible robot is similar to the form of the dynamic equation of a rigid robot.

3. Numerical examples

In this section, to validate the dynamic modeling method proposed in this study, the behavior of the forward dynamics of two different flexible robots is analyzed and simulated. The first robot has one flexible link, and the second robot has two flexible links. In the examples, the numerical solution of the dynamic equation derived by using the proposed method is compared with the solutions of the dynamic equations established by using the previous method (Usoro, Nadira, and Mahil1986; Al-Bedoor and Almusallam2000; Tokhi, Mohamed, and Shaheed2001; Mahto 2014;

My et al.2019).

3.1. Example 1: Dynamics of a robot with one flexible link

In this example, as shown inFigure 3, the robot has two links, but the first link is rigid. Only the second link is flexible. The parameters of the robot are as follows. Link 1 has a length of 0.2 m and a mass of 2.0 kg. The length of link 2 is 1.0 m. The mass per meter of link 2 is 0.706 kg/m.

Link 2 has a cross-sectional area which is 910⁵ m², and it is divided into five elements.

When the force/torque (bang-bang rule) was applied as shown in Figures 4 and 5 for the inputs of the simulation, the two joints of the robot were displaced as shown in Figures 6and7.

Figure 8shows the flexural displacement at the end effector of the robot. These graphs show that the time evolution of the displacement curves computed by using the method proposed in this study having a close resemblance to the curves computed by using the previous methods. Note that the simulation curves of the two numerical solutions continue to match well even when the parameters of the robot are changed.

Figure 3. A flexible robot for numerical simulation.

3.2. Example 2: Dynamics of a robot with two flexible links

In order to demonstrate the proposed dynamic formulation with a more complex flexible robot, a two-link flexible robot with two rotational joints as shown inFigure 9is analyzed. The lengths of flexible link 1 and flexible link are 0.8 and 0.3 m, respectively. The cross-sectional areas of the two links are equal as 310⁵ m². Each flexible link is divided into two elements. The mass of the payload at the end effector is 50 g.

The applied torques at joint 1 and joint 2 are shown in Figures 10 and 11, respectively. The displacements of the joints are described in Figures 12 and 13. The flexural and slope displace- ment at the end effector is presented inFigures 14and15.

Figure 4. The input of applied force on the prismatic joint.

Figure 5. The input of applied torque on the revolution joint.

These figures show that, with a more complex flexible robot, the shape and the value of the displacement curves computed by using the method proposed in this study have a close resem- blance to the curves computed by using the previous methods.

4. Experimental results

In the previous section, the proposed method has been validated by simulation scenarios. In this section, the results of the experiments that were conducted for validation of the dynamic analysis of the flexible robot are presented. Figures 15 and 16 show the robot prototype and the system diagram of the robot used in the experiment. The two joints of the robot are actuated by a motor

Figure 6. The displacement of the prismatic joint.

Figure 7. The displacement of the revolute joint.

GB 37-3530 DC 12 V (1) and a NEMA 17 step motor 200 steps/rev 12 V (4), respectively. To col- lect the feedback signals, two rotary encoders LPD3806-600BM-G5-24V-2P-AB are used. The experimental setup also includes two flex sensors of 4.5-inch FSL0095-103-ST. A Terminal Board STM32 F407/417 with the Cortex^TM-M4 core running at 168 MHz was selected for the control hardware. The LabVIEW software interface was used to display the readings of the experiment.

Figure 8. The flexural displacement at the end effector.

Figure 9. A two-link flexible robot with two rotational joints.

The two flex sensors 112.24 mm in length were adhered toward both ends of the flexible link.

The maximum resistance of the sensors was 110 kX.

Figures 17and18show the bang-bang input signals (the applied force/torque) that were trans- formed into voltage and pulse signals for the motors.Figures 19and20 show the displacement of the joints produced by experiments and by simulation, respectively. It is observed that, in the steady state, the deviation between the experimental curve and the computational curve is very small (about 3.5% in magnitude).Figure 21presents the flexural displacement at the second node of the second element of link 2.Figure 22 shows the flexural displacement at the arm tip. It can be seen that the shape and the value of the simulation curves were calculated by both the

Figure 10. The input of applied torque on the rotational joint 1.

Figure 11. The input of applied torque on the rotational joint 2.

proposed method and the previous method closely matching the experimental readings. These results are a good validation of the dynamic modeling method proposed in this study.

6. Discussion and conclusion

The simulation and experimental results have demonstrated the effectiveness and efficiency of the dynamic modeling method newly proposed in this study. In particular, as compared with the pre- vious methods, the proposed method has two main advantages. The first advantage is that the dynamic equation of a flexible robot is derived in a simplified and explicit manner. In particular, the expression of the global mass matrices of the dynamic equation for a flexible robot and a rigid robot is similar. The second advantage of the method is that all the steps of the dynamic modeling procedure for a flexible robot are similar to the commonly used steps of the dynamic

Figure 12. The displacement of the rotational joint 1.

Figure 13. The displacement of the rotational joint 2.

Figure 14. The flexural displacement at the end effector.

Figure 15. The flexible robot prototype for the experiment: (1) DC motor, (2) Lead screw, (3) Rotary encoder 1, (4) Step motor, (5) Flex sensor, (6) Rotary encoder 2, (7) Flexible link.

Figure 16. The systematic diagram of the robot for the experiment.

modeling procedure for the conventional rigid robot. Thus, the proposed method is very useful and helpful to users who have been familiar with the dynamic modeling and control of the rigid robots. A detailed comparison of the proposed method with the previously used methods is pre- sented inTable 1.

It can be seen from Table 1 that, in the previous methods (Kermanian, Kamali, and Taghvaeipour 2019; My et al. 2019; Shabana 2020), each elemental mass matrix is usually com- puted and implicitly expressed in terms of nodal elastic deformations of a link because the sym- bolic integral of the element kinetic energy is employed. Not all the moment of inertia of an element and a link are included in the formulation. Moreover, the construction of the global mass matrix requires complex and implicit computational transformations, whereas, in the pro- posed method, the mass and all the moment of inertia of all elements are taken into account. In addition, because the symbolic formulation of the dynamics model is simplified, the calculation of the symbolic integrals and the implicit expressions of the elemental global mass matrices are not required. Furthermore, the global mass matrices for a flexible robot and a rigid robot are

Figure 17. The applied force for the experiments.

Figure 18. The applied torque for the experiments.

similar, which can be expressed in the same compact form ready for symbolic programming.

Besides, the proposed method is even more efficient for the case when the number of links of a flexible robot increases and a large number of finite elements on the links are considered.

Based onEq. (23), the mass matrix for a general flexible linki that includesnj elements is cal- culated as

Mi¼Xⁿⁱ

j¼1

ðmilieJ^Tij Tð ÞJijðTÞþJ^Tijð ÞRIijJijðRÞÞ (26)

Figure 19. The prismatic joint displacement.

Figure 20. The revolute joint displacement.

Figure 21. The flexural displacement at the second node of the second element of link 2.

Figure 22. The flexural displacement at the end effector.

Table 1. A comparison of the proposed method with the previously used methods.

Flexible robots Rigid robots

The previous methods(Kermanian, Kamali, and Taghvaeipour2019; My et al.2019;Shabana2020)

The proposed method The conventional methods

Mij¼Ð

Vij

@r0ij

@q

h iT @r0ij

@q

h i

dVij Mij¼milieJ^T_{ij Tð Þ}JijðTÞþJ^T_{ij Rð Þ}IijJijðRÞ Mij¼mliJ^T_{i Tð Þ}JjðTÞþJ^T_{i Rð Þ}IiJiðRÞ

mliandIiare the mass and the matrix of the moment of inertia of linki

Complex and implicit transformations are needed to formulateMijand constructM:

M¼Pn i¼1Pnj

j¼1Mij M¼Pn

i¼1Mi

In the case that a linkiis rigid, all the elements are rigid, and the componentsJ^f_{ij Tð Þ} andJ^f_{ij Rð Þ} that characterize the elasticity effects of linkivanish. SubstitutingJ^f_{ij Tð Þ}¼0 andJ^f_{ij Rð Þ}¼0 intoEqs.

(20)and(21), with respect to the constraint j_i¼u_{ð Þ1i} u_{ð Þ2i} ::: u_{ð Þð2ni iþ1Þ} u_{ð Þð2ni iþ2Þ} T

, Eq.

(26)can be transformed to obtain the mass matrix for a rigid linkias follows:

Mi¼mliJ^T_{i Tð Þ}JiðTÞþJ^T_{i Rð Þ}IiJiðRÞ (27) It shows clearly that Eq. (26) can be used for the mass matrix formulation for both a flexible and a rigid link. Therefore, by usingEqs. (25) and(26) the dynamic equation for a robot having multirigid/flexible links can be derived effectively.

One more feature of the dynamic modeling method proposed in this study is that the dynamic equation derived by using the algorithm in Figure 2 can be used further for control design of a flexible robot. Since the form of the dynamic equation of a flexible robot is similar with that of a rigid robot, the control design framework applying for a rigid robot control synthesis can be extended effectively for a flexible robot. Based on the dynamic modeling method presented in this article, the control methods and techniques that were reviewed in Lochan, Roy, and Subudhi (2016) and Sayahkarajy, Mohamed, and Mohd Faudzi (2016) can be employed to design a desir- able control law for a given flexible robot in practice.

Note that, in our previous work (My et al.2019), we proved a generic transformation matrix that characterizes the kinematics of a serial flexible robot constructed with three types of joints:

the revolute joint, the fixed prismatic joint, and the sliding prismatic joint. In addition, we addressed a new and effective algorithm of symbolic manipulation for constructing recursively the global mass and stiffness matrices of the dynamic equation for the flexible robots. In this pro- posed article, we continue developing a mathematical scheme to formulate more effectively the dynamic equation of a flexible robot without implicit mathematical transformations and expres- sions. As a result, the methodology proposed in this study has some the advantages over the pre- vious methods, as discussed earlier.

In conclusion, a new mathematical approach for the dynamic modeling of flexible robots and flexible/rigid robots has been presented. The dynamic modeling method proposed in this study has been validated through numerical simulations and experimental results. A close match between the simulation and the experimental result has been shown clearly. The proposed method is more effi- cient and advantageous in comparison with the previous ones. It has also been shown that the pro- posed method is more useful when dealing with not only the flexible robots but also the hybrid flexible/rigid robots that consist of rigid links and flexible links simultaneously.

To apply the proposed methodology to analyze the dynamics and design control laws for more complex flexible robots is the future works of this investigation.

Funding

This work was supported by the Vingroup Innovation Foundation (VINIF) annual research support program under Grant VINIF.2019.DA08.

ORCID

T. A. Nguyen http://orcid.org/0000-0002-5961-3050 Michael Packianather http://orcid.org/0000-0002-9436-8206

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