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Advances in Mechanical Engineering
Volume 2013 (2013), Article ID 216014, 15 pages
Recursive Formulation for Dynamic Modeling and Simulation of Multilink Spatial Flexible Robotic Manipulators
School of Sciences, Nanjing University of Science and Technology, Nanjing, Jiangsu 210094, China
Received 25 August 2013; Accepted 11 October 2013
Academic Editor: Xiaoting Rui
Copyright © 2013 Zhenjie Qian et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The dynamics for spatial manipulator arms consisting of n flexible links and n flexible joints is presented. All the transversal, longitudinal, and torsional deformation of flexible links are considered. Within the total longitudinal deformation, the nonlinear coupling term, also known as the longitudinal shortening caused by transversal deformation, also is considered here. Each flexible joint is modeled as a linearly elastic torsional spring, and the mass of joint is considered. Lagrange’s equations are adopted to derive the governing equations of motion of the system. The algorithmic procedure is based on recursive formulation using homogenous transformation matrices where all the kinematical expressions as well as the final equations of motion are suited for computation. A corresponding general-purpose C++ software package for dynamic simulation is developed. Several examples are simulated to illustrate the performance of the algorithm.
Flexible multibody dynamics has become a key methodology for various engineering fields, such as robotic manipulator arms, large radar antennas, solar panels, transportation vehicles, and manufacturing equipment as well as flexible ligament in human musculoskeletal system , and so forth.
In particular, the flexible robotic manipulators are typically precise and complex operated at high speed. They are designed to be light with low inertia in order to achieve cost-reduction, energy-saving, and high-performance. Therefore, the dynamic analysis of flexible manipulator arms is complicated due to the coupling between the large rigid body motion and deformation.
A large number of the literature related to the so-called dynamic stiffening effects have been published, which is first proposed by Kane et al.  due to link’s high-order coupling flexibility. It was observed that an industrial link under high-speed rotational motion would exhibit instability problems when its angular velocity exceeds a certain limit. The incorrect simulation solutions are attributed to neglecting the high-order coupling deformation terms in the dynamic equations. Wu and Haug  modeled a flexible multibody system by means of substructure synthesis formulation to account for structural geometric non-linear effects. Xi and Fenton  studied a manipulator consisting of one flexible link and one flexible joint based on the assumption that the link is constrained to move only in a horizontal plane, whereas the gravity and some coupling terms between the equations of motions have been dropped. It has been demonstrated by Wallrapp and Schwertassek , that the so-called “geometric stiffening” problem can be solved by keeping second order terms in the expression of the deformations of the material, and the amplitude of the flexible motions must remain small in those formalisms which are generally based on kinematic restrictions as regards the flexibility.
Low  presented vibration analysis of a rotating beam carrying a tip mass at its end by using Hamilton’s principle and the associated boundary conditions. Yoo et al.  used a non-Cartesian variable along with two Cartesian variables to describe the elastic deformation and investigate the dynamic stiffness effect. Ryu et al.  put forward a criterion on inclusion of stiffening effects such that it clarifies the limit of the validity of the linear modeling method. El-Absy and Shabana  introduced the effect of longitudinal deformation due to bending and studied the influence of geometric stiffness on instability problem of nonlinear elastic model. Al-Bedoor and Hamdan , based on the condition of inextensibility to relate the axial and transverse deformations of the material point, studied a rotating flexible arm deformation undergoing large planar motion. Hong et al. [11, 12] established higher-order rigid-flexible coupling dynamic model of rotating flexible beam undergoing large overall motion. When the deformation of flexible beam or the deformation rate is larger, the first-order or zero-order approximation coupling model will appear with divergent results, nevertheless the complete high-order coupling model’s results are still converged. Schwertassek et al.  presented the fundamental shape functions choice in floating frame of reference formulation, by separating the flexible body motion into a reference motion and deformation.
Book  used the 4 × 4 homogenous transformation matrices and assumed modes to describe the kinematics of the rigid-joint and flexible-link robots, whereas the dynamic model cannot deal with the torsional deformation of links. A recursive formulation for the spatial kinematic and dynamic analysis of open chain mechanical systems containing interconnected deformable bodies is given by Changizi and Shabana , and Kim and Haug . Jain and Rodriguez  developed new spatially recursive dynamics algorithms for flexible multibody systems by using spatial operators, which is based on Newton-Euler factorization and innovations factorization of the system mass matrix. Hwang  developed a recursive formulation for the flexible dynamic manufacturing analysis of open-loop robotic systems with the generalized Newton-Euler equations. Zhang and Zhou [19, 20] did a further work based on Book’s work in . Both the bending and torsional flexibility of links were taken into account. Dynamic simulation of a spatial flexible manipulator arm was given as an example to validate the algorithm. However, the dynamic stiffening effect was not considered yet.
When the system’s operating speed becomes high, neglecting the flexibility of joints is quite devastating, which will usually give rise to the errors of position precision. A typical dynamic model for taking into account the joint flexibility was presented by Spong . In his work, the joint flexibility is modeled as a torsional spring with more emphasis on simplifying the equations of motion for control purposes. Other dynamic analyses of robots with joint flexibilities can be seen in the work of Wasfy and Noor , Dwivedy and Eberhard , and Na and Kim .
As mentioned above, there are a lot of research work in the dynamic modeling and simulation. But it is still very difficult for us to deal with the dynamics of the complex multibody systems, such as the spatial flexible-link and flexible-joint robotic manipulators with consideration of the axial, bending, and torsional deformation for links, and the flexibility and mass effects for joints, and the so-called “dynamic stiffening” effects. In this paper, we will present the dynamic modeling methodology to include all such terms. In the following section, the kinematics of the system are presented, in which coordinate frames are established, and 4 × 4 homogeneous transformation matrices are used to describe the kinematics of flexible links and flexible joints. The approach of assumed modes is employed to describe the deformation of the flexible links. Section 3 firstly deals with the description of the kinetic and elastic potential energy as well as the gravitational potential energy of system, and then focuses on the derivation of the recursive rigid-flexible coupling dynamic equations of the system. In the modeling, the high-order coupling terms related to the non-linear geometry deformation are retained and the recursive strategy for kinematics is adopted. In Section 4, several examples are simulated to illustrate the performance of the algorithm proposed in the paper. Finally, Section 5 summarizes the results and draws conclusions from them.
2. Kinematics of Flexible Robots
As a point of departure, the system considered here is an assembly of n flexible links connected by n rotary joints, as shown in Figure 1.
2.1. Simplified Model of Flexible Joint
Figure 2 shows the flexible joint model. We can model the flexibility of joint as a linear torsional spring with stiffness is the moment of inertia of rotor about its spinning axis. And is the torque exerted at joint . For simplicity, we neglect friction or damping in the flexible joint. Let be the theoretical rotational angle of link , be the real rotational angle of link , be the torsional angle of joint , be the angular displacement of rotor , and be the gear ratio, respectively. The relationships among them are as follows:
2.2. Simplified Model of Flexible Link
Assume that the links are slender beams. Analysis here is based on the Euler-Bernoulli beam theory in the elastic small displacements field.
2.3. Coordinate Systems and Transformation Matrices
To express the transformation between different coordinate systems clearly, we establish four coordinate systems for link . Fix the coordinate system at the proximal end of link (oriented so that the coincides with the neutral axis of link in undeformed shape). This will be referred to as the base reference frame of link . Fix the coordinate system at the distal end of link . This is the distal frame of link . When link is in its undeformed state, the distal frame can be located by a pure translation of the base reference frame along the length of link . Let and be two Denavit-Hartenberg (D-H) frames fixed at the proximal end (at joint ) and the distal end (at joint ) of link , respectively. When joint is motionless, is coincident with , and matrix , that is, the transformation matrix between them, is the function of . Matrix , the 4 × 4 homogeneous transformation matrix between frames and , is a constant matrix. The transformation matrix of and is also a constant matrix. Define the joint-transformation matrix of joint to be the transformation matrix from to . Then, Obviously, is a function of . Define to be the link-transformation matrix of link from to . According to the assumption of small deformation of the links, the small deformable angles can be added vectorially. can be written as in which Here, , , and are the , and components of the elastic linear displacement mode of link at the origin of the coordinate , respectively. , , and are the , and rotation components of the elastic angular displacement mode of link at the origin of the coordinate , respectively. is the time-varying amplitude of mode of link , and is the number of modes used to describe the deformation of link .
Define or to be the 4 × 4 homogeneous transformation matrix from the base coordinate frame to . Then, we have where is the 4 × 4 homogeneous transformation matrix from the base coordinate frame to the distal coordinate system of link .
2.4. Velocity of a Point of Link
Let be the homogeneous coordinates in the system of a point of the deformed link at position with the link under an undeformed condition from the origin of . Then, can be approximated as Here, is the nonlinear strain coupling term, also known as the axial shortening due to the bending deformations of the link. When the flexible links are undergoing a high speed, this term will bring the so-called dynamic stiffening effect which will have a great influence on the dynamic behavior of flexible arms. In the dynamic modeling presented here, the high-order terms related to the non-linear coupling term are retained, which are ignored in the first-order and zero-order approximation coupling modeling.
In terms of the fixed inertial coordinates of the base , the position or of the point is given as Taking the time derivative of the position , we have the velocity of the point as To accelerate the computation of the matrices or , we use the recursive kinematics method. By differentiating (7), one obtains where Here, , , and is the joint variable of joint . Thus, and can be computed recursively from and its derivatives. Here, one additionally needs and its derivatives. These can be computed recursively from and its derivatives as follows: where
3. Dynamics of Flexible Robots
To use Lagrange’s equations, we need the kinetic and potential energy of the system.
3.1. The System Kinetic Energy
The system kinetic energy contains two parts: the links kinetic energy and the joints kinetic energy as follows:
Assume that the links are slender beams, so the rotary inertia and shear effects can be neglected. Therefore, the present analysis is based on the Euler-Bernoulli beam theory. Also assume that the links can undergo a large overall rigid motion, however the elastic displacements are small. The kinetic energy of the th link is where is the trace operator; and are the mass per unit length and the polar moment of inertia per unit length of the link about the neutral axis , respectively. For slender beams with uniform cross section area along axis, . The first term in (16) is the kinetic energy of link accounting for the rigid-body motion and the lateral and longitudinal deformation due to flexibility, whereas the second term is the kinetic energy accounting for the torsional deformation. Note that the torsional angle of link , , is expressed as where and are mentioned in (4) and (6), respectively. is the function of time, whereas is the function of position .
Substituting (10) and (17) into (16), expanding it, and then summing over all links, one finds the links’ kinetic energy to be where With the consideration of expressed in (8) and its derivative, consider the matrices , , and can be written as where in which
It should be noted that the link shape mentioned above is restricted to be the slender beam type. In fact, the link shape can further be extended to the other cases.
Case 1. Link is the rigid-body with irregular shape. In this case, , , and . The term is the equivalent of the inertia moment of the rigid-link. It is actually the pseu-matrix of the inertia moment in  with its more complex form compared with that of (23). Thus, for rigid-link, the link shape can be arbitrary, and one should only input the corresponding matrix .
Case 2. Link consists of a flexible beam with the concentrated mass at its proximal end. To account for the contribution of the concentrated mass to the kinetic energy of link , the extra term should be added to the matrix , where
Should the concentrated mass locate at the position near the proximal end, the extra matrix is modified to be
Case 3. Link consists of a flexible beam with the concentrated mass at its distal end. Considering the contribution of the concentrated mass to the kinetic energy of link , the extra terms , , , , , and should be added to the corresponding matrix , , , , , and , respectively. Here,
For the calculation of the kinetic energy of the joint , we can lump its mass to link in accordance with the assumptions made in  for simplicity. Thus, the kinetic energy needs to be included only in the part accounting for spinning kinetic energy of the rotor of joint as follows:
Considering the relationship in (2). We can obtain the kinetic energy of the joints as follows:
3.2. The System Potential Energy
The potential energy of the flexible manipulators mainly includes the elastic potential energy of flexible joints , the elastic potential energy of the flexible links , and their gravitational potential energy . Therefore, the whole system potential energy is
Firstly, the elastic potential energy of joint is
Secondly, the elastic potential of the flexible links considered here have three parts: one by bending about the transverse and axes, one by compressing about the longitudinal axis, and one by twisting about the longitudinal axis. Along an incremental length , the elastic potential energy is Here, , , and are the th link’s rotations of the neutral axis at the point in the , , and directions, respectively; is Young’s modulus of material of link ; is the shear modulus of material of link ; is the th link’s area of cross section about the axis; and are the area moment of inertia of th link’s cross section about the and axes, respectively; and is the polar area moment of inertia of the th link’s cross section about the neutral axis. Similar to the torsional angle of (17), and can be expressed as where and are mentioned in (6). By integrating of (33) over the link, and summing over all links, one can obtain as where
Finally, by using the same procedure given by Book , the gravity potential of the system is in which is the gravity vector with respect to the inertial base, and it has the following form: Here, is the total mass of link and is the homogenous coordinates of the gravity center of link (undeformed) in the frame , Note that and can be found in the top row of and , respectively.
3.3. Dynamic Equations of the System
We use the Lagrange method to derive the dynamics of the system and accord with the method of . Then, the form of Lagrange’s equations will be as follows. For the joint variable , the following simply joint variable equation: For the joint variable , the following simply joint variable equation: For the deformation variable , the following simply deformation variable equation:
Upon the substitution of the system kinetic energy of (15), the system elastic potential energy of (31), and the gravity energy of (37) into (40), (41), and (42), thus becoming as follows. For the joint variable , For the joint variables , For the deformation variables , Here,
Finally, by complicated derivation and assembling, and using the recursive scheme to reduce the number of calculation, the dynamic equations of the system are obtained in the following formulation: where the generalized coordinates column is defined aswhere is the inertia matrix of the system composed of coefficients of the generalized acceleration in the system dynamic equations. It is positive-definite and symmetric. The elements of are arranged in the order of the generalized coordinate column , the are diagonal elements, the are off-diagonal elements, and these elements are given as follows:
(I) In the joint variable equation , the inertia coefficient of the second derivative of the joint variable is where Here, is the 4 × 4 homogeneous transformation matrix from the proximal reference frame of link to the proximal reference frame of link . From (52), we have (symmetric).
(II) In the joint variable equation , the inertia coefficient of the second derivative of the joint variable (for ) is expressed as follows:
for , for , for , Here, where is the homogeneous transformation matrix from the distal coordinate system of link to the proximal coordinate system of link . Note that the inertia coefficient for the deformation variable in the joint equation is the same as that for the joint variable in the deformation equation , . This further extends the symmetry of the inertia matrix and reduces the computation necessary.
(III) In the joint variable equation , the inertia coefficients of the second derivative of the joint variable are expressed as follows:
for , for , for ,
(IV) In the deformation variable equation , the inertia coefficients of the second derivative of the deformation variable () are expressed as follows:
for , for , for , for , for , for , Here,
For the symmetry of the coefficients, we have .
(V) is generalized force, containing the remaining dynamics and external forcing terms except impact force as follows:
The is the remains of (43) with the second derivatives removed as follows:
The is the remains of (44) with the second derivatives removed as follows:
for , for ,
The is the remains of (45) with the second derivatives removed as follows.
For , for , Here, where the value of is the remainder of by only eliminating the terms involving and , and it can be calculated recursively by The value of can be obtained similarly as follows: where
As for , for , for , for , And for ,
As for , only if , calculate
As for , it can be obtained from easily as follows:
As for , only if and , calculate
4. Example of Dynamic Simulation
A general-purpose software package for dynamic simulation of flexible-links and flexible-joints robotic manipulators based on the said algorithm is written in C++. The data structure is pointers and linked lists, so that the data length can be changed based on different initial conditions of the system. The first step is setting up a given interface to input system parameters via reading files. Then, the basic module matrix can be generated automatically. With that, reducing a second order differential equation to a first order linear ordinary differential equation. Eventually, a process based on Adams predictor-corrector method for solving ordinary differential equation calculates till the end time, with an output file per step. It is universal for usage, computationally efficient, and numerically stable.
4.1. Flexible Single-Link Undergoing Low/High Rotational Speed
Ignoring the gravity and the flexible effect of the joint, here, consider the flexible single-link undergoing low/high rotational speed in order to discover the applicability of the high-order geometric nonlinearity coupling model. The properties of the flexible beam are the same as those in [3, 8], and they are given as follows. The length , the cross-section area , the area moment of inertia , the mass density , and the elasticity modulus . For the flexible cantilever beam without large overall motion, the fundamental frequency is 0.464 Hz. The large rotating motion law of the system adopted in [3, 8] is given by where , and is chosen here to be 0.6, 2, 3, 8, and 10 rad/s, respectively, in the following numerical simulations. The tip deformations of link are shown from Figures 3, 4, and 5.
It can be observed that, undergoing a low rotating speed (0.6 rad/s or 2 rad/s), there is an obvious difference between zero-order and high-order coupling model, and the difference becomes larger and larger with the increasing of the rotating speed. When the high-order coupling model is at the rotating speed 3 rad/s (≈0.477 Hz), closing to the fundamental frequency (0.464 Hz), the vibrating amplitude of the link tip is aroused much higher, but it is convergent. Whereas, the results form zero-order coupling model turn to be divergente. When reaching a high rotating speed, 8 rad/s (≈1.272 Hz) or 10 rad/s (≈1.59 Hz), the zero-order coupling model will fail to get the convergent results. These phenomena are coincided with the numerical result in [2, 11].
4.2. Spatial Flexible-Link and Flexible-Joint Robotic Arm
In order to validate the algorithm and package presented in this paper, the dynamic simulation of Canadarm2 (the Space Station remote manipulator system serving on the international space station) is given as an example.
As shown in Figure 6, Canadarm2 consists of four parts: shoulder, upper arm boom, lower arm boom, and wrist, in which the shoulder and the wrist both have 3 rotational degrees of freedom (DOF), and the elbow connecting the upper arm and the lower arm has 1 rotational DOF. Hence, the total system can be regarded as a 7-DOF manipulator with seven links and seven revolute joints. The links making up the shoulder and the wrist are short and thick in shape; therefore, they can be regarded as rigid links, whereas the upper and lower arms are assumed to be flexible due to their slender beam type. Therefore, Canadarm2 system is simplified to contain two flexible links and three rigid links which are connected through flexible joints, as shown in Figure 7. In this example, there is a distance d = 0.475 m between the 2nd link and 1st link along the Z-axis, which consequently arouse the rotation of the flexible links. And the longitudinal stretching (along X-axis), transversal bending (along Y-axis and Z-axis), and torsion (around X-axis) are considered here. Correspondingly, a total of 12 eigenfunctions (the first three preorientation of deformation) of a clamped-free beam are utilized to represent the mode shapes of each link deformation. The links parameters are shown in Table 1. The material of the links is aluminum, cross-section parameter , , , and . Initial joint angle is , , , and , respectively. Initial joint angle velocity is zero.
4.2.1. Dropping Motion in the Gravity Field
Figure 8 describes the motion of the high-order coupling flexible five-links system dropping in the gravity field.
4.2.2. Effects of Joint Flexibility on the System
Figure 9 shows the torsional angle of the 1st joint in the high-order coupling model with different mass , 5, 10, or 15, respectively. Figures 10, 11, 12, and 13 show the X, Y, and Z deformation , , , and torsional angle of 1st link tip with different joint mass. The larger joint’s mass is, the larger joint angles of motion the system has. The same phenomena can be seen form Figures 10 to 13.
This paper aims at building a comprehensive model of spatial manipulator arms consisting of n flexible links and n flexible joints. The deformations of stretching, bending, and torsion of the links are all considered. And the flexibility and the mass of the joints are also included. In the modeling, the so-called dynamic stiffening effects are considered via the adoption of nonlinear deformation description. Based on the presented method, a general-purpose software package of multilink manipulator arms is developed. The results demonstrate that dynamic stiffening effects cannot be neglected. The coupling effects of flexible links and flexible joints on dynamic characteristics are significant to the system’s accurate dynamic response predictions, performance evaluation, and implementation of procedure control.
This work is supported by the National Natural Science Foundations of China (11272155, 11132007, and 10772085), 333 Project of Jiangsu Province (BRA2011172), and the Fundamental Research Funds for Central Universities (30920130112009).
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