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Latin American applied research

versión On-line ISSN 1851-8796

Lat. Am. appl. res. vol.42 no.2 Bahía Blanca abr. 2012

 

Analysis of flexible mobile manipulators undergoing large deformation with stability consideration

M. H. Korayem, M. Haghpanahi and H. R. Heidari

Robotic Research Laboratory, Center of Excellence in Experimental Solid Mechanics and Dynamics,
School of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran.
hkorayem@iust.ac.ir

Abstract — Mobile manipulators operating in field environments will be required to manipulate large loads, and to perform such tasks on uneven terrain which may cause the system to reach dangerous tipover instability. Therefore, this paper presents a method for finding the Maximum Allowable Dynamic Load (MADL) of geometrically nonlinear flexible link mobile manipulators with stability consideration. Moment-Height Stability (MHS) criterion is used as an index for the system stability. The dynamic model for links in most mechanisms has often based on small deflection theory but for applications like light-weight links, high-precision or high speed, it is necessary to capture the deflection caused by nonlinear terms. Hence, the equations of motion are derived taking into account the nonlinear strain-displacement relationship. Then, a method for determination of the maximum allowable loads is described. In order to verify the effectiveness of the presented algorithm, several simulation studies and experiments are carried out and the results are discussed.

Keywords — Flexible Link; Nonholonomic Mobile Manipulator; Large Deflection; Stability Constraint.

I. INTRODUCTION

The research interest in flexible mobile manipulator, i.e., light and large dimension robotic manipulator, has increased significantly during the last few years. Flexible mobile manipulators have important application in space exploration, manufacturing automation, construction, undersea, nuclear contaminated environments, and many other areas. Major advantages of flexible mobile manipulator include, but not limited to, small mass, fast motion, and large force to mass ratio, which are reflected directly in the reduced energy consumption, increased productivity, and enhanced payload capacity. On the other hand, the effect of the manipulator and the payload movement, fast maneuver and moving on an uneven terrain may cause the base to turn over, particularly for flexible nonholonomic mobile manipulator when operating high speed with long arm. Therefore, the stability of the mobile manipulator is the most important constraints for finding the MADL or enhancement of the payload capacity on a given trajectory.

The MADL of a flexible mobile manipulator is defined as the maximum load that can be carried by the system with an acceptable precision for a given end effector path. The emphasis on the tracking accuracy is due to relaxation of the rigid body assumption and to the fact that one of the main reasons for the deviation from the desired trajectory is the flexibility in links. These systems are usually "power on board" with limited torque capacity. Hence, using light and small platforms and motor actuators will help a mobile manipulator to work in an energy-efficient manner. On the other side, smaller actuators reduce the torque capacity of the actuators and limit the capability of carrying heavy loads. Consequently, a more successful approach should maximize the load carrying capacity of the flexible mobile manipulator, subject to the constraints imposed by torque capacity of joint actuators and the motion accuracy and stability of the system on a given trajectory.

Many approaches have been taken to the development of mobile manipulators (Seraji, 1998; Yamamoto and Yun, 1994; Chen and Zalzala, 1997). The dynamic model for links in most of these approaches has often based on rigid or small deflection theory but for applications like light-weight links, high-precision elements or high speed, it is necessary to capture the deflection caused by nonlinear terms. Seraji (1998) presented a simple on-line approach for motion control of mobile manipulators using augmented Jacobian matrices. The approach used additional kinematic constraints to be met for the manipulator configuration, which can be equally applied to nonholonomic and holonomic mobile robots. Yamamoto and Yun (1994) focused their research on the modeling and compensation of the dynamic interaction between the manipulator and the mobile platform of a mobile manipulator and developed a coordination algorithm based on the concept of a preferred operating. Chen and Zalzala (1997) have offered an approach for the modeling and motion planning of a mobile manipulator system with a nonholonomic constraint. The Newton-Euler equations are used to obtain the complete dynamics of the system.

Bakr and Shabana (1986) expanded a formulation based on identifying the configuration of each elastic body in order to account for geometric nonlinearity and shear deformation in nonlinear modeling of flexible multibody systems. Simo and Vu-Quoc (1987) showed that for rotating structures, the appropriate accounting of the influence of centrifugal force on the bending stiffness and proposed a formulation based on the fully geometrically nonlinear beam theory. They demonstrated analytically the use of a first-order linear beam theory cannot account for the complete inertia effects to predict the influence of centrifugal stiffening. Shaker and Ghosal (2006) considered nonlinear modeling of planar flexible manipulators with one and two revolute joints using the nonlinear finite element mathematical models. The motion equations of the systems are derived taking into account the nonlinear strain-displacement relationship.

Several researches have tried to establish the maximum load-carrying capacity of mobile manipulators. Wang and Ravani (1988) studied and formulated the load carrying capacity of a manipulator on a given a dynamic robot trajectory. The formulation associated with the dynamic load carrying capacity (DLCC) is considered with both joint actuator torque and joint variable constraints throughout the trajectory. Korayem and Ghariblu (2003) developed an algorithm in order to find the DLCC of rigid mobile manipulators on a given trajectory. Korayem and Heidari (2007) computed the maximum allowable dynamic load (MADL) of a moving base flexible manipulator. The accuracy and torque constraints are taken into account for the proposed algorithm during motion for a given trajectory.

On the other point of view, some researchers have studied the stability of mobile manipulators. For instance, Arakawa and Fukuda (1997) presented a center of gravity control method to prevent a mobile manipulator from falling down, but the method discusses only the static stability. Ghasempoor and Sepehri (1998) extended the energy stability method by Messuri and Klein (1985) to quantitatively show the effects of top-heaviness and sloping ground. The extension can be used as an off-line tool to provide the designers with an inexpensive and fast method of maintaining the stability of mobile manipulators. Papadopoulos and Rey (1998) have proposed a new tip over stability measure, the Force-Angle stability measure, which is easily computed and sensitive to top-heaviness. The proposed measure has a simple geometric interpretation and applicable to dynamic systems subject to inertial loads and external forces. Li (2002) has used the normal forces between the rear/front tires and ground for checking the static and dynamic stability of mobile manipulators.

The other approach is using zero moment point (ZMP), which, for the first time, Huang and Sugano (1996) employed for tip over prevention of mobile manipulators. The valid stable region has been investigated for a mobile manipulator based on the ZMP criterion. But the inertia effect of rigid body has not been considered. Kim et al. (2002) have proposed an online compensation algorithm to include the mass moment of inertia the original formulation of ZMP for the dynamic stability of the mobile manipulators. It should be mentioned that in mobile manipulators, especially in manipulating heavy objects, the center-of-mass (C.M.) can be variable. But the ZMP criterion, in its basic format is insensitive to the system C.M. height. Moreover, the ZMP disregards the significant factor of mass moment of inertia of the mobile base and does not provide any specific indication about severity of the system instability at all (Goswami, 1999a, b). More recently, an efficient metric has been suggested by Moosavian and Alipour (2007) which is called as moment-height stability (MHS) measure. This measure is based on stabilizing and destabilizing moments exerted on the moving base which provides the system mobility. Also, the proposed metric is compared with other measures, such as Energy-Equilibrium Plan, Force-Angle, ZMP, and the efficiency of the MHS is superior over the others.

This paper presents a more accurate way of computing the "maximum load" of the nonholonomic mobile manipulator considering the flexibility and the dynamic stability. First, the equations of motion are derived taking into account the nonlinear strain-displacement relationship using finite element method approaches. The strain energy is expressed according to the slender beam theory and various non-linear terms are identified. Finite element method, which can consider the full nonlinear dynamic of mobile manipulator, is applied to derive the kinematic and dynamic equations. The additional constraint functions and the augmented Jacobian technique are used for motion planning and redundancy resolution. Then, a method for determination of the maximum allowable dynamic load for geometrically nonlinear elastic robot is described with attention to accuracy, actuator and overturning stability constraints on a given trajectory. MHS criterion is used as an index for the system stability. Finally, experiments on a three dimensional flexible link manipulator and numerical examples considering a flexible two-link mobile manipulator while traveling on a clothoid path and slope terrain are presented. The obtained simulation results demonstrate the accuracy and merits of the proposed method.

II. KINEMATIC AND DYNAMIC MODEL OF FLEXIBLE MOBILE MANIPULATOR

In this section, a flexible mobile manipulator comprising a manipulator arm mounted on a nonholonomic mobile base is considered which has two rear driving wheels and two castor wheels, as shown in Fig.1. The motion of the system has to be decomposed into the motion of the manipulator and the motion of the base. That is, the desired position of the manipulator's end effector can be defined as follows:

xe = xb(qb) + xm/b(qm), (1)


Fig. 1: Schematic view of a flexible mobile manipulator.

where xe and xb are the position of the end effector and the base in the inertial coordinate system. xm/b is the position vector of manipulator with respect to the base, also qb and qm are generalized variables of the base configuration space and arms space. The Jacobian equation of the mobile manipulator can be expressed as:

(2)

In the case of the wheeled mobile platforms, the rolling without slipping of the wheels on the ground introduces a specific non-integrable kinematic constraint in the modeling. The base, which cannot move instantly in any arbitrary direction, is then said to be nonholonomic. As a result the base must be moved in the direction of its main axis of symmetry. The compact form of the nonholonomic constraint equation can be written as:

(3)

where is the system's motion variable vector and Jc is the corresponding coefficient. Equations (2) and (3) are combined to obtain the differential kinematic model of the mobile manipulator system as

(4)

On the other hand, the manipulator/vehicle system has redundant degrees of freedom with the degree of redundancy r=n-m on its motion. n and m are respectively the dimensions of the generalized spaces associated to the mobility of the system and to the working Cartesian space. There are different types of constraints which can be applied to a robotic system to solve the redundancy resolution. However, as explained by Seraji (1998) the well-known method which uses r additional user-defined kinematic functions can be given with general form as follows:

xz = g(q), (5)

The additional task variables (5) can be expressed in the velocity form

(6)

where Jr is the Jacobian matrix associated with the kinematic functions . Therefore, in this study to resolve the redundancy, the user-defined additional constraints are considered as the base velocity. On combining the kinematic Eq. (4) and the user-defined additional task variables (6), can be obtained the integrated model of mobile manipulator as

(7)

where, the expression of Jij can be obtained using kinematic equation and θ0 denote the base angle. Note that the expression represents the velocity of the rear wheel in the orthogonal direction to the main axis; that is, along the y-axis in Fig. 1.

The Lagrangian method is utilized to formulate the dynamic equations of the mobile manipulator systems. The mobile manipulator system consists of a mobile platform and a multi-link manipulator. Each link of the manipulator composed of ni elements of length li. The finite element method, which can consider the full nonlinear dynamic of mobile manipulator, is applied to discrete the manipulator flexible links.

In order to derive dynamic equations, the kinetic energy and the potential energy are computed for the entire system. The kinetic energy for the overall system is obtained by computing the kinetic energy for each element ij and then summing over all the elements in terms of a selected system of n generalized variables q = (q1, q2, ..., qn) and their rate of change . Also, the potential energy of the manipulator is obtained by computing the strain energy for each element ij due to elasticity of any link. In this investigation, the non-linear strain - displacement relations are employed for the elasticity of link. For in-plane bending of beams, only the normal strain εxx needs to be considered, and the full nonlinear strain-displacement relationship for εxx (assuming a 2D problem) is given by

(8)

where the variables uy* and ux* denote the field-displacement variables defined over the entire domain.

After calculation these energies, by applying the Lagrangian procedure and performing some algebraic manipulations, the compact form of the governing equations of motion can be obtained (Korayem et al., 2010)

(9)

where [M] is the system mass matrix, [KL] is the conventional stiffness matrix, [KNL] is the geometrically nonlinear stiffness matrix. considers the contribution of other dynamic forces such as centrifugal, Coriolis and gravity forces while τ consists of generalized external forces / torques.

III. MADL FORMULATION FOR A GIVEN TRAJECTORY

The MADL of a flexible mobile manipulator is often defined as the maximum payload that can be carried by the mobile manipulator with an acceptable tracking accuracy. A number of factors limit a flexible mobile manipulator for computing the maximum allowable dynamic load during a given trajectory. The major limiting factors are; the joint actuator capacity, accuracy and overturning stability.

The actuators in a robot are key factors in determining its dynamic performance. Also, emphasis on the tracking accuracy is due to relaxation of the rigid body assumption and to the fact that one of the main reasons for the deviation from the desired trajectory is the flexibility in links. In this investigation, the actuator torque and the accuracy constraints are formulated on the computational procedure by Korayem et al. (2010). In order to determine the MADL, by considering the actuator and the accuracy constraints, it introduces the concept of the load coefficient for each point along the given dynamic trajectory.

A. Formulation of Stability Constraint

In order to successfully carry out tasks especially in carrying heavy loads on its trajectory for flexible mobile manipulators, the concepts of the stability degree and the valid stable region become very important to avoid tipping over. Therefore, estimation and evaluation of dynamic stability with an appropriate criterion throughout the motion of such systems is a fundamental necessity. In this investigation, a new tip-over stability measure named as Moment-Height Stability (MHS) measure will be used, which is given by Moosavian and Alipour (2007) for wheeled mobile manipulators. The MHS is defined as an amount of moment that causes the moving base to rotate around each edge of stability margin while considering all forces and torques exerted to the base body due to manipulator motion, gravitational forces, inertial force and external forces/ torques.

As shown in Fig. 2, the support boundary with four edges and the base frame x0, y0, z0 has been considered for computational aspects. The support boundary polygon (stability margin) is specified as the outermost of the base contact points with the ground. Then, to apply the MHS measure, the resultant moments of all forces and torques exerted to the base about different edges of support polygon are computed. These moments about edges 1, 2, ..., and n are named M1, M2, ..., Mn, respectively. The dynamic MHS measure, α, is defined as follows:

α = min(αi), i = {1, 2,..., n} (10)


Fig. 2: The support boundary and base frame.

where αi indicates the dynamic stability margin around the ith boundary edge and is calculated as

(11)

where is the unit vector for each edge of the support polygon and Ivi denotes the base moment of inertia around the ith edge of the rectangular boundary, and

(12)

When α is positive, the system is stable. On the contrary, negative values of α indicate that a tip-over instability is in progress. Also, critical tipover stability occurs when α is equal to zero.

For a mobile manipulator which is able to varying its C.M. height, or carrying a variable payload, the tip-over stability margin must be sensitive to the reduced stability associated with an increase in the C.M. height. The MHS measure in the above form is not sensitive to changes in system C.M. height. Therefore, the MHS measure α is improved by making it sensitive to changes in system C.M. height, thereby fully capturing the susceptibility to tip-over of the entire system as follows:

α = (hc.m.)λ. min(αi), i = {1, 2,..., n}, (13)

where hc.m. indicates the system C.M. height, and

(14)

In order to better condition the computational problem and to facilitate interpretation of the measure, the stability measure should be normalized with respect to its nominal value. In this study, the initial situation is considered as nominal configuration. The stability index α can be normalized as follows:

(15)

where represents the normalized dynamic stability margin and subscript "nom" refers to the corresponding nominal value. The algorithm of computational procedure for determining maximum allowable load is outlined and also shown in the flowchart in Fig. 3.


Fig. 3: Flowchart of determining the MADL.

IV. SIMULATION AND EXPERIMENTAL RESULTS

In this section, numerical and experimental results are presented to show the validity and effectiveness of the geometrically nonlinear flexible link and computed the MADL of a given trajectory with stability consideration. In order to initially check the validity of the dynamic equations, the response of the system, and compute the MADL of a given trajectory, several experiments on a three-dimensional flexible link manipulator have been carried out. Figure 4 shows the flexible manipulator used for the experiments. The experimental platform is constructed of two DC motors, θ and rotations, that drive a flexible link made of carbon fiber. This configuration allows a spherical motion for flexible link. The flexible link can be change with different lengths and diameters. The sensor systems consist of two incremental encoders for measuring the motors angle and a F/T sensor in order to estimate all six components of force and torque.


Fig. 4: Experimental setup.

An Optotrak motion-measurement system with three infrared cameras is used to capture x, y and z coordinate data at a sampling rate of 120 Hz and precision 0.3 mm. An aluminum sphere at the tip of the flexible carbon fiber link acts as the payload. The parameters of the flexible manipulator are given in Table 1. The system is initially at rest, thus the initial conditions are θ(0)=0, (0)=0. The final time is set to tf =0.5 s and the final conditions are θ( tf)=60°, ( tf) =60°.

Table 1. Physical properties of the flexible manipulator and payload

The permissible error bound for the end-effector motion around the desired path is Rp=400 mm. Both actuators of the robot are considered to be the same with τs=0.2N.m and ωs=90min-1. If only the joint actuator torque constraint was imposed, the maximum dynamic load carrying capacity for a given trajectory was found to be m=5.74×10-3kg. However, if the accuracy constraint were added to the above calculation, the MADL would reduce to m=4.21×10-3kg. Therefore, the maximum allowable dynamic load of the flexible manipulator by considering both constraints is calculated to be m=4.21×10-3kg. Figures 5-7 show the tip position of the flexible arm in x, y and z coordinates, respectively. The corresponding applied torques to the manipulator actuators within an upper and lower bounds of the available torques illustrate in Figs. 8 and 9 which depend on the joint velocities. The simulation results indicate that the absolute error of the tip will reduce by increasing the number of elements.


Fig. 5: The tip position in X direction.


Fig. 6: The tip position in Y direction.


Fig. 7: The tip position in Z direction.


Fig. 8: Actuator torque of the θ motor.


Fig. 9: Actuator torque of the motor.

Two additional simulations of the flexible mobile manipulator to illustrate the efficiency of the proposed model are performed. One is used to simulate the nonholonomic mobile manipulator with elastic links that the end-effector and its load track a predefined trajectory, as shown in Fig. 10. In this case, the mobile base of manipulator moves along a clothoid path. In the second one, MADL is found for the flexible mobile manipulator in which the mobile base moves along a straight line on sloped terrain. Parameters of the mobile manipulator are given in Table 2.


Fig. 10: The response of the system, with various Young's modulus E.

Table 2 Simulation of parameters

A. MADL of a Flexible Mobile Manipulator Moving on a Clothoid Path

This simulation study is carried out to illustrate the efficiency of the procedure presented in Fig. 3 for determining the maximum allowable load of a two-links manipulator. The parameters of a simulated two-link flexible mobile manipulator are shown in Table 2.

In order to initially check the validity of the dynamic equations, the response of the system, with various Young's modulus E, has been simulated. It can be seen that the obtained path of the end-effector, considering link flexibility reaches to the desired path with increasing E, as shown in Fig. 10.

The desired and the simulated trajectories of the end-effector and its payload within upper and lower bounds is shown in Fig. 11 which starts from the coordinate (x1=0.78, y1=0.35, z1=1.64 m) to the coordinate (x1=3.64, y1=0.11, z1=1.9 m) with zero initial and final velocity.


Fig. 11: The desired and the actual load path.

As shown in Fig. 12, a clothoid path is planned for the vehicle, which is very useful for smoothing the motion of a mobile robot moving along a trajectory. An arc of a clothoid has variable curvature, in every point proportional to the arc length, and it provides the smoothest connection between a straight line and a circular curve. The centrifugal force along a clothoid curve actually varies in proportion to the time, at a constant rate, from zero value (along the straight line) to the maximum value (along the curve) and back again. The parametric equations of the clothoid are given by the so-called "Fresnel Integrals":

(16)


Fig. 12: The planned clothoid path for the vehicle.

Now, the MADL of the mobile manipulator can be determined with respect to pervious section. The allowable error bound for the end-effector motion around the desired path is limited to Rp=12cm.. Both actuators of the robot are taken into account to be the same with Ts=385N.m and ωn1=10rad/s. If only the joint actuator torque constraint was imposed, the maximum dynamic load carrying capacity for a given trajectory was found to be m=2.52kg. Figures 13 and 14 show the corresponding applied torques to the manipulator actuators within an upper and lower bounds of the available torques. From these results, it can be seen that the loads so determined uses the joint 1 to its maximum extent while the bounds of joint 2 are not reached during the course.


Fig. 13: Actuator torque at the first joint against torque bounds.


Fig. 14: Actuator torque at the second joint.

However, if the accuracy constraint were added to the above calculations, the MADL would reduce to m=1.82kg. Therefore, the maximum allowable dynamic load of the flexible manipulator by considering both constraints is calculated to be m=1.82kg without considering stability. In order to be stable, α, must have a positive value for the mobile manipulator. Therefore, the MADL by imposing the stability criterion of platform must reduce to the safe maximum load to carry, which is 1.5 kg for the given trajectory. The magnitude of a positive α=278 describes magnitude of the tip-over stability margin of a stable system. Variation of the MHS measure during such a maneuver, between t=0 and t=1.8s, is shown in Fig. 15. Also, the variation of stability index for different payloads is illustrated in Fig. 16.


Fig. 15: The MHS criteria associated to the different edges.


Fig. 16: The variation of normalized stability index for different payloads.

B. MADL of a Flexible Mobile Manipulator Moving on a Slope

In most of industrial applications such as factory automation, a mobile manipulator has been used in a flat plane without slope. From a practical point of view, however, it is important to deal with various conditions of the floor surface to increase the reliability and the flexibility of the locomotion. When a mobile robot performs some special tasks, such as carrying a heavy load, moving on a slope or rough terrain, it may become unstable even overturn. In this case study, the computation of the MADL for the mobile manipulator in the motion on a slope (Fig. 17) is presented. The end-effector trajectory, considering link flexibility is compared with the desired path in Fig. 18. A linear path is planned for the vehicle, while it is on an inclined plane and the gradient angle of the slope is supposed to be 30 degrees. The link parameters and inertia properties of the mobile manipulator were given in Table 1.


Fig. 17: Schematic of a flexible mobile manipulator moving on a slope terrain.


Fig. 18: The desired path and the end-effector trajectory with link flexibility consideration.

In this case, the allowable error bound for the end-effector motion around the desired path is limited to Rp=12cm. Both actuators of the robot are taken into account to be the same with Ts=450N.m, ωn1=10rad/s.

If only the joint actuator torque was imposed, the maximum dynamic load carrying capacity for a given trajectory was computed to be m=4.52kg. However, if the accuracy constraint were added to the above calculations, the MADL would reduce to m=2.83kg. Therefore, the maximum allowable dynamic load of the flexible mobile manipulator with considering both constraints is calculated to be m=2.83kg in which the stability criterion is satisfied. Variation of the MHS measure during such a maneuver, between t=0 and t=1.8s, is shown in Fig. 19. Also, the variation of stability index for different slope angles is illustrated in Fig. 20.


Fig. 19: Variation of the MHS measures related to the different edges of support polygon.


Fig. 20: The variation of normalized stability index for different slope angles.

The results depicted the importance of the all constraints. According to the accuracy and tracking, they show which one would be the main. As shown in Figures 10 to 20, the main reason of the manipulator deviation is its major link's flexibility.

V. CONCLUSIONS

The main objective of this investigation was determining the maximum allowable dynamic load for geometrically nonlinear flexible-link mobile manipulators for a given trajectory using the finite element method. The effect of the dynamic interaction between the manipulator and the mobile platform is considered to characterize the motion of a nonholonomic mobile manipulator with compliant link capable of large deflection. The MADL can be obtained by a flexible mobile manipulator during a predefined trajectory are restricted by the number of factors. This was achieved by imposing actuator torque capacity, end effector accuracy and overturning stability constraints to the problem formulation. Designer should be paying attention to their priority constraints in order to present an ideal path manipulator. The proposed approach has been implemented and tested on a three-dimensional flexible link manipulator and the MADL was obtained. In the two case studies a two-link flexible mobile manipulator was considered for carrying a load on the clothoid path and slope terrain. The Numerical and experimental results obtained indicate the effect of introducing geometric elastic nonlinearities in the mathematical model whose neglect will affect the overall behavior of the robot. The results of the case study are shown by changing the vehicle motion for a pre-defined end effector trajectory or initial configuration of the mobile manipulator, the stability condition is assured and the motion accuracy constraint in large deformation dominates in comparison to the motor torque constraint for calculating the MADL. Therefore, the permissible error bound for the constraints in large deformation are sensitive to calculating the MADL. In addition, the formulation is more stable and efficient than most alternatives and has the added advantage that it can be calculated stability constraint. Moreover, the MHS stability criterion is sensitive to such variation of C.M. position and provides the specific indication of instability occurrence, especially in manipulating heavy objects.

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Received: June 9, 2010.
Accepted: May 18, 2011.
Recommended by Subject Editor Jorge Solsona.