* Corresponding Author, Associate Professor. ** Graduate Student

Cubic-Spline Trajectory Planning of a Constrained Flexible Manipulator

Atef A. Ata* Habib Johar** Mechatronics Engineering Department Mechatronics Engineering Department

Faculty of Engineering, IIUM Faculty of Engineering, IIUM Jalan Gombak, Kuala Lumpur 53100, Malaysia Jalan Gombak, Kuala Lumpur 53100, Malaysia

atef@iiu.edu.my

Abstract- A cubic-splines trajectory planning of a constrained rigid-flexible manipulator to minimize the contact force is considered. Based on the general dynamic model of the chain of flexible links, equations of motion for the manipulator are derived using the extended Hamilton’s principle. An analytical solution for the inverse dynamics problem using the assumed modes method is presented to compute the required joint torques for the tip mass to move along the constrained surface. Cubic-splines interpolation has been adjusted for the joint motion profiles to make sure that the end-effector will follow the prescribed trajectory without degradation. The knot points of the trajectories have been chosen on the constrained surface to achieve minimum contact force as possible.

Index Terms- Rigid-flexible manipulator, Cubic-spline,

trajectory planning, constrained motion, analytical, assumed modes

I. INTRODUCTION

Many of today’s flexible robots are required to perform tasks that can be considered as constrained motions. A robot is considered to be in constrained maneuvering when its end-effector contacts and/or interacts with the environment as the robot moves. These applications include grinding, deburring, cutting, polishing, and the list continues. For the last two decades many researchers have investigated the constrained motion problem through different techniques such as hybrid position/force control [1 and 2], force control [3] , nonlinear modified Corless-Leitman controller [4], two-time scale force position control [5], multi-variable controller [6], parallel force and position control [7] , and multi-time-scale fuzzy logic controller [8]. Luca et al. [9] introduced rest-to-rest motion of a two-link robot with a flexible forearm. The basic idea is to design a set of two outputs with respect to which the system has no zero dynamics. Ata and Habib have investigated the open loop performance of a constrained flexible manipulator in contact with a rotary surface [10] and arbitrary shaped fixed surfaces [11]. They have simulated the effect of the contact force on the required joint torques for different surfaces based on prescribed joint profiles. The present study aims at designing cubic-spline end-effector trajectory to move so close to the contact surface and thus minimizing the contact force and the hub torques. The Cartesian path of the end-effector is to be converted into joint trajectories at intermediate points (knots). The trajectory segments between knots are then interpolated by cubic-splines to ensure the smoothness of the trajectory for each joint. To avoid separation between the end-effector

and the constraint surface and to minimize the contact surface as well, these knots are chosen at the contact surface itself. Simulation results have been carried out for circular and inclined contact surfaces. This paper contains six sections. Section II introduces the dynamic modeling of the rigid-flexible manipulator during constrained motion. Section III illustrates the analytical approach applied to obtain the elastic deflection of the flexible link by solving the inverse dynamics problem. In section IV a verification of the dynamic model has been carried out and section V presents the selection of Cubic-splines trajectory for the end-effector as well as the simulation results for the contact force and the required hub torques. Conclusions are presented at section six.

II. DYNAMIC MODELING

Consider the two-link planar manipulator shown in Figure (1). The first link is rigid while the second link is flexible and carries a tip mass m3. This tip mass always moves in contact with a constrained surface. The mass density of the flexible link is 2ρ and Euler-Bernoulli’s beam theory is employed to describe the flexural motion of the flexible link. The manipulator moves in the horizontal plane so that the effect of gravity is not considered. The flexible link is described by the Virtual Link Coordinates System (VLCS) in which the deformation is measured relative to the link that joins the end-points and the angle θ is measured with respect to the horizontal axis [12].

Figure (1) A two link rigid-flexible manipulator with tip mass

In general, the dynamic equation of motion for a rigid-flexible robot can be written as.

Proceedings of the 2005 IEEEInternational Conference on Robotics and AutomationBarcelona, Spain, April 2005

0-7803-8914-X/05/$20.00 ©2005 IEEE. 3126

( ) ( ) r,VM τ+θθ+θθ=τ &&& (1)

where θ is 1n × vector of joint positions; θ& is 1n × vector of joint velocities; θ&& is 1n × vector of joint accelerations;

( )θM is the nn× inertia matrix; ( )θθ &,V is the 1n × Coriolis and centripetal terms of the manipulator; τ is

the 1n × vector of joint torques, n is the number of degree of freedom Based on the dynamical model of an open chain multi-link flexible manipulator using the extended Hamilton’s principle [11 and 12], the elements of the inertia and Coriolis matrices of the rigid/flexible manipulator considered here can be given by :

)lmlmII(M 213

2121h111 +++=

dx)-sin(wl-)-cos()llmxlm(M 12

l

011221321212

2

θθρθθ+= ∫1221 MM =

∫ρ++++=2l

0

222332h222 dxw)lmIII(M

∫ θθθθθθρ+

θθθ+=2l

012212

2212

122221321211

dx)]-sin()w(2-)-cos()w-w[(l

)-sin()llmxlm(-v

&&&&&

&

∫ θθθ++θρ+

θθθ+=2l

012

21122

122121321221

dx)}-cos(wlwxww2-{

)-sin()llmxlm(v

&&&&&

&

where ∫ρ=2l

0222 xdxxm and nI and hnI are the inertia of the

links and actuators respectively.

rτ is the 1n × interaction joint torques due to the force exerted at the end-effector and is given by:

==τ

y

xTTr F

FJFJ

where TJ is the transpose Jacobian of the system.

θθ

θ+θθθ=

2222

22112211T

cosl sinl-coslcosl sinl-sinl-

J

xF and yF represent the applied forces by the end-effector

on the constrained surface and it includes two parts. The first component is due to the difference between the proposed coordinates of the end-effector in free motion state and the corresponding point in the contact surface and can be modelled as mass-spring system [1]. The second part arises from the inertia of the tip mass while manoeuvring. Then the components of the contact force can be given as:

)coslsinlcoslsinl(m

)cosl-cosl-r(KF

22222221

2111113

2211xsx

θθ+θθ+θθ+θθ+

θθ=&&&&&&

(2)

)cosl-sinlcosl-sinl(m

)sinl-sinl-r(KF

22222221111

2113

2211ysY

θθθθ+θθθθ+

θθ=

&&&&&&(3)

Where sK is the spring stiffness (N/m), xr , and yr are the

coordinates of the contact point. For the second link, the equation due to the flexibility effect can be written as:

0wIE -)]}-cos()-sin([l-x-w-w{

22

121122112

222

=′′′′θθθ+θθθθθρ &&&&&&&&

(4) where 22IE is the flexural stiffness of the flexible link. Equation (4) describes the vibration of the flexible link subject to four boundary conditions [12].

0)t,0(w = (5a) 0)t,l(w 2 = (5b)

22

2

IE)t(-)t,0(w τ

=′′ (5c)

0)t,l(w 2 =′′ (5d)

III. INVERSE DYNAMICS ANALYSIS

The main purpose of this analysis is to derive the elastic deflection )t,x(w to be able to calculate the rigid and flexible torques from equation (1). By introducing a new variable )t,x(y which represents the total deflection of the flexible link as:

)t(x)t,x(w)t,x(y 2θ+= (6) The effect of the first nonlinear term of equation (4) appears only at very high speed, so it can be neglected. Then equation (4) can be written in terms of )t,x(y as:

0yIE -)]}-cos()-sin([l-y-{ 22121122112 =′′′′θθθ+θθθρ &&&&& (7)

Accordingly, the modified boundary conditions are: 0)t,0(y = (8a)

)t(l)t,l(y 222 θ= (8b)

22

2

IE)t(-)t,0(y τ

=′′ (8c)

0)t,l(y 2 =′′ (8d) Solving equation (7) analytically subject to the time-dependent inhomogeneous boundary conditions (8b and c) is quite a difficult task since the flexible torque that should be obtained is already included in the boundary conditions for the total deflection )t,x(y . To avoid this difficulty, the flexible torque )t(2τ can be assumed as rigid torque without any elastic effect and can be substituted into (8c). The boundary condition (8c) can then be evaluated and equations (7) and (6) can be solved to get )t,x(w . Finally upon substitution into equation (1) one can get the flexible torque of the second link. The solution for )t,x(y can be obtained using the assumed modes method [8 and 9]in the form:

[ ] )t(f)x(h)t(e)x(g)t()x(v)t,x(y nn1n

n ++ζ=∑∞

=

where: )t(l)t(e 22θ= (10a)

IE)t(-)t(f22

2τ= (10b)

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)x(g and )x(h are functions of the spatial coordinate

alone to satisfy the homogeneous boundary conditions for )x(vn and they are given by ,

x)x(g = (10c)

3

2

22 x

l61-x

21xl

31-)x(h += (10d)

and )t(nζ is the time function. The subscript n represents the number of modes taking into considerations. In this study, five modes of vibration of the flexible link have been taken into consideration. The corresponding eigenvalue problem for equation (7) can be written as:

0)x(vdx

)x(vdIE 24

4

22 =ρω+ (11)

Then )x(vn can be obtained in the form [13]:

....3,2,1n,xlnsin

l2)x(v

22n =

πρ

= (12)

The last two nonlinear terms inside the parentheses in equation (6) can be regarded as distributed excitation force with unit density. This effect can be compensated for the time function as [11, 14 and15]:

τττω

=ζ ∫ d)-tsin()(N1 )t(t

0n

nn (13)

The convolution integral in equation (13) can be evaluated using Duhamel integral method [13]. Then, the rigid and flexible torques incorporated with vibration effects can be obtained from equations (1). For more details, the reader is referred to [11].

IV. MODELING VERIFICATION

To verify the proposed dynamical model a simulation of the joint torques through the solution of the inverse dynamics problem has been carried out for different values of the hub to beam inertia ratio (IR). The inertia ratio is defined as the percentage of the actuator’s inertia to the link inertia that, i. e., inertia ratio = (Hub inertia / Link inertia). The joint profiles are assumed as a sine rest-to-rest maneuver for both links as:

))T

T2sin(2Tt(

Tnd

nπ

π−

θ=θ

where ndθ =1 rad is the desired final angle and T=5 sec. is the duration time. The system parameters used in this study [9] are summarized in Table (1)

TABLE 1 System parameters

Rigid link Flexible link Length (m) l1=0.5 l2=0.75 Mass density (kg/m)

ρ2=0.5

Mass (kg) m1=0.5 m2=L2*ρ2

Link inertia (kgm2) I1=0.0834 I2=( l23*ρ2/3)

Hub inertia (kgm2) 10I1 10I2 Flexural rigidity (Nm2)

22IE =2.4507

Tip mass 0.15 Kg

Four different inertia ratios (1,5,10, and 15) have been used to simulate and compare the joint torque profiles and two cases (IR=1 and 5) are sampled in Figures 2(a and b) for unconstrained motion . The end-effector is maneuvering freely in space and thus, no contact force is included in joint torque simulation.

(2-a) IR =1.0 (2-b) IR =5.0

Fig. 2 (a-b): Joint torque profiles for different inertia ratios It is clear from Figures (2- a and b) that fluctuations arise in joint torques due to the vibrations of flexible link while the manipulator is manoeuvring. It is clear that as the inertia ratio increases, the magnitude of the joint torques increases and the amplitudes of fluctuations decrease. It should be noticed that the amplitudes of fluctuations in joint torques resulting from inertia ratio of 10 have significantly diminished as compared to those of inertia ratio of 1 and 5. This implies that vibration effects on joint torques can be damped to certain extent with proper selection of the inertia ratio. The situation is even better for an inertia ratio of 15 where the fluctuations are very much significantly reduced but the price is a bigger actuator size. However, since we desire to design light weight manipulator, inertia ratio of 10 is more preferable to that of 15. This conforms with the stability analysis of stable and asymptotically stable flexible manipulator [16].

V. CUBIC-SPLINE TRAJECTORY INTERPOLATION

Generally, the trajectory of the robot arm is specified in the task space, while the robot is controlled in the joint space. Using inverse kinematics and Jacobian, the desired path and velocity can be converted to joint trajectories and velocities. Among the interpolation path planning methods, the piecewise cubic and B-spline polynomial joint paths are often adopted due to their easy and fast mathematic manipulation. Unfortunately, a B-spline does not go through the knots of the path though they are suitable for on-line computation. In case of cubic-spline trajectory interpolation, joint angles are well calculated beforehand for certain points on the surface as such the end effector strictly remains in contact with the surface at those particular points while manoeuvring [17]. Then, the obtained joint angles through the solution of the inverse kinematics problem are interpolated using cubic-spline for entire surface such that

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the end effector closely manoeuvres along the surface. To study the variation of the contact force and its effect on the joint torques, two different contact surfaces are introduced here; circular and inclined. For each surface, the force exerted at the end-effector and the required joint torques for the rigid and flexible links are plotted in Figures 3 and 4 respectively.

(3-a) (3-b)

(3-c) (3-d) Figure (3) Circular surface: (a) End-effector trajectory, (b) Joint angles (c)

Force distribution (d) Joint torques profiles

(4-a) (4-b)

(4-c) (4-d) Figure (4) Inclined Surface: (a) End-effector trajectory, (b) Joint angles (c)

Force distribution (d) Joint torques profiles

Simulation of force distribution in Figures 3(c) and 4(c) verifies that designing the joint trajectories based on cubic-splines ensures that the end effector closely maneuvers along the desired trajectory. However, in the case of circular contact surface, the initial force distribution in Figure 3(a) is relatively high within the time interval of 0 to 1 second and then gradually reduces as the end effector moves along the surface. This force is expected to be small since the end-effector is controlled to pass closely to the constraint surface. This implies that the cubic spline interpolation technique may have a good command on the end-effector to maneuver closely along the entire contact surface only for certain shapes of surface. Perhaps for complicated shapes of contact surface, the cubic spline interpolation technique may not work well to compel the end effector to maneuver closely along the entire region of the contact surface. The simulation of the joint torque profiles in Figures 3(b) and 4(b) shows that irregular fluctuations can appear in the joint torque profiles due to the effect of cubic-spline interpolation.

VI. CONCLUSIONS

Cubic-spline end-effector trajectory of a constrained motion of a planar rigid-flexible manipulator is considered in this study. An analytic approach to solve the fourth order differential equation with time dependent, nonhomogeneous boundary conditions including the nonlinear terms is presented to compute the elastic deflection )t,x(w of the flexible link. The joint motion profiles have been designed based on the cubic-splines interpolation to ensure that the end-effector will move along the constrained surface without separation and to minimize the contact force exerted at the end-effector as well. The proper hub to beam inertia ratio of 10 has been selected based on a comparison study

ACKNOWLEDGMENT

The authors would like to thank the Research Center, International Islamic University Malaysia for supporting this work.

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