A Design Methodology for the Compensation of Positioning Deviation in Gantry Manipulators

A Design Methodology for the Compensation of Positioning Deviation in Gantry Manipulators

C. V. Ferreira

[e-mail]

V. F. Romano

Universidade Federal do Rio de Janeiro

Departamento de Engenharia Mecânica Laboratório de Robótica

Centro de Tecnologia, bloco G, sala 204 P. Box – 68503

21945-970 Rio de Janeiro, RJ. Brazil

[e-mail]

This work presents recent results concerning a design methodology used to estimate the positioning deviation for a gantry (Cartesian) manipulator, related mainly to structural elastic deformation of components during operational conditions. The case-study manipulator is classified as gantry type and its basic dimensions are 1,53m x 0,97m x 1,38m. The dimensions used for the calculation of effective workspace due to end-effector path displacement are: 1m x 0,5m x 0,5m. The manipulator is composed by four basic modules defined as module X, module Y, module Z and terminal arm, where is connected the end-effector. Each module controlled axis performs a linear-parabolic positioning movement. The planning path algorithm has the maximum velocity and the total distance as input parameters for a given task. The acceleration and deceleration times are the same. Denavit-Hartemberg parameterization method is used in the manipulator kinematics model. The gantry manipulator can be modeled as four rigid bodies with three degrees-of-freedom in translational movements, connected as an open kinematics chain. Dynamic analysis were performed considering inertial parameters specification such as component mass, inertia and center of gravity position of each module. These parameters are essential for a correct manipulator dynamic modelling, due to multiple possibilities of motion and manipulation of objects with different masses. The dynamic analysis consists of a mathematical modelling of the static and dynamic interactions among the modules. The computation of the structural deformations uses the finite element method (FEM).

Keywords: Mechatronics design, robotics, Gantry manipulator, FEM modelling

Introduction

The main catalogue specifications of any industrial robot, provided and guaranteed by its manufacturer for normal operational conditions, are related to its workspace and other performance parameters such as payload, maximum velocity and acceleration, positioning accuracy and repeatability (Rivin, 1988; ABB, 1996; DEA, 1995 and Warneck et all, 1985). The usual control techniques applied to these manipulators are mainly based on kinematics variables.

In the case that the value of payload, velocity or acceleration is above the specification, dynamic effects will cause positioning deviations on accuracy and repeatability. These deviations can be mainly associated to structural deformations. To avoid these problems, the most adopted solution by robot manufacturers is to increase the rigidity of the manipulator, resulting in a high ratio between manipulator mass and working load.

The design of less rigid manipulators in relation to payload requires real time monitoring of positioning deviations for control system compensation, in order to obtain the operational conditions previously specified (Yong et all, 1985). Another limiting aspect is related to its natural frequency behavior, whose numerical value is reduced when the structural rigidity is decreased.

Manipulator Characteristics

The design parameters used in this article are based on a manipulator prototype designed at LabRob COPPE-PEM/UFRJ. This case-study manipulator, see figure 1, is classified as a gantry type, with three degrees of freedom (not considering the end-effector) and four structural modules: Module X, Module Y, Module Z and Arm, besides the End-effector that can be coupled to the Arm.

The basic dimensions are 1530mm x 972mm x 1379mm and the dimensions used for the calculation of effective workspace due to end-effector path displacement are: 1000mm x 500mm x 500mm. The actuation system is composed by three actuators, in this case three step motors, which are controlled by a microstep control card. The transmission elements consists of ball screws, linear guides and bearings.

Inertial parameters specifications such as components mass, inertia and centre of gravity (cg) position of each module, are essential for a correct manipulator FEM and dynamic modelling. The manipulator estimated mass is 42.5 Kg, not taking in account the end-effector and payload masses. The calculated mass for module X is 19.9 Kg, for module Y is 13.7 Kg, for module Z is 7.8 Kg and for the arm is 1.1 Kg. The masses of the end-effector and payload can vary according to a specific task.

The centre of gravity position of manipulator is related to the values of end-effector and payload masses and the coordinates assumed by the end-effector during its planned motion. In this paper all component masses are considered. The position of cg module X, module Y and module Z are only dependent of the respective module component masses. Equationos (1), (2) and (3) describe the manipulator cg coordinates (in millimeters) referred to an inertial reference frame, {I}, located at module X base, see figure 1.

where:

m_load = end-effector and payload masses, Kg.

x = end-effector extremity coordinate in direction X, mm;

y = end-effector extremity coordinate in direction Y, mm;

z = end-effector extremity coordinate in direction Z, mm.

Manipulator Kinematics Analysis

The kinematics analysis of the manipulator, which has a fixed base (Module X) and 4 links (Module Y, Module Z, Arm and End-effector) with three degrees-of-freedom in translational movements, connected as an open kinematics chain, as can be seen in figure 2, is referred in the Joints Space Coordinates.

Denavit-Hartenberg (D-H) parameterization method (Sciavicco and Siciliano, 1995) is used in the manipulator kinematics model. In order to optimize D-H parameters the Module Z was subdivided in two links, Link 2 and Link 3. Besides this, since the End-effector is fixed to the Arm, both of them can be represented as one link (Link 4) as presented in figure 3 and table 1.

Local homogeneous transformation matrices ⁰T₁, ⁰T₂, ²T₃ and ³T₄ are calculated based on D-H parameters. From the figure 3 is obtained the homogeneous transformation matrix ^IT₀, relating the local {0} to the global {I} inertial reference frames. The matrices are the following:

The global homogeneous transformation matrix ^IT₄, which describes the end-effector position and orientation in the global inertial reference {I} is given by equation 4:

where:

ai = D-H geometrical parameter (body i length), mm;

di = D-H geometrical parameter (joint i variable), mm;

iT_i-1 = homogeneous transformation matrix, relating the reference frames {i-1} and {i}.

The linear velocity extremity can be obtained as a function of the homogeneous transformation matrices. The equation of the linear velocity of the end-effector for the gantry manipulator studied here is (Fu et all, 1987):

where:

Qi = link matrix for prismatic joints, (dimensionless);

4P_E = end-effector position in the reference frame {4}, (m).

The solution of equation 5 yields:

where the Jacobian obtained from

has the following elements:

This Jacobian matrix is invertible. Therefore it has no singular points and the manipulator can move into any point of its workspace.

Dynamic Analysis

The manipulator dynamics relates the kinematics of the links composing the kinematics chain, with the loads applied at the joints, through the actuators. The generic dynamic equation for a manipulator with n degrees of freedom and joint vector q(t) can be expressed as (Sciavicco and Siciliano, 1995):

where:

t = torque/force vector aplied in the joints, (N m) or (N);

Õ = external torque/force aplied in the ith link, (N m) or (N);

M = manipulator inertia matrix, (Kg m²) or ( Kg m);

C = Coriollis and centrifuge term, (Kg m²/s) or (Kg m/s);

t_V = viscosity friction term, (Kg m²/s) or (Kg m/s);

t_C = Coulomb friction term, (N m) or (N);

G = gravity term, (N m) or (N).

The Newton-Euler equations are used for the calculation of static and dynamic load interactions among the modules of the case-study manipulator. These interactions occur at the joints and transmissions elements (ball screws, linear guides and bearings). The analysis is referred in the Actuators Space Coordinates and friction is taken into consideration (Ferreira, 2000).

Motor Z

The forces considered in the calculation of the motor Z total torque are represented in figure 4. This torque is calculated by equations 10 through 13, where the signal (-) is used for movements in Z positive direction and (+) otherwise.

where:

h_Z = mechanical efficiency coeficient, (dimensionless);

d_Z = distance between the guide Z and screw Z, (mm);

m_Z = frictional coeficient, (dimensionless);

p_Z = screw Z lead, (mm);

I_Rotorz = mass moment of inertia of rotor z, (Kg m²);

I_Eqz = mass moment of inertia of moving components in respect to screw Z, (Kg m²);

M_RLZ = bearing friction torque, (N m);

F_aZ = guide frictional force, (N);

m_A/EL = mass of End-effector, Arm and Load, (Kg);

g = gravity acceleration, (m/s²).

Motor Y

In figure 5 are represented all the forces considered for the calculation of the motor Y total torque given by equations 14 through 17, where the signal (+) is used when the movement is in Y positive direction and (-) otherwise.

where:

d_Y = distance between the guide Y and screw Y, (mm);

h_Y = mechanical efficiency coefficient, (dimensionless);

p_Y= screw Z lead, (mm);

I_Roly = mass moment of inertia of the wheel, (Kg m²);

I_Rotory = mass moment of inertia of rotor Y, (Kg m²);

I_Eqy = mass moment of inertia of moving components in respect to screw Y, (Kg m²);

M_RLy = bearing friction torque, (N m);

F_aY = guide frictional force, (N);

m_Y = frictional coefficient, (dimensionless);

m_Z = module Z mass, (Kg).

Motor X

The forces taken into consideration to calculate the motor X total torque are represented in figure 6. This torque is calculated by equations 18 through 21, where one has to use the signal (-) for movement in X positive direction and (+) otherwise.

where:

dY = distance between the guide Y and screw Y, (mm);

h_Y = mechanical efficiency coefficient, (dimensionless);

p_Y = screw Y lead, (mm);

I_Roly = mass moment of inertia of the wheel, (Kg m²);

I_Rotory = mass moment of inertia of rotor Y, (Kg m²);

I_Eqy = mass moment of inertia of moving components in respect to screw Y, (Kg m²);

M_RLy = bearing friction torque, (N m);

F_aX = guide frictional force, (N);

m_Y = frictional coefficient, (dimensionless);

m_ZY = mass of modules Z and Y, (Kg).

Finite Element Model

A commercial FEM program, ANSYS^â5.4, is used for the calculation of the structural deformations. The element "3-D Elastic Beam" (ANSYS-PC, 1991) is used to model the manipulator structural components, since it can support tension, compression, torsion and flexion. This element also has six degrees-of-freedom in each node (three translations and three rotations referred to X, Y and Z axis).

Figure 7 presents the model of the manipulator used to calculate the structural deformations, with all the loads applied at the contact surfaces, the acceleration gravity vector (node 1) and the restrictions to the six degrees of freedom at the manipulator base (nodes 1, 10, 11 and 20). Figure 8 represents the deformed condition of the manipulator calculated by the FEM program ANSYS^â .

End-effector Positioning Deviation

The values of the positioning deviations of the end-effector extremity are essentially dependent on the elastic deformations caused in the manipulator during the realization of a given task. In the FEM modeling the module Z, figure 9, was defined by the elements connected by the nodes 51, 52 and 53; and nodes 62 and 63, which are defined to locate point 60. This node (Node 60) is the point where the arm containing the end-effector is assumed to orthogonally intercept the plane p , see figure 10. The planes p and p_D represent, respectively, the undeformed and deformed conditions of the manipulator. In order to obtain the end-effector extremity coordinates is used the D-H parameter d4 (defined in the kinematics analysis) which represents the arm extended length. The End-effector position deviation D is given by equation 22, where x, y and z are the desired End-effector extremity coordinates in the reference systems{I}.

where:

x_ijD = position of Node ijD in the X direction, (mm); see Fig. (10).

y_ijD = position of Node ijD in the Y direction, (mm);

z_ijD = position of Node ijD in the Z direction, (mm);

n_Dk = component k of the plane p_D normal vector.

Results

The manipulator performance is related to the specification of its tasks. A task can be interpreted as a temporal relation between kinematics and inertial parameters. After the definition of the manipulator movement, the kinematics parameters are obtained for the specified path and the values of the mechanical loads applied at the elements are calculated. Then these data are used in the FEM modeling to calculate the deformation of the manipulator, which results are used to determine the end-effector extremity position deviation.

In this article, the tasks are defined as:

·Task one: simultaneous movement of the manipulator through the directions X (d₁=0 to 1m), Y (d₂=0 to 0,5m) and Z (d₄=0,5 to 0m), where d1, d2 and d4 are the DH parameters defined in the kinematics analisys, figure 11;

·Task two: simultaneous movement of the manipulator through the directions X (d₁=0 to 1m) and Z (d₄=0,5 to 0m), whith d₂=0,25m, figure 12;

·Task three: simultaneous movement of the manipulator through the directions Y (d₂=0 to 0,5m) and Z (d₄=0,5 to 0m), whith d₁=0,5m, figure 13;

·Task four: movement of the manipulator through the direction X (d₁=0 to 1m), whith d₂=0 and d₄=0,25m, figure 14;

·Task five: movement of the manipulator through the direction X (d₁=0 to 1m), whith d₂=0,25m and d₄=0,25m, figure 15;

·Task six: movement of the manipulator through the direction X (d₁=0 to 1m), whith d₂=0,5m and d₄=0,25m, figure 16;

·Task seven: movement of the manipulator through the direction Y (d₂=0 to 0,5m), whith d₁=0,5m and d₄=0,5m, figure 17;

·Task eight: movement of the manipulator through the direction Y (d₂=0 to 0,5m), whith d₁=0,5m and d₄=0,25m, figure 18;

·Task nine: movement of the manipulator through the direction Y (d₂=0 to 0,5m), whith d₁=0,5m and d₄=0,5m, figure 19;

·Task ten: movement of the manipulator through the direction Y (d₂=0 to 0,5m), whith d₁=0,5m and d₄=0,5m, figure 20.

The loads (payload and end-effector masses) used for all tasks are 0, 1, 5 and 10 Kg. Each step motor is assumed to perform a linear-parabolic movement, due to the fact that this kind of motion is the most employed for tasks such as manipulation of objects. The planning path algorithm has the maximum velocity and the total distance as input parameters for a given task. The acceleration and deceleration times are the same. For the case study manipulator the maximum velocities are: 0,41m/s at direction X and 0,20 m/s at directions Y and Z.

To study the end-effector extremity position deviation of the manipulator when submitted to a specified task, the analysis is done according to the following criteria: five points at the accelerated interval, seven points at the constant velocity interval and five points at the decelerated interval.

Conclusions

It can be observed for a given task the proportionality between the end-effector extremity position deviation and the payload/end-effector masses, due to the elastic nature of the element used in the FEM modelling, beside this, the hypothesis of the friction coefficients to be constant implies in the elimination of the non-linearities due to friction in the torque/loads equations.

Experimental analysis of the manipulator is of fundamental importance to provide the necessary modelling adjustments and data validation, due to non-linearities and non-precise components assembly.

The design methodology developed by this research can be used as an optimization tool for a conventional application of gantry manipulators, since it is able to identify structural limits during the design process. The most significant phases are the following:

a) determination of inertial and geometrical properties of each component;

b) kinematics modelling based on homogeneous transformation matrices;

c) dynamic modelling based on Newton-Euler equations;

d) definition of the planned workspace where tasks will be performed (i.e., CIM environment);

e) determination of the desired paths and kinematics parameters;

f) calculation of the manipulator static and dynamic loads (based on phases b to e);

g) structural deformations mapping (based on phase f);

h) calculation of the deviation-function at the manipulator end-effector.

At this point, the modal analysis and the interpolation functions for positioning deviation are being implemented for different boundary conditions. The manipulator is actually under construction and further results will be presented in the near future.

Acknowledgements

The authors would like to express their gratitude to FAPERJ for the financial support provided during the course of this present research through grant 170.818/95. Additional partial financial support was provided by CNPq through grant 520864/97.

Presented at COBEM 99 - 15th Brazilian Congress of Mechanical Engineering, 22-26 November 1999, São Paulo. SP. Brazil technical Editor: José Roberto F. Arruda.

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Publication Dates

• Publication in this collection
  21 Aug 2002
• Date of issue
  May 2002