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Workspace analysis and trajectory tracking of a planar hybrid manipulator with ball screw feed drive

  • Rashmi AroraEmail author
  • Dharmveer Agarwal
  • Tarun Kumar Bera
Technical Paper
  • 35 Downloads

Abstract

The hybrid manipulators have benefits of both serial as well as parallel manipulators. In this paper, a planar hybrid manipulator with six degrees-of-freedom is proposed; it consists of two parallel manipulators each having three degrees-of-freedom. The models for forward and inverse dynamic analysis of this manipulator are presented with bond graph approach. The actuators of the manipulator have ball screw feed drive mechanism to convert rotary motion into linear motion. There are basically three problems addressed in the paper. First, the models are simulated for trajectory tracking of a semi-circular path traced by the lower manipulator initially and then tracking of another semi circular path by upper manipulators. The second problem addressed is to develop the bond graph model of the hybrid manipulator with proportional control to reach a desired position from an initial position with an unity slope after few seconds. Finally, workspace analysis for given semi-circular trajectory is done with calculation of area of the workspace of lower, upper and hybrid manipulators. It is observed from the simulation that the response follows the desired trajectory within the permissible limit. The result indicates that manipulator follows nearly tangential path. It is seen from the result that the desired trajectory lies within the workspace boundary.

Keywords

Hybrid manipulator Ball screw Bond graph Trajectory tracking Target reaching Workspace analysis 

Abbreviations

C

Capacitance

De

Effort detector

Df

Flow detector

DOF

Degrees-of-freedom

I

Inertance

MSe

Modulated effort source

MSf

Modulated flow source

MTF

Transformer with varying modulus

List of symbols

A

Cross-sectional area of shaft

Ca

Overall rotational damping

Cc

Damping of coupling

Cn

Nut damping

Cr

Overall rotational damping

Csr

Rotational damping of shaft

Cx, Cy

Current positions in x and y directions, respectively (m)

d

Shaft diameter (m)

E

Modulus of elasticity (N/m2)

F

Input force (N)

Fx, Fy

Force in respective x and y directions (N)

g

Acceleration due to gravity (m/s2)

G

Shear modulus (N/m2)

Gx, Gy

Gain in respective x and y directions

h

Travel distance during shaft’s one revolution (m)

J

Mass moment of inertia (kg m2)

ka

Overall axial stiffness (N/m)

kb

Axial rigidity of bearing (N/m)

kc

Torsional rigidity of coupling (N/m)

kn

Rigidity of ball screw nut (N/m)

kr

Overall rotational stiffness (Nm/rad)

ksa

Axial stiffness of shaft

ksr

Rotational stiffness of shaft

K

Stiffness (N/m)

l

Length of shaft (m)

leff

Effective length (m)

\( L_{i} \)

Length of legs (m) (i = 1…3)

\( L_{i}^{\hbox{min} } ,L_{i}^{\hbox{max} } \)

Length of legs for minimum and maximum radii of concentric circles (m) (i = 1…3)

\( \dot{L}_{i} \)

Rate of change of leg length of manipulator (m/s) (i = 1…3)

m, M

Mass (kg)

Q1, Q2

Difference between target and current positions in respective x and y directions (m)

r

Radius of shaft (m)

R

Damping (Ns/m)

S1

Length of platform AG of hybrid manipulator (m)

S2

Length of platform GB of hybrid manipulator (m)

t

Time (s)

x, y, X, Y

Displacement in x and y directions respectively (m)

\( x_{{a_{i} }} ,y_{{a_{i} }} \)

Position components of point Ai in fixed coordinate system attached to manipulator base (i = 1…3)

\( x_{{b_{i} }} ,y_{{b_{i} }} \)

Position components of point Bi in mobile coordinate system (i = 1…3)

\( x_{{c_{i} }} ,y_{{c_{i} }} \)

Coordinates of centre of circle (i = 1…3)

xd, yd

Target positions in respective x and y directions (m)

\( \dot{x}_{{}} \), \( \dot{y}_{{}} \)

Velocities in respective x and y directions (m/s)

Greek characters

α

Controller gain

θ

Rotation about the z-axis

\( \theta_{{a_{i} }} \)

Motion of ith leg with respect to base (i = 1…3)

\( \theta_{{a_{i} }}^{\hbox{min} } ,\theta_{{a_{i} }}^{\hbox{max} } \)

Motion of ith leg w.r.t. base for minimum and maximum radii of concentric circles

\( \dot{\theta }_{{}} \)

Rotational velocity about z-axis (rad/s)

µi

Modulus of transformer (i = 1…6)

µH, µL

High and low gain, respectively

ɷ

Angular velocity (rad/s)

ρ

Density (kg/m3)

Subscripts

A, B, C, D, E

Platform’s geometric locations

c

Controller

G

Centre of gravity

m

Motor

n

Nut

p

Pad

P

Plant

P*

Virtual system

s

Shaft

t

Table

vel

Velocity

Notes

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

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Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2019

Authors and Affiliations

  1. 1.Mechanical Engineering DepartmentChandigarh UniversityGharuan (Mohali)India
  2. 2.Department of Mechanical EngineeringIIT PatnaPatnaIndia
  3. 3.Mechanical Engineering DepartmentThapar Institute of Engineering and Technology (Deemed to be University)PatialaIndia

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