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

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## 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

*C*_{a}Overall rotational damping

*C*_{c}Damping of coupling

*C*_{n}Nut damping

*C*_{r}Overall rotational damping

*C*_{sr}Rotational damping of shaft

*C*_{x}, C_{y}Current positions in

*x*and*y*directions, respectively (m)*d*Shaft diameter (m)

*E*Modulus of elasticity (N/m

^{2})*F*Input force (N)

*F*_{x},*F*_{y}Force in respective

*x*and*y*directions (N)*g*Acceleration due to gravity (m/s

^{2})*G*Shear modulus (N/m

^{2})*G*_{x},*G*_{y}Gain in respective

*x*and*y*directions*h*Travel distance during shaft’s one revolution (m)

*J*Mass moment of inertia (kg m

^{2})*k*_{a}Overall axial stiffness (N/m)

*k*_{b}Axial rigidity of bearing (N/m)

*k*_{c}Torsional rigidity of coupling (N/m)

*k*_{n}Rigidity of ball screw nut (N/m)

*k*_{r}Overall rotational stiffness (Nm/rad)

*k*_{sa}Axial stiffness of shaft

*k*_{sr}Rotational stiffness of shaft

*K*Stiffness (N/m)

*l*Length of shaft (m)

*l*_{eff}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)

*Q*_{1},*Q*_{2}Difference between target and current positions in respective

*x*and*y*directions (m)*r*Radius of shaft (m)

*R*Damping (Ns/m)

*S*_{1}Length of platform

*AG*of hybrid manipulator (m)*S*_{2}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

*A*_{i}in fixed coordinate system attached to manipulator base (*i*= 1…3)- \( x_{{b_{i} }} ,y_{{b_{i} }} \)
Position components of point

*B*_{i}in mobile coordinate system (*i*= 1…3)- \( x_{{c_{i} }} ,y_{{c_{i} }} \)
Coordinates of centre of circle (

*i*= 1…3)*x*_{d},*y*_{d}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

*i*th leg with respect to base (*i*= 1…3)- \( \theta_{{a_{i} }}^{\hbox{min} } ,\theta_{{a_{i} }}^{\hbox{max} } \)
Motion of

*i*th leg w.r.t. base for minimum and maximum radii of concentric circles- \( \dot{\theta }_{{}} \)
Rotational velocity about

*z*-axis (rad/s)*µ*_{i}Modulus of transformer (

*i*= 1…6)*µ*_{H},*µ*_{L}High and low gain, respectively

*ɷ*Angular velocity (rad/s)

- ρ
Density (kg/m

^{3})

## 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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