Question

The dynamics of a single-link manipulator can be described by ?ӫ+?ө̇+?sinө= ? where m, d and k are positive physical parameters, ө is the joint angle and ? the control torque. i) Verify that the PI control shown in Figure C1b, where ө? is a constant reference, achieves the objective of position control if the parameters ?? and ?? are properly chosen. ii) Comment on the robustness of the PI control to a disturbance in the form of an unknown constant torque acting on the manipulator.

+ T Single-link Manipulaori k sin(-)

Answer 1

The dynamics of a single-link manipulator can be described by ?ӫ+?ө̇+?sinө= ? where m, d, and k are positive physical parameters, ө is the joint angle and ? the control torque:

i) The dynamic of a single link manipulator equations governing the behavior of a single link robot with the flexible joint is traditionally obtained from lagrangian dynamics considerations. The simple robot system under study is shown in the attached image.

Let x1= θmx1=θm be the motor angular position, the corresponding angular velocity.

x2=dθ/dtx2=dθ/dt be the corresponding angular velocity.

x3=ks(θt-θm)x3= ks(θt-θm) be the elastic force     

x4= {dθ/dt}/ρx4={dθ1/dt-dθm/dt}/ρ where ρ2=1/ksρ2=1/ks.

The state variable representation is :

x˙1(t)=x2(t)

x˙2(t)=−a5x2(t)+a1x3(t)+a1u(t)

x˙3(t)=x4(t)/ρ

x4(t)/ρ={−a2a3sin[ρ2x3(t)+x1(t)]−a4x3(t)−a7x2(t)−a6ρx4(t)−a1u(t)}/ρ

Hence we Verified that the PI control shown in Figure C1b, where ө? is a constant reference, achieves the objective of position control if the parameters ?? and ?? are properly chosen through this variable representation.

ii)  Robot manipulators play an important part in the modern industry by providing lower production costs, enhanced precision, quality, productivity, and efficiency.since linear control methods are not suitable for strong coupled, nonlinear, and time-varying rigid robot manipulator systems much nonlinear control system based on conventional PI control theory have been proposed to improve the control performance.

• In the global asymptotic stability of a class of nonlinear PI-type controllers for position and motion control of a robot, manipulators are analyzed, and a global regulator constrained to deliver torques within the prescribed limits of the actuator's capabilities is proposed.
• This class of controllers, when rule-based or gain scheduling approaches rare used, can get high-performance control systems.
• The PI control scheme can eliminate steady-state errors, but it can only ensure local asymptotic stability.
• The gain matrices must satisfy complicated inequalities a new variable structure PI control scheme is designed for robot manipulators.
• Even though the global asymptotic stability of the controlled robot system is analyzed, the bounds of system parameter matrices need to be known in the controller design.
• So the robustness of the PI control to a disturbance in the form of an unknown constant torque acting on the robot manipulator.

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