代写TRC4800/MEC4456 Robotics PC 9: Multi-Variable Control代写留学生数据结构程序

Department of Mechanical and Aerospace Engineering

TRC4800/MEC4456 Robotics

PC 9: Multi-Variable Control

Problem 1. Derive the computed-torque-control (CTC) scheme and write the control law for the robot in Figure 1. Assume point masses. The dynamics of the system is given by:

Figure 1: An RP manipulator

Problem 2. Specify the feedforward and feedback components of the CTC law derived in problem 1 and explain their roles.

Problem 3. Consider a linear control law (PD controller) based on linearization of the system about an equilibrium point is directly used for a robotic manipulator.

Where Kv and Kp are positive definite matrices and e  = θ − θd.

Prove that if θ̇d  = 0, the control law applied to the system renders the equilibrium point θd = 0 globally asymptotically stable.

The Lypunov function is defined as:

Where P is the potential energy.

Problem 4. Suppose a PID computed torque controller law is applied to eliminate nonzero steady-state error, and the error is defined as:

e =  qd  − q

The error dynamics is given by:

Draw the block diagram of the proposed PID-CTC law and derive the system dynamics equation with PID-CTC law, using the parameters below:

m1  = m2  =  1kg         l1  = l2   = 1m

The dynamics of the system (Figure 2) is given by:

Figure 2: An RR manipulator

Problem 5. Since we have proved that the PD control law is asymptotically stable at the equilibrium point θ  = 0, when θ̇d  = 0. Now let’s consider a modified version of the PD control law:

We call it augmented PD control law. Prove that the control law applied to the system is asymptotically stable if, kv > 0, kp > 0, and explain how you would choose E. The Lyapunov function candidate is chosen as: