Advanced Robotics - Computed Torque Control by Sangsin Park, Ph.D.
Explore the concept of Computed Torque Control in Advanced Robotics through an in-depth explanation by Sangsin Park, Ph.D. This involves feedback linearization, feedforward and feedback loops, derivation of control laws, and PD/PID controllers for outer loop stability.
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ME729 Advanced Robotics - Computed-Torque Control 4/16/2018 Sangsin Park, Ph.D.
Sangsin Park, Ph. D. Computed-Torque Control Introduction A special application of feedback linearization of the nonlinear system. Feedforward loop : for eliminating nonlinear terms of the system. Feedback loop : for tracking a reference input. Linear system Nonlinear inner loop ?(?, ?) + ?, ? + + ? ?(?) ?? Arm ? Outer loop feedback
Sangsin Park, Ph. D. Computed-Torque Control Derivation of inner feedforward loop The robot dynamics: ? + ? ?, ? + ? ? = ? ? ? For convenience, ? ?, ? + ? ? = ? ?, ? : ? + ? ?, ? = ? ? ? Define tracking error: ? ? = ??? ? ? where, ??? is a desired trajectory. Differentiate the error twice: ? = ?? ?,and ? = ?? ? To eliminate nonlinear term and track a desired trajectory, define the computed-torque control law: ? = ? ?? ? + ? , where ? is the outer loop feedback control input. Substituting into the robot dynamics: ? ? + ? = ? ?? ? + ? ? ?? ? = ?? ? = ? Therefore, if we select a control ? that stabilizes ? = ?, then ? goes to zero.
Sangsin Park, Ph. D. Computed-Torque Control PD controller for the outer loop One way to select the outer loop feedback control input ?is the PD feedback, ? = ?? ? ??? Then the control law becomes ? = ? ??+ ?? ? + ??? + ? The closed-loop error dynamics are ? + ?? ? + ??? = 0 The error system is asymptotically stable as long as the ?? and ?? are all positive.
Sangsin Park, Ph. D. Computed-Torque Control PD controller for the outer loop Block diagram for PD computed-torque controller. ?(?, ?) ?? + + + + ? + ?? ?(?) ?? Arm + ? + ?? ?? ? Feedback loop (or linear controller) Feedforward loop (or linearization)
Sangsin Park, Ph. D. Computed-Torque Control PID controller for the outer loop PD computed-torque control is very effective if all the arm parameters are known and there is no disturbance. it gives a nonzero steady-state error. Consequently, we add an integral controller in the feedback loop. ? = ?? ? ??? ?? ? Then the control law becomes ? = ? ??+ ?? ? + ??? ?? ? + ?
Sangsin Park, Ph. D. Computed-Torque Control PID controller for the outer loop Block diagram for PID computed-torque controller. ?(?, ?) ?? + + + + ? + ?? ?(?) ?? Arm + ? + + ?? ?? + ?? ?? ? Feedback loop (or linear controller) Feedforward loop (or linearization)
Sangsin Park, Ph. D. Computed-Torque Control For example The two-link planar manipulator. Take the link masses as 1 kg and their lengths as 1 m. ?2 ?2 ?2 ?2 ?1 ?2 ?1 ?1 ?? ?1 ?1 ?? ?? ? The manipulator s equations of motion ? = ? ? ? + ? ?, ? 2 ?1 ?2+ 2?2 ?2 ?1 ?2 3 + 2?2 1 + ?2 1 + ?2 1 ?1 ?2 = + 2?2 ?1
Sangsin Park, Ph. D. Computed-Torque Control For example Rewrite the manipulator s equations of motion ? = ? ? 1{? ? ?, ? } The PD computed-torque control law is given as ??+ ?? ? + ??? + ? ?, ? ? = ? ? ? = ?? ? Let the desired trajectory ?? ?1?= 0.1sin?? ?2?= 0.1cos?? Differentiate the desired trajectory ?1?= 0.1?cos?? ?2?= 0.1?sin?? ?1?= 0.1?2sin?? ?2?= 0.1?2cos?? Set the gains as ??= 100,??= 20
Sangsin Park, Ph. D. Computed-Torque Control For example Simulink blocks System - Manipulator theta1_d_ddot theta_d_ddot y u q theta2_d_ddot M_inv q_dot u y u y y u M V1 V Matrix Multiply 1 s 1 s Matrix Multiply tau theta1_d_dot q_ddot q_dot 20 theta_d_dot Product Integrator Integrator1 Joint Angles (rad) Product1 kd theta2_d_dot Switch into matrix multiplication Switch into matrix multiplication Set initial value as [0; 0] Torque inputs (Nm) Set initial value as [0; 0] theta1_d 100 theta_d kp theta2_d Tracking error (rad)
Sangsin Park, Ph. D. Computed-Torque Control For example Simulation results Yellow line is regarding to ?1. Blue line is regarding to ?2. [Joint angles (rad)] [Tracking error (rad)] [Computed torque input (Nm)]
