OutlineIndependent Joint Control

EE 451 - Independent Joint Control

H.I. Bozma

Electric Electronic Engineering

Bogazici University

November 22, 2017

H.I. Bozma EE 451 - Independent Joint Control

OutlineIndependent Joint Control

Independent Joint ControlIntroductionActuator DynamicsSet-Point TrackingPID ControlFeedforward ControlState Space Control

H.I. Bozma EE 451 - Independent Joint Control

OutlineIndependent Joint Control

IntroductionActuator DynamicsSet-Point TrackingPID ControlFeedforward ControlState Space Control

Control Problem

I Motion – Specified as a sequence of end-effector positions andorientations

I Joint inputs – Forces or torques or voltage inputs

I Control problem – Time history of joint inputs reqd to movethe end effector to execute a necessary motion

I Simplest approach: Independent joint control

H.I. Bozma EE 451 - Independent Joint Control

OutlineIndependent Joint Control

IntroductionActuator DynamicsSet-Point TrackingPID ControlFeedforward ControlState Space Control

Single-Input Single-output (SISO) with Feedback Control

Problem: How to design the compensator so that the error

btw output and reference is minimal or hopefully 0?

H.I. Bozma EE 451 - Independent Joint Control

OutlineIndependent Joint Control

IntroductionActuator DynamicsSet-Point TrackingPID ControlFeedforward ControlState Space Control

Actuation

I Permanent magnet DC-motor – Simplest to analyzeF = i × φ!

I AC motors

I Brushless DC motors

H.I. Bozma EE 451 - Independent Joint Control

OutlineIndependent Joint Control

IntroductionActuator DynamicsSet-Point TrackingPID ControlFeedforward ControlState Space Control

Permanent Magnet DC motor

H.I. Bozma EE 451 - Independent Joint Control

OutlineIndependent Joint Control

IntroductionActuator DynamicsSet-Point TrackingPID ControlFeedforward ControlState Space Control

At Work - I

H.I. Bozma EE 451 - Independent Joint Control

OutlineIndependent Joint Control

IntroductionActuator DynamicsSet-Point TrackingPID ControlFeedforward ControlState Space Control

At Work - II

H.I. Bozma EE 451 - Independent Joint Control

OutlineIndependent Joint Control

IntroductionActuator DynamicsSet-Point TrackingPID ControlFeedforward ControlState Space Control

Torque Speed Curves

H.I. Bozma EE 451 - Independent Joint Control

OutlineIndependent Joint Control

IntroductionActuator DynamicsSet-Point TrackingPID ControlFeedforward ControlState Space Control

Circuit Diagram → Transfer Function

H.I. Bozma EE 451 - Independent Joint Control

OutlineIndependent Joint Control

IntroductionActuator DynamicsSet-Point TrackingPID ControlFeedforward ControlState Space Control

Single Link Actuation with Motor

H.I. Bozma EE 451 - Independent Joint Control

OutlineIndependent Joint Control

IntroductionActuator DynamicsSet-Point TrackingPID ControlFeedforward ControlState Space Control

Single Link Actuation with Motor & Gear

H.I. Bozma EE 451 - Independent Joint Control

OutlineIndependent Joint Control

IntroductionActuator DynamicsSet-Point TrackingPID ControlFeedforward ControlState Space Control

System Block Diagram

H.I. Bozma EE 451 - Independent Joint Control

OutlineIndependent Joint Control

IntroductionActuator DynamicsSet-Point TrackingPID ControlFeedforward ControlState Space Control

Transfer Function

Assuming LR<< Jm

Bm

With τl = 0

θmV

=Km/R

s(Jms + Bm + KbKm/R)

With V = 0

θmτl

=−1/r

s(Jms + Bm + KbKm/R)

H.I. Bozma EE 451 - Independent Joint Control

OutlineIndependent Joint Control

IntroductionActuator DynamicsSet-Point TrackingPID ControlFeedforward ControlState Space Control

Open-Loop Control System

H.I. Bozma EE 451 - Independent Joint Control

OutlineIndependent Joint Control

IntroductionActuator DynamicsSet-Point TrackingPID ControlFeedforward ControlState Space Control

Control Problems

I Point-to-point motion: Set-Point Tracking: Tracking aconstant (step reference) comment θd

I Time varying trajectory: Feedforward control

I Joint flexibility: State space design

H.I. Bozma EE 451 - Independent Joint Control

OutlineIndependent Joint Control

IntroductionActuator DynamicsSet-Point TrackingPID ControlFeedforward ControlState Space Control

Set-Point Tracking

I PD Compensator

I PID Compensator

H.I. Bozma EE 451 - Independent Joint Control

OutlineIndependent Joint Control

IntroductionActuator DynamicsSet-Point TrackingPID ControlFeedforward ControlState Space Control

PD Control

H.I. Bozma EE 451 - Independent Joint Control

OutlineIndependent Joint Control

IntroductionActuator DynamicsSet-Point TrackingPID ControlFeedforward ControlState Space Control

Second Order Systems

H.I. Bozma EE 451 - Independent Joint Control

OutlineIndependent Joint Control

IntroductionActuator DynamicsSet-Point TrackingPID ControlFeedforward ControlState Space Control

PID Control

I With PD control, ess =DKp

I Note that ess → 0 if Kp → ∞

I Proportional-Integral-Derivative (PID) Control

I U(s) = (Kp +KI

s(θd − θ)− Kdsθ

H.I. Bozma EE 451 - Independent Joint Control

OutlineIndependent Joint Control

IntroductionActuator DynamicsSet-Point TrackingPID ControlFeedforward ControlState Space Control

PID Control

H.I. Bozma EE 451 - Independent Joint Control

OutlineIndependent Joint Control

IntroductionActuator DynamicsSet-Point TrackingPID ControlFeedforward ControlState Space Control

Routh-Hurwitz Criterion

I P(s) = a2s2 + a1s + a0 = 0 - All the coefficients satisfyan > 0.

I P(s) = a3s3 + a2s

2 + a1s + a0 = 0, an > 0, and a2a1 > a3a0

H.I. Bozma EE 451 - Independent Joint Control

OutlineIndependent Joint Control

IntroductionActuator DynamicsSet-Point TrackingPID ControlFeedforward ControlState Space Control

Comparison of PD and PID Controls

H.I. Bozma EE 451 - Independent Joint Control

OutlineIndependent Joint Control

IntroductionActuator DynamicsSet-Point TrackingPID ControlFeedforward ControlState Space Control

Feedforward Control

I Time-varying reference

H.I. Bozma EE 451 - Independent Joint Control

OutlineIndependent Joint Control

IntroductionActuator DynamicsSet-Point TrackingPID ControlFeedforward ControlState Space Control

Feedforward Control

H.I. Bozma EE 451 - Independent Joint Control

OutlineIndependent Joint Control

IntroductionActuator DynamicsSet-Point TrackingPID ControlFeedforward ControlState Space Control

Feedforward Control with Disturbance

H.I. Bozma EE 451 - Independent Joint Control

OutlineIndependent Joint Control

IntroductionActuator DynamicsSet-Point TrackingPID ControlFeedforward ControlState Space Control

Feedforward Control

H.I. Bozma EE 451 - Independent Joint Control

OutlineIndependent Joint Control

IntroductionActuator DynamicsSet-Point TrackingPID ControlFeedforward ControlState Space Control

State Space Control

I Controllability: ∀xt0 and xtf , ∃ u : t0 → tf such that thesystem is transferred from xt0 → xtf

I Linear systems: det[

b,Ab,A2b, . . .An−1b]

6= 0

I Optimal control: J =∫

∞

0xTQx + utRudt where Q is a sym,

pd matrix and R is psd.

I Soln to Riccati equation ATK + KA− 1RKbbTK + Q = 0 for

K .I u = −k(K )T x where k(K ) = 1

RbTK

H.I. Bozma EE 451 - Independent Joint Control