ee 451 - independent joint control
TRANSCRIPT
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