Classical Motion Controllers: Trajectory Generation and PID Control, Study Guides, Projects, Research of Control Systems

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Design of

classical

motion

controllers

SARV PARTEEK SINGH

JULY-08-

Problem

Statement

Move a system (control center of mass) or a

single axis of motion from one point to another

under

A. Velocity limits (VL)

B. Acceleration/deceleration limits (AL)

C. Final position constraint (PC)

D. Final(initial) velocity constraint(s) (VC)

C1(SW)

C2 +

power

P

H

High-level

control laws

Drive

amplifier Plant

Feedback

y

r

e = r - y

Waypoint

generator

Targets and

constraints

(I) PID control

with step

inputs

P control - simulations

(I)

P = 2.

dt [s] ] ]

Simulation parameters used

throughout

• (^) Trapezoidal velocity profile for P gain = 2.
• (^) Optimal (minimum time) motion profile

P =

P control – gain variation

(I)

P = 2.0, state evolution: q k

. dt P = 1.5, state evolution: q k . dt

Phase lag in the forward loop is equivalent to a lower controller gain with no phase lag in the

forward loop. Both prolong settling and motion profile is no longer minimum time / optimal.

143% increase in settling

time

118% increase in settling

time

P control – with Velocity

Constraint



Modified control law implementation to respect velocity

constraint

1. e p,k

= r p,k

• y p,k

2. c k

= P. e p,k

• v des

5. if | > ||

c k,a_clamped

6. c k,a_v_clamped

= clamp(

(I)

AL

VL

PC, VC

P control – simulations

(I)

P = 2.0, control law: c k

= P. e p,k

v des

,v des

Feeding forward desired velocity in the control law allows reaching the target position close(?)

to the desired velocity, but perfect tracking requires fine-tuning

P = 2.0, control law: c k

= P. e p,k

1.4 m/s at target position

(desired = 1 m/s)

0 m/s at target position

PD/PI/PID control



In practice, settling with conservative P gains results in a long

tail, and in presence of stiction, results in a steady state errors.



This can be resolved with integral gain, but high gain results in

overshoot and needs anti-windup control



D gain can resolve oscillations, but requires a filter (a single-pole

IIR implemented as EMA does a reasonable job) but addition of a

filter erodes phase further.

(I)

Flaws in PID + step ref.

architecture



Sensitivity to controller gain and plant dynamics

 Low gain => suboptimal settling; high gain => ringing (time-domain) / peaking

(frequency domain)

 Variation in plant inertia (due to material load/unload etc.), friction (due to

inclement whether, wet surfaces etc.) exacerbates the sensitive nature of

forward loop gain



Inability to accurately achieve both position and velocity targets

 Fine-tuned gains are required to achieve this, and the same set cannot be then

used to achieve a position with zero velocity optimally.



Inability to handle discrete jumps in reference/feedback

 Controller is directly exposed to these changes, and transfers them to drive.

Adding a filter to reference/feedback could resolve this.

(I)

C1(SW)

C2 +

power

P

H

High-level

control laws

Drive

amplifier Plant

Feedback

y

r

e = r - y

Waypoint

generator

Targets and

constraints Trajectory reference

architecture

Trajectory

generator

(II) Trajectory

generation

Trajectory design - derivation



Assume only 2 segments (1/accel and 3/decel – no coasting)



Segment 1



Segment 2





Segment 3

(II)

Notation

Trajectory design - derivation



Set. This gives



, where

 discriminant can be proved non-negative

 will have only one positive root



Velocity constraint violation check

 With calculated, compute

 If , then coasting segment is required

 Recompute :

 Recompute. Then,

(II)

𝑚𝑎𝑥