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Selective Laser Sintering 3D Printer
Part 1
ELEC 341 Design Project
Muchen He 44638154
Ou Liu 18800152
Coordinates (X, Y, Z)
Inverse Kinematic
Controller
Motors
Direct Kinematic
Laser Control
System OverviewThe 3D printer uses a laser to build parts. The laser is
oriented using a two-axis motor assembly.
The inverse kinematic module converts the
Cartesian coordinates to the desired motor angle.
The joint controller module handles the system
response and controller.
The direct kinematic converts actual motor angles
back to Cartesian coordinates
The laser control module adjusts the intensity of the
laser based on how far the target is
The motors control the orientation of the laser
Sensor & Feedback
I.K. implementation in SimuLink D.K. implementation in SimuLink
π0: angle of motor 0
π1: angle of motor 1
π₯
βπ¦
π§
Desired π₯
Desired π§
Desired π¦
π§β²: Distance between laser and top of part
Inverse & Direct KinematicThe inverse kinematic module converts the
Cartesian coordinates to the desired motor angle.
π1 = tanβ1
π¦
π§β²π0 = tan
β1π₯
π§β²π§β² = height β π§
The direct kinematic converts actual motor angles
back to Cartesian coordinatesπ₯ = π§β² tan π0π¦ = π§β² tan π1
ππ
ππPower AmplifierThe power amplifier takes the role
of a filter and a voltage follower.
The transfer function is the output
divided by input.
Since there is negative feedback
coupled by the capacitor, ππ = ππ.
MNA is used to solve the circuit
Transfer Function: π(π )
π(π )=
Vout
Vin=
CR2R1βL
LCR1s+CR1R2
MNA:Vin β Vn
πΏπ =Vnπ 2
Vin β Vnπ 1
= Vn β Vout Γ sπΆ
Amplifier circuit SimuLink implementation
Electric Motor Dynamics
π = πΎππThe DC motor has relationships between the
torque and the current
ππππ = πΎππ
π =πππ β πππππΏπ + π
The two motors for π0 πππ π1 are identical, so the relationships are the
same for both motors
The current is given by Ohmβs law
The back-EMF is proportional to the
motor speed
Equivalent circuit
πππ ππππ
π πΏ
Implementation
ππ ππ
Response
Torque Gain
πΌ π
Back EMF Gain
π π
π(π )
π(π )=
1
πΏπ + π
π(π )
π(π )=π
π=
π
π½π 2 + π΅π + πΎ
Mechanical Motor Dynamics
Implementation
Ο Ο
Mechanical motor dynamics is a transfer function that converts
torque to angular speed. It is given as:
Where π½ is moment of inertia, π΅ is the kinetic friction constant, and πΎ is the spring constant.
Kinetic friction is the due to the mass imposed on the
motion given as
π΅ =πΌnoLoad πΎππnoLoad
Spring constant is 0 for motor 1 and positive for motor 0
Moment of inertia for motor 1 is π½rotor since the laser has negligible mass
π½π1 = π½rotor
Mechanical Motor DynamicsThe moment of inertia component for motor 1 is the
superposition of four parts
π1
π2βmotorπ
βlink
Motor 0 rotor: Inertia due to the internal rotor of the outer motor
π½link =masslink
123 π2
2 + π12 + βlink
2
Aluminium link: Hollow cylinder. Mass is its volume multiplied by
6061 aluminium density
Motor 1 & counter weight: massπ.π€. = massπ1
massπ1
massπ.π€.
β π
Extended
Imaginary
π½π0 = π½link + π½rotor + π½π0weight + π½counter weight
π½π1weight + π½counter weight = π½extended β π½imaginary
Motor 1
Counter weight
Static Friction
πΉπ π‘ππ‘ππ = ππ πΉπΉππΉπ π‘ππ‘ππ = ππ πΉπ(masslink + massq1 +massc.π€.)
Static friction works against the applied force and turns into
dynamic friction after motor starts moving. It is given by:
Motor 1 does not experience any noticeable static
friction since massrotor β 0
anglevoltage
Sensor Feedback
If πapplied < πstatic then πnet = 0
Implementation
The sensor maps the actual angle of the motors to voltage linearly
Gain: βπ, π β [β5,5]
5V
2.5V
π
2
π
Electrical Response Amplifier Response
The step response
suggests that the
integrating amplifier act
as a voltage follower with
a gain of 1 π
π
Performance
Rise time: 3 Γ 10β4s
The electrical system
produces current given a
voltage
π(π )
π(π )=
2762.4
π + 1.489 Γ 104
Internal inductances of
the motors causes
exponential behavior
Given a sudden jolt of
voltage, the current
spikes, but quickly
dissipates
π(π )
π(π )=
49.14
π + 41.17
Performance
Rise time: 0.1s
π(π )
π(π )=
2.294 Γ 106π
π (π + 0.3856)
Motor 0 Response
Motor 1 Response
Motor 0 transfer function outputs angular velocity given
some torque
π(π )
π(π )=
12644π
π 2 + 0.00213π + 14.11
Oscillation occurs due to the spring behavior
Angular velocity decays to 0 due to friction
Given a impulse torque, the motor 1 angular velocity
decays exponentially due to friction
With a constant torque, motor 1 speeds up to its
maximum angular velocity
Root Locus
Power amplifier
Motor 1Motor 0
Electrical
The electrical system, amplifier , and
motor 1 are stable according to the
root locus
To be continued in part 2 (PID Control)
Motor 0 has complex root locus, which
explains the damped oscillation
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