Transformation Matrices for Robotic Arms

Python functions for serial manipulators.

# -*- coding: utf-8 -*-
"""
Functions for calculating Basic Transformation Matrices in 3D space.
"""
from math import cos, radians, sin
from numpy import matrix

def rotate(axis, theta, angular_units='radians'):
    '''Compute Basic Homogeneous Transform Matrix for
    rotation of "theta" about specified axis.'''
    #Verify string arguments are lowercase
    axis=axis.lower()
    angular_units=angular_units.lower()
    #Convert to radians if necessary
    if angular_units=='degrees':
        theta=radians(theta)
    elif angular_units=='radians':
        pass
    else:
        raise Exception('Unknown angular units.  Please use radians or degrees.')
    #Select appropriate basic homogenous matrix
    if axis=='x':
        rotation_matrix=matrix([[1, 0, 0, 0],
                               [0, cos(theta), -sin(theta), 0],
                               [0, sin(theta), cos(theta), 0],
                               [0, 0, 0, 1]])
    elif axis=='y':
        rotation_matrix=matrix([[cos(theta), 0, sin(theta), 0],
                               [0, 1, 0, 0],
                               [-sin(theta), 0, cos(theta), 0],
                               [0, 0, 0, 1]])  
    elif axis=='z':
        rotation_matrix=matrix([[cos(theta), -sin(theta), 0, 0],
                               [sin(theta), cos(theta), 0, 0],
                               [0, 0, 1, 0],
                               [0, 0, 0, 1]])
    else:
        raise Exception('Unknown axis of rotation.  Please use x, y, or z.')
    return rotation_matrix

def translate(axis, d):
    '''Calculate Basic Homogeneous Transform Matrix for
    translation of "d" along specified axis.'''   
    #Verify axis is lowercase
    axis=axis.lower()
    #Select appropriate basic homogenous matrix
    if axis=='x':
        translation_matrix=matrix([[1, 0, 0, d],
                                  [0, 1, 0, 0],
                                  [0, 0, 1, 0],
                                  [0, 0, 0, 1]])
    elif axis=='y':
        translation_matrix=matrix([[1, 0, 0, 0],
                                  [0, 1, 0, d],
                                  [0, 0, 1, 0],
                                  [0, 0, 0, 1]])
    elif axis=='z':
        translation_matrix=matrix([[1, 0, 0, 0],
                                  [0, 1, 0, 0],
                                  [0, 0, 1, d],
                                  [0, 0, 0, 1]])
    else:
        raise Exception('Unknown axis of translation.  Please use x, y, or z.')
    return translation_matrix

if __name__=='__main__':
    #Calculate arbitrary homogeneous transformation matrix for CF0 to CF3
    H0_1=rotate('x', 10, 'degrees')*translate('y', 50)
    H1_2=rotate('y', 30, 'degrees')*translate('z', 10)
    H2_3=rotate('z', -20, 'degrees')*translate('z', 10)
    H0_3=H0_1*H1_2*H2_3
    print(H0_3)

Also available on GitHub.

Time-lapse Camera with Raspberry Pi

Building a Time-lapse Camera with Raspberry Pi.

Recently I built a time-lapse camera with a Raspberry Pi.  Here’s how:

Bill of Materials

  1. Raspberry Pi 3
  2. Power Supply
  3. Camera Mount
    • This ended up having a slightly different mounting hole pattern than the Arducam.
  4. Arducam Camera

During initial setup, you’ll also want to have a HDMI Cable, Keyboard, Mouse, and Monitor for your Pi.

Hardware

  1. Fasten the Arducam to the camera mount.
  2. Connect the Arducam ribbon cable to the Pi’s CSI port.
  3. Download the python code.
    • Update start time, end time, and sleep interval as desired.
  4. (Optional) Update rc.local as mentioned below.

Code

from time import sleep
from picamera import PiCamera
from datetime import datetime

MORNING_START_HOUR=7
EVENING_END_HOUR=19

def day_or_night(datetime):
    hour=datetime.hour
    if hour&gt;=MORNING_START_HOUR and hour<EVENING_END_HOUR:
        return 'day'
    else:
        return 'night'
    
def take_picture():
    camera.start_preview()
    sleep(2)
    now=datetime.now().strftime("%Y-%m-%d-%H-%M")
    label='timelapse_' + now + '.jpg'
    camera.capture('/home/pi/Pictures/'+label)
    camera.stop_preview()
    print('Image captured at '+now)
    
if __name__=='__main__':
    camera = PiCamera()
    
    while True:
        now=datetime.now()
        if day_or_night(now) is 'day':
            try:
                take_picture()
            except:
                pass
		sleep(900) #15 minutes

Also on GitHub.  I was able to get the code to execute upon startup by updating the Pi’s rc.local file.  I followed the rc.local method shown here.

The images are saved in /home/pi/Pictures/ on the Pi.  I used ImageMagick to create the GIF of the plant shown above.

Future Improvements

  • Saving the files to Google Drive to avoid file storage limitations.  Also, you can view the images without disturbing the camera system.  Looks like this article points us in the right direction.
  • Utilizing a portable power supply.

Hole/Shaft Tolerance Calculator

Calculate Hole/Shaft Tolerances using Excel.

Here’s is an Excel-based calculator for determining the tolerances required to achieve standard metric hole/shaft fits.  Download the calculator here.

As shown in the image above, the user inputs a basic size (Cell B3) and selects the desired fit from the dropdown menu (Cell A3).  The upper and lower limits are instantly calculated for both the shaft and bore.  The results are displayed in the cells highlighted green.

All calculations are based on the methods described in Shigley’s Mechanical Engineering Design (9th Edition).