| #!/usr/bin/python3 |
| |
| # This code was used to select the gear ratio for the distal arm. |
| # Run it from the command line and it displays the time required |
| # to move the distal arm 60 degrees. |
| # |
| # Michael Schuh |
| # January 20, 2018 |
| |
| import math |
| import numpy |
| import scipy.integrate |
| |
| # apt-get install python-scipy python3-scipy python-numpy python3-numpy |
| |
| pi = math.pi |
| pi2 = 2.0 * pi |
| rad_to_deg = 180.0 / pi |
| inches_to_meters = 0.0254 |
| lbs_to_kg = 1.0 / 2.2 |
| newton_to_lbf = 0.224809 |
| newton_meters_to_ft_lbs = 0.73756 |
| run_count = 0 |
| theta_travel = 0.0 |
| |
| |
| def to_deg(angle): |
| return (angle * rad_to_deg) |
| |
| |
| def to_rad(angle): |
| return (angle / rad_to_deg) |
| |
| |
| def to_rotations(angle): |
| return (angle / pi2) |
| |
| |
| def time_derivative(x, t, voltage, c1, c2, c3): |
| global run_count |
| theta, omega = x |
| dxdt = [omega, -c1 * omega + c3 * math.sin(theta) + c2 * voltage] |
| run_count = run_count + 1 |
| |
| #print ('dxdt = ',dxdt,' repr(dxdt) = ', repr(dxdt)) |
| return dxdt |
| |
| |
| def get_180_degree_time(c1, c2, c3, voltage, gear_ratio, motor_free_speed): |
| #print ("# step time theta angular_speed angular_acceleration theta angular_speed motor_speed motor_speed_fraction") |
| #print ("# (sec) (rad) (rad/sec) (rad/sec^2) (rotations) (rotations/sec) (rpm) (fraction)") |
| global run_count |
| global theta_travel |
| |
| if (False): |
| # Gravity is assisting the motion. |
| theta_start = 0.0 |
| theta_target = pi |
| elif (False): |
| # Gravity is assisting the motion. |
| theta_start = 0.0 |
| theta_target = -pi |
| elif (False): |
| # Gravity is slowing the motion. |
| theta_start = pi |
| theta_target = 0.0 |
| elif (True): |
| # Gravity is slowing the motion. |
| theta_start = -pi |
| theta_target = 0.0 |
| |
| theta_half = 0.5 * (theta_start + theta_target) |
| if (theta_start > theta_target): |
| voltage = -voltage |
| theta = theta_start |
| theta_travel = theta_start - theta_target |
| if (run_count == 0): |
| print( |
| "# Theta Start = %.2f radians End = %.2f Theta travel %.2f Theta half = %.2f Voltage = %.2f" |
| % (theta_start, theta_target, theta_travel, theta_half, voltage)) |
| print( |
| "# Theta Start = %.2f degrees End = %.2f Theta travel %.2f Theta half = %.2f Voltage = %.2f" |
| % (to_deg(theta_start), to_deg(theta_target), to_deg(theta_travel), |
| to_deg(theta_half), voltage)) |
| omega = 0.0 |
| time = 0.0 |
| delta_time = 0.01 # time step in seconds |
| for step in range(1, 5000): |
| t = numpy.array([time, time + delta_time]) |
| time = time + delta_time |
| x = [theta, omega] |
| angular_acceleration = -c1 * omega + c2 * voltage |
| x_n_plus_1 = scipy.integrate.odeint(time_derivative, |
| x, |
| t, |
| args=(voltage, c1, c2, c3)) |
| #print ('x_n_plus_1 = ',x_n_plus_1) |
| #print ('repr(x_n_plus_1) = ',repr(x_n_plus_1)) |
| theta, omega = x_n_plus_1[1] |
| #theta= x_n_plus_1[0] |
| #omega = x_n_plus_1[1] |
| if (False): |
| print ("%4d %8.4f %8.2f %8.4f %8.4f %8.3f %8.3f %8.3f %8.3f" % \ |
| (step, time, theta, omega, angular_acceleration, to_rotations(theta), \ |
| to_rotations(omega), omega*gear_ratio*60.0/pi2, omega*gear_ratio/motor_free_speed )) |
| if (theta_start < theta_target): |
| # Angle is increasing through the motion. |
| if (theta > theta_half): |
| break |
| else: |
| # Angle is decreasing through the motion. |
| if (theta < theta_half): |
| break |
| |
| #print ("# step time theta angular_speed angular_acceleration theta angular_speed motor_speed motor_speed_fraction") |
| #print ("# (sec) (rad) (rad/sec) (rad/sec^2) (rotations) (rotations/sec) (rpm) (fraction)") |
| #print ("# Total time for 1/2 rotation of arm is %0.2f" % (time*2)) |
| return (2.0 * time) |
| |
| |
| def main(): |
| gravity = 9.8 # m/sec^2 Gravity Constant |
| voltage_nominal = 12 # Volts |
| |
| # Vex 775 Pro motor specs from http://banebots.com/p/M2-RS550-120 |
| motor_name = "Vex 775 Pro motor specs from http://banebots.com/p/M2-RS550-120" |
| current_stall = 134 # amps stall current |
| current_no_load = 0.7 # amps no load current |
| torque_stall = 710 / 1000.0 # N-m Stall Torque |
| speed_no_load_rpm = 18730 # RPM no load speed |
| |
| if (False): |
| # Bag motor from https://www.vexrobotics.com/217-3351.html |
| motor_name = "Bag motor from https://www.vexrobotics.com/217-3351.html" |
| current_stall = 53.0 # amps stall current |
| current_no_load = 1.8 # amps no load current |
| torque_stall = 0.4 # N-m Stall Torque |
| speed_no_load_rpm = 13180.0 # RPM no load speed |
| |
| if (True): |
| # Mini CIM motor from https://www.vexrobotics.com/217-3371.html |
| motor_name = "Mini CIM motor from https://www.vexrobotics.com/217-3371.html" |
| current_stall = 89.0 # amps stall current |
| current_no_load = 3.0 # amps no load current |
| torque_stall = 1.4 # N-m Stall Torque |
| speed_no_load_rpm = 5840.0 # RPM no load speed |
| |
| # How many motors are we using? |
| num_motors = 2 |
| |
| # Motor values |
| print("# Motor: %s" % (motor_name)) |
| print("# Number of motors: %d" % (num_motors)) |
| print("# Stall torque: %.1f n-m" % (torque_stall)) |
| print("# Stall current: %.1f amps" % (current_stall)) |
| print("# No load current: %.1f amps" % (current_no_load)) |
| print("# No load speed: %.0f rpm" % (speed_no_load_rpm)) |
| |
| # Constants from motor values |
| resistance_motor = voltage_nominal / current_stall |
| speed_no_load_rps = speed_no_load_rpm / 60.0 # Revolutions per second no load speed |
| speed_no_load = speed_no_load_rps * 2.0 * pi |
| Kt = num_motors * torque_stall / current_stall # N-m/A torque constant |
| Kv_rpm = speed_no_load_rpm / ( |
| voltage_nominal - resistance_motor * current_no_load) # rpm/V |
| Kv = Kv_rpm * 2.0 * pi / 60.0 # rpm/V |
| |
| # Robot Geometry and physics |
| length_proximal_arm = inches_to_meters * 47.34 # m Length of arm connected to the robot base |
| length_distal_arm = inches_to_meters * 44.0 # m Length of arm that holds the cube |
| mass_cube = 6.0 * lbs_to_kg # Weight of the cube in Kgrams |
| mass_proximal_arm = 5.5 * lbs_to_kg # Weight of proximal arm |
| mass_distal_arm = 3.5 * lbs_to_kg # Weight of distal arm |
| mass_distal = mass_cube + mass_distal_arm |
| radius_to_proximal_arm_cg = 22.0 * inches_to_meters # m Length from arm pivot point to arm CG |
| radius_to_distal_arm_cg = 10.0 * inches_to_meters # m Length from arm pivot point to arm CG |
| |
| radius_to_distal_cg = ( |
| length_distal_arm * mass_cube + |
| radius_to_distal_arm_cg * mass_distal_arm) / mass_distal |
| J_cube = length_distal_arm * length_distal_arm * mass_cube |
| J_proximal_arm = radius_to_proximal_arm_cg * radius_to_proximal_arm_cg * mass_distal_arm # Kg m^2 Moment of inertia of the proximal arm |
| J_distal_arm = radius_to_distal_arm_cg * radius_to_distal_arm_cg * mass_distal_arm # Kg m^2 Moment of inertia of the distal arm |
| J = J_cube + J_distal_arm # Moment of inertia of the arm with the cube on the end |
| |
| error_margine = 1.0 |
| voltage = 10.0 # voltage for the motor. Assuming a loaded robot so not using 12 V. |
| # It might make sense to use a lower motor frees peed when the voltage is not a full 12 Volts. |
| # motor_free_speed = Kv*voltage |
| motor_free_speed = speed_no_load |
| |
| print("# Kt = %f N-m/A\n# Kv_rpm = %f rpm/V\n# Kv = %f radians/V" % |
| (Kt, Kv_rpm, Kv)) |
| print("# %.2f Ohms Resistance of the motor " % (resistance_motor)) |
| print("# %.2f kg Cube weight" % (mass_cube)) |
| print("# %.2f kg Proximal Arm mass" % (mass_proximal_arm)) |
| print("# %.2f kg Distal Arm mass" % (mass_distal_arm)) |
| print("# %.2f kg Distal Arm and Cube weight" % (mass_distal)) |
| print("# %.2f m Length from distal arm pivot point to arm CG" % |
| (radius_to_distal_arm_cg)) |
| print("# %.2f m Length from distal arm pivot point to arm and cube cg" % |
| (radius_to_distal_cg)) |
| print( |
| "# %.2f kg-m^2 Moment of inertia of the cube about the arm pivot point" |
| % (J_cube)) |
| print( |
| "# %.2f kg-m^2 Moment of inertia of the arm with the cube on the end" % |
| (J)) |
| print("# %.2f m Proximal arm length" % (length_proximal_arm)) |
| print("# %.2f m Distal arm length" % (length_distal_arm)) |
| |
| print( |
| "# %.2f kg-m^2 Moment of inertia of the proximal arm about the arm pivot point" |
| % (J_proximal_arm)) |
| print( |
| "# %.2f kg-m^2 Moment of inertia of the distal arm about the arm pivot point" |
| % (J_distal_arm)) |
| print( |
| "# %.2f kg-m^2 Moment of inertia of the distal arm and cube about the arm pivot point" |
| % (J)) |
| print("# %d Number of motors" % (num_motors)) |
| |
| print("# %.2f V Motor voltage" % (voltage)) |
| for gear_ratio in range(30, 181, 10): |
| c1 = Kt * gear_ratio * gear_ratio / (Kv * resistance_motor * J) |
| c2 = gear_ratio * Kt / (J * resistance_motor) |
| c3 = radius_to_distal_cg * mass_distal * gravity / J |
| |
| if (False): |
| print("# %.8f 1/sec C1 constant" % (c1)) |
| print("# %.2f 1/sec C2 constant" % (c2)) |
| print("# %.2f 1/(V sec^2) C3 constant" % (c3)) |
| print("# %.2f RPM Free speed at motor voltage" % |
| (voltage * Kv_rpm)) |
| |
| torque_90_degrees = radius_to_distal_cg * mass_distal * gravity |
| voltage_90_degrees = resistance_motor * torque_90_degrees / ( |
| gear_ratio * Kt) |
| torque_peak = gear_ratio * num_motors * torque_stall |
| torque_peak_ft_lbs = torque_peak * newton_meters_to_ft_lbs |
| normal_force = torque_peak / length_distal_arm |
| normal_force_lbf = newton_to_lbf * normal_force |
| time_required = get_180_degree_time(c1, c2, c3, voltage, gear_ratio, |
| motor_free_speed) |
| print ("Time for %.1f degrees for gear ratio %3.0f is %.2f seconds. 90 degree torque %.1f N-m and voltage %.1f Volts. Peak torque %3.0f nm %3.0f ft-lb Normal force at distal end %3.0f N %2.0f lbf" % \ |
| (to_deg(theta_travel),gear_ratio,time_required,torque_90_degrees,voltage_90_degrees, |
| torque_peak,torque_peak_ft_lbs,normal_force,normal_force_lbf)) |
| |
| |
| if __name__ == '__main__': |
| main() |