Formula library | Simbotics Engineering

FRC

RPM to m/s

Surface speed of a wheel, roller or pulley from its RPM and diameter.

v = \omega \, \frac{d}{2} = \frac{2\pi \, \text{rpm}}{60} \, \frac{d}{2}

Encoder to distance

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Distance travelled from encoder counts, plus the conversion factors to put in code.

n_\text{wheel} = \frac{\text{counts}}{\text{CPR} \times R}, \qquad x = n_\text{wheel} \, \pi d

k_\text{rot} = \frac{\pi d}{R}, \qquad k_\text{count} = \frac{\pi d}{\text{CPR} \times R}

Swerve drive

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Top linear speed, drive-base radius and maximum rotation rate of a swerve drive.

v_\text{max} = k \, \frac{\omega_\text{free}}{R} \, \frac{d}{2}

r = \sqrt{(t/2)^2 + (b/2)^2}, \qquad \omega_\text{max} = \frac{v_\text{max}}{r}

Projectile trajectory

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Path of a launched game piece, whether it reaches a target, and the speed needed to hit it.

y(x) = h_0 + x \tan\theta - \frac{g \, x^2}{2 v^2 \cos^2\theta}

t = \frac{x}{v \cos\theta}, \qquad h_\text{apex} = h_0 + \frac{v^2 \sin^2\theta}{2g}

v_\text{needed} = \frac{d}{\cos\theta} \sqrt{\frac{g}{2\,(d \tan\theta - (h_t - h_0))}}

Robot stability

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Tip angle and the acceleration a robot can take before it tips, from its centre of gravity.

\theta_\text{tip} = \arctan\frac{b}{h}

a_\text{max} = g \, \frac{b}{h}

Electrical

Ohm's law

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Enter any two of voltage, current and resistance to find the third, plus the power.

V = I R

P = V I

Power, current and voltage

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Enter any two of power, voltage and current to find the third.

P = V I

R_\text{eq} = \frac{V}{I}

Voltage drop

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Voltage lost in a copper wire run, as volts, a share of supply, and heat.

d = 0.127\,\text{mm} \times 92^{(36 - n)/39}, \qquad A = \frac{\pi d^2}{4}

R = \frac{\rho \, 2L}{A}, \qquad V_\text{drop} = I R, \qquad P_\text{loss} = I^2 R

Current budget

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Total robot current against the 120 A main breaker, and the battery voltage it leaves.

I_\text{total} = n_d I_d + n_m I_m + I_\text{other}

V_\text{bus} = V_\text{oc} - I_\text{total} \, R_\text{int}

Battery capacity

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Charge used in a match, share of the battery, and runtime at an average current.

Q_\text{match} = I \, t

t_\text{run} = \frac{u \, C}{I}, \qquad E = V \, C

Physics

Constant acceleration

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Final velocity, displacement and distance travelled under constant acceleration.

v = v_0 + a t

s = v_0 t + \tfrac{1}{2} a t^2

t_\text{stop} = -\frac{v_0}{a}, \qquad d = \frac{v_0^2 + v^2}{2|a|} \quad \text{if it reverses, else } d = |s|

Energy and momentum

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Kinetic energy, potential energy and momentum of a moving mass.

E_k = \tfrac{1}{2} m v^2, \qquad E_p = m g h, \qquad p = m v

Controls

Feedforward gains

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Theoretical kV and kA for a drivetrain, elevator or roller, from motor specs and gearing.

k_V = \frac{V_\text{nom}}{v_\text{free}}, \qquad v_\text{free} = \frac{\omega_\text{free}}{R} \frac{d}{2}

k_A = \frac{V_\text{nom}}{a_\text{stall}}, \qquad a_\text{stall} = \frac{n \, \tau_\text{stall} \, R}{m \, d/2}

Trapezoidal motion profile

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Time to move a distance with velocity and acceleration limits, with the velocity curve.

t_a = \frac{v_\text{max}}{a_\text{max}}, \qquad d_a = \frac{v_\text{max}^2}{2 a_\text{max}}

t = 2 t_a + \frac{d - 2 d_a}{v_\text{max}} \quad \text{if } 2 d_a < d, \qquad \text{else } t = 2\sqrt{d / a_\text{max}}

Manufacturing

Feeds and speeds

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Spindle RPM and feed rate for an end mill or router bit from surface speed and chip load.

n = \frac{v_c}{\pi D} \quad (\text{revolutions per unit time; RPM} = 60\,n \text{ with } n \text{ in rev/s})

v_f = n \times z \times f_z

Sheet metal bend

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Bend allowance and bend deduction for a sheet metal flat pattern.

BA = \theta \, (R + K t)

OSSB = \tan(\theta/2) \, (R + t), \qquad BD = 2 \, OSSB - BA