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Robotics & Control

Making machines move: PID control, kinematics, torque, gear reduction and encoders

PID Controller

NEWAdvanced
u(t)=Kpe+Ki ⁣edt+Kddedtu(t) = K_p e + K_i \!\int e\,dt + K_d \frac{de}{dt}

Control output from proportional, integral and derivative of the error.

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Degrees of Freedom

NEWIntermediate
DoF=6ncDoF = 6n - \sum c

Independent motions of a mechanism from bodies and constraints.

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Gear Reduction Torque

NEWIntermediate
τout=τin×GR×η\tau_{out} = \tau_{in} \times GR \times \eta

Output torque after gear reduction, allowing for efficiency.

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Encoder Resolution

NEWBasic
θ=360N\theta = \frac{360}{N}

Smallest angle a rotary encoder can resolve.

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DC Motor Torque

NEWIntermediate
τ=ktI\tau = k_t I

Torque of a DC motor is proportional to current.

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Motor Back-EMF

NEWAdvanced
Vb=keωV_b = k_e \omega

Voltage a spinning motor generates against its supply.

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Forward Kinematics (2-Link)

NEWAdvanced
x=L1cosθ1+L2cos(θ1+θ2)x = L_1\cos\theta_1 + L_2\cos(\theta_1+\theta_2)

End-effector position from joint angles of a two-link arm.

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Angular Acceleration Torque

NEWIntermediate
τ=Iα\tau = I\alpha

Torque needed to angularly accelerate a rotating mass.

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Stepper Motor Step Angle

NEWBasic
θ=360S\theta = \frac{360}{S}

Angle turned per step from steps per revolution.

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Microstepping Resolution

NEWIntermediate
θm=θm\theta_m = \frac{\theta}{m}

Finer step angle from microstepping a stepper motor.

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Wheel Odometry Distance

NEWIntermediate
d=2πrCNd = 2\pi r \frac{C}{N}

Distance a wheeled robot travels from encoder counts.

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Settling Time (approx.)

NEWAdvanced
ts4ζωnt_s \approx \frac{4}{\zeta \omega_n}

Time for a control response to settle within 2% of target.

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Control Loop Rate

NEWBasic
f=1Tf = \frac{1}{T}

Frequency of a control loop from its cycle time.

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Payload Holding Torque

NEWBasic
τ=mgL\tau = m g L

Torque a joint must hold to support a payload at reach L.

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Robot Battery Life

NEWBasic
t=QIt = \frac{Q}{I}

Runtime from battery charge and current draw.

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