Understanding the fundamental physics principles of force, torque, and power in robotic systems

Force, Torque, and Power

Force, torque, and power are the three fundamental quantities that drive all mechanical motion in robotic systems. Understanding their relationships is crucial for selecting appropriate motors, designing mechanisms, and controlling robot behavior.

Force

Force is a push or pull acting upon an object, causing it to accelerate or deform.

Newton's Second Law

F = m × a

Where:

F = Force (Newtons, N)
m = Mass (kilograms, kg)
a = Acceleration (m/s²)

Key Implications for Robotics:

Heavier robots require more force to accelerate at the same rate
Acceleration is directly proportional to applied force
Forces in opposite directions subtract from each other

Types of Forces in Robotics

Force Type	Description	Example in Robotics
Applied Force	External force on robot	Motor pushing wheel
Friction Force	Opposes motion	Wheel-ground interaction
Normal Force	Perpendicular to surface	Robot weight on ground
Tension	Force in cables/belts	Cable-driven mechanisms
Gravitational	Weight of robot	Lifting mechanisms
Centripetal	Keeps object in circular path	Turning robots

Force Diagram Example

Example: Mobile Robot Acceleration

A 10 kg robot experiences:

Applied motor force: 50 N
Friction force: 20 N
Net force: 50 - 20 = 30 N

Acceleration:

a = F_net / m = 30 N / 10 kg = 3 m/s²

Torque

Torque is the rotational equivalent of force. It causes objects to rotate around an axis and is essential for all rotating mechanisms in robots.

Torque Formula

For perpendicular force application:

τ = r × F

General formula:

τ = r × F × sin(θ)

Where:

τ (tau) = Torque (Newton-meters, N·m)
r = Distance from axis (lever arm length, meters)
F = Applied force (Newtons, N)
θ = Angle between force and lever arm

Key Principle: Lever Arm Effect

The same force applied farther from the pivot produces greater torque:

Torque Applications in Robotics

Joint Motors in Robotic Arms

Each joint motor must provide sufficient torque to:

Support the weight of all links and payload below it
Accelerate/decelerate the arm segments
Overcome friction in bearings and joints

Example: 6-axis Robot Arm

Base joint torque calculation for lifting a 5 kg payload at 0.4 m distance:

Force needed = m × g = 5 kg × 9.81 m/s² = 49.05 N
Torque needed = r × F = 0.4 m × 49.05 N = 19.62 N·m

Plus additional torque for:

Arm link weight: +5 N·m
Acceleration reserve: +5 N·m
Safety factor: 1.5-2.0×

Total required: ~50 N·m

Motor selection: Use motor rated for at least 50 N·m continuous torque

Gripper Torque Requirements

Gripping force depends on:

Gripper finger length from pivot
Friction coefficient between gripper and object
Object weight and geometry

Gripper Torque Calculation:

For a parallel gripper with fingers 0.05 m long gripping a 2 kg object:

Normal force on each finger = Object weight = ~9.81 N
Torque per finger = 0.05 m × 9.81 N = 0.49 N·m
Motor torque needed = ~0.5 N·m per finger

Important: Gripper motor must provide enough torque WITHOUT damaging delicate objects (friction-limited gripping)

Wheel Drive Motors

Wheel motors need torque to overcome:

Static friction (getting the robot moving)
Rolling resistance (maintaining motion)
Climbing inclines (overcoming gravity component)
Acceleration (speeding up)

Example: Climbing an Incline

For a 20 kg robot on a 30° incline:

Gravity component = m × g × sin(30°)
                  = 20 × 9.81 × 0.5
                  = 98.1 N

For 0.1 m radius wheels:
Torque needed = 98.1 N × 0.1 m = 9.81 N·m

Add friction losses and acceleration buffer → select 15+ N·m motor

Gear Reduction and Torque Multiplication

Gears allow us to trade speed for torque using the gear ratio:

Output Torque = Input Torque × Gear Ratio
Output Speed = Input Speed ÷ Gear Ratio

Example: 50:1 Gearbox

Motor specifications:

Speed: 3000 RPM
Torque: 2 N·m
Power: ~630 W

After 50:1 gearbox:

Speed: 3000 ÷ 50 = 60 RPM
Torque: 2 × 50 = 100 N·m (assuming 90% efficiency)
Power: ~590 W (slightly reduced due to friction)

Torque-Speed Trade-off

Power remains approximately constant through gearing (minus losses). You can't get "more power" from gearing - you're just trading speed for torque or vice versa.

For picking tasks: Use high torque, lower speed (geared down) For fast movement: Use lower torque, higher speed (geared up or direct drive)

Power

Power is the rate at which energy is transferred or work is performed.

Power Formulas

Linear Motion:

P = F × v

Rotational Motion:

P = τ × ω

Where:

P = Power (Watts, W)
F = Force (N)
v = Velocity (m/s)
τ = Torque (N·m)
ω = Angular velocity (rad/s)

Converting RPM to Angular Velocity

To use rotational power formula, convert RPM to rad/s:

ω (rad/s) = (RPM × 2π) / 60

Example:

3000 RPM = (3000 × 2π) / 60 = 314.16 rad/s

Power Calculation Examples

Motor Power Calculation

A motor rated at 5 N·m torque spinning at 3000 RPM:

Step 1: Convert RPM to rad/s

ω = (3000 × 2π) / 60 = 314.16 rad/s

Step 2: Calculate power

P = τ × ω = 5 N·m × 314.16 rad/s = 1,570.8 W ≈ 1.57 kW

This motor consumes approximately 1.57 kilowatts of electrical power (before accounting for motor efficiency, typically 80-90%).

Lifting Power

To lift a 100 kg payload vertically at 0.5 m/s:

Force needed:

F = m × g = 100 × 9.81 = 981 N

Power required:

P = F × v = 981 N × 0.5 m/s = 490.5 W

This means the motor must provide at least 490.5 W to lift the payload at this speed. In practice, add 20-30% for efficiency losses.

Actual power needed: ~600-640 W

Driving Power for Mobile Robot

A robot pushing forward against 50 N of resistance at 2 m/s:

Power required:

P = F × v = 50 N × 2 m/s = 100 W

Battery life calculation:

For a 1500 W·h battery (common large robot battery):

Runtime = 1500 W·h ÷ 100 W = 15 hours

At higher speed (4 m/s) with same resistance:

P = 50 N × 4 m/s = 200 W
Runtime = 1500 W·h ÷ 200 W = 7.5 hours (half!)

This shows why faster movement dramatically reduces battery life.

Power Relationship Triangle

Power Dissipation

When a motor stalls (torque applied but no motion), all electrical power is converted to heat! This can destroy motors instantly if stalled for too long.

Stall current protection:

Electronic current limiting
Thermal cutoff
Mechanical overload release
Torque limiting firmware

Integration: Force, Torque, and Power

All three concepts work together:

Force causes linear acceleration
Torque causes rotational acceleration
Power limits how fast energy can be delivered

System Design Process:

Key Formulas Summary:

Concept	Formula	Units
Force	F = m × a	N = kg × m/s²
Torque	τ = r × F	N·m = m × N
Linear Power	P = F × v	W = N × m/s
Rotational Power	P = τ × ω	W = N·m × rad/s
Gear Output	τ_out = τ_in × ratio	N·m

Next Steps:

Learn about friction and traction in the next section
Understand rotational motion and RPM
Apply these concepts to your robot design

Force, Torque, and Power

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