Understanding center of mass, stability principles, and optimization techniques for stable robot design

Center of Gravity and Stability

The Center of Gravity (COG), also called Center of Mass, is the point where the entire weight of an object can be considered to act. For robots, COG position is critical to stability, balance, and performance. Understanding and optimizing COG is essential for designing robots that won't tip over and perform reliably.

Understanding Center of Gravity

What is Center of Gravity?

Definition: The point representing the average position of all mass in an object, weighted by the mass of each element.

Physical Meaning:

If you could balance the robot on a single point, that point would be the COG
Gravity appears to act on this single point
All weight calculations can use this point

Calculating COG Position

2D Calculation:

X_cog = Σ(m_i × x_i) / Σ(m_i) = Σ(m_i × x_i) / M_total
Y_cog = Σ(m_i × y_i) / Σ(m_i) = Σ(m_i × y_i) / M_total

3D Calculation:

Simply add Z coordinate:

Z_cog = Σ(m_i × z_i) / M_total

Practical Example

Calculate COG of a simple robot:

Component	Mass (kg)	X (m)	Y (m)
Chassis	8.0	0.25	0.10
Battery	3.0	0.25	0.05
Motor 1	0.5	0.10	0.15
Motor 2	0.5	0.40	0.15
Electronics	0.5	0.25	0.20
Total	12.5	-	-

Calculation:

X_cog = (8.0×0.25 + 3.0×0.25 + 0.5×0.10 + 0.5×0.40 + 0.5×0.25) / 12.5
      = (2.0 + 0.75 + 0.05 + 0.20 + 0.125) / 12.5
      = 3.125 / 12.5
      = 0.25 m

Y_cog = (8.0×0.10 + 3.0×0.05 + 0.5×0.15 + 0.5×0.15 + 0.5×0.20) / 12.5
      = (0.80 + 0.15 + 0.075 + 0.075 + 0.10) / 12.5
      = 1.2 / 12.5
      = 0.096 m ≈ 0.1 m

Result: COG is at (0.25 m, 0.10 m) - center X, very low Y (good for stability!)

Stability Principles

Stability depends on the relationship between COG position and the robot's support base.

Support Polygon

Support Polygons for Different Robots:

Robot Type	Support Polygon	Typical Margin
2-wheel (balanced)	Line between wheels	0% (no margin)
3-wheel	Triangle	10-30% typical
4-wheel (car)	Rectangle	20-40% typical
Quadruped	Triangle (3 legs stance)	15-25% typical
Hexapod	Polygon (3+ legs)	25-40% typical

Static Stability

Condition for Static Stability:

The vertical projection of the COG must fall within (or very close to) the support polygon.

Stability Margin:

Margin = Minimum distance from COG projection to polygon edge

Typical Safety Margins:

Industrial robots: 20-25% of base width
Competition robots: 10-15% (more aggressive)
Legged robots: 15-30% depending on gait

Zero Margin = Unstable

A robot with zero stability margin is at tipping threshold. Any small disturbance causes tipping. Always design with safety margin.

Dynamic Stability During Acceleration

When a robot accelerates, inertia shifts the effective COG:

COG_effective = COG_actual + (a × h) / g

Where:

a = Linear acceleration (m/s²)
h = COG height above ground (m)
g = Gravity (9.81 m/s²)

Example: Robot Braking

A robot with COG at 0.3 m height decelerating at 5 m/s²:

Forward shift = (5 × 0.3) / 9.81 = 0.153 m

Effective COG shifts 15 cm forward!

If the robot only has 10 cm stability margin at the rear, it will tip backward when braking hard!

Turning Stability

When turning, centrifugal effects shift the effective COG:

Critical Turning Speed:

v_critical = √((g × w) / (2 × d_cog_to_outer_wheel))

Where:

g = Gravity (9.81 m/s²)
w = Width between wheels (m)
d_cog_to_outer_wheel = Distance from COG to outer wheel (m)

Example: Robot Turning

Wheel width: 0.4 m
COG to outer wheel: 0.2 m (centered)
COG height: 0.2 m

v_critical = √((9.81 × 0.4) / (2 × 0.2))
           = √((3.924) / (0.4))
           = √9.81
           = 3.13 m/s

At speeds above 3.13 m/s, this robot will tip when turning!

How to increase turning speed:

Lower COG height (reduces rotation effect)
Increase wheel width (larger support polygon)
Reduce turn radius (less centrifugal force)

COG Optimization Strategies

Practical COG Determination Methods

CAD-Based Analysis

Using Modern CAD Software:

Model all components with accurate dimensions
Assign material properties (density) to each part
Software automatically computes COG position

Advantages:

Fast and accurate
Can test multiple configurations quickly
Includes complex shapes easily
Accounts for manufacturing variations

Disadvantages:

Requires detailed CAD model
Material densities must be accurate
Doesn't account for internal components (wires, etc.)

Manual Calculation Method

Create spreadsheet with all components:

Part	Mass (kg)	X (m)	Y (m)	Z (m)	m×X	m×Y	m×Z
Chassis	5.0	0.25	0.10	0.15	1.25	0.50	0.75
Battery	2.0	0.25	0.05	0.08	0.50	0.10	0.16
Motor	0.8	0.20	0.15	0.10	0.16	0.12	0.08
...	...	...	...	...	...	...	...
TOTAL	12.5	-	-	-	3.12	1.22	1.50

COG_X = 3.12 / 12.5 = 0.250 m
COG_Y = 1.22 / 12.5 = 0.098 m ≈ 0.10 m
COG_Z = 1.50 / 12.5 = 0.120 m

Measuring COG on Built Robot

Equipment needed:

Digital scale (or 2 scales)
Level surface
Robot in complete assembled state

2-Scale Method (most accurate):

Place all wheels on left scale
Record weight: W_left
Place all wheels on right scale
Record weight: W_right
Total: W_total = W_left + W_right

Calculate horizontal COG:

For wheelbase L:

Distance from left wheel to COG = (W_right × L) / W_total

Example:

Wheelbase: 0.5 m
Weight on left wheels: 4.5 kg
Weight on right wheels: 5.5 kg
Total: 10 kg

COG_distance = (5.5 × 0.5) / 10 = 0.275 m from left wheels
OR 0.225 m from right wheels (0.5 - 0.275)

Vertical COG (tilting method):

Tilt robot forward by height h
Measure angle θ
Calculate COG height

COG_height = (h × cos(θ)) / sin(θ)

Or using length L and height change h:

COG_height = (W_new × L) / (W_original × h_tilt)

Advantages:

Measures actual built robot
Includes all components and wiring
Accounts for manufacturing variations
Real-world accuracy

Disadvantages:

Destructive (may need to tilt robot)
Takes more time
Less precise than CAD
Requires measurement equipment

Plumb Line Method

For small robots (< 20 kg):

Procedure:

Suspend robot from known point A (e.g., top corner)
- Use string or rope
- Robot hangs freely
Hang plumb line from same point
- Marks vertical reference
- Mark this line on robot with tape
Suspend from different point B (e.g., opposite corner)
- Repeat with plumb line
- Mark second vertical line
Find intersection
- Draw both lines on robot
- Intersection = COG location!

For 3D verification:

Suspend from 3rd point at different height to confirm Z-coordinate.

Advantages:

Simple, uses basic tools
Works for complex shapes
Physically intuitive
Can validate calculations

Disadvantages:

Only works for relatively small robots
Not suitable for large/heavy robots
Time-consuming
Labor-intensive

COG Checklist for Stable Robots

Stability Design Checklist

During Robot Design:

✓ Calculate target COG position (before building!)
✓ Plan component placement for low COG
✓ Sketch support polygon relative to expected COG
✓ Document target stability margin (20% typical)

After Assembly:

✓ Measure actual COG position
✓ Compare to design target (± 5% tolerance)
✓ Verify stability with physical test
✓ Document final COG for control algorithms

For Operation:

✓ Know COG position during different configurations
✓ Apply dynamic COG offsets in motion planning
✓ Monitor payload effects on COG
✓ Reduce speed for high-COG configurations
✓ Include anti-tip mechanisms if needed

Special Conditions:

✓ Test on inclines (especially with arm extended)
✓ Verify turning stability at max speed
✓ Check braking/acceleration behavior
✓ Validate with loads off-center

Summary

Center of Gravity determines robot stability:

✓ Lower COG = More stable, faster turning, better off-road performance ✓ Centered COG = Balanced handling, symmetric characteristics ✓ Dynamic COG shifts = Account for accelerations and extensions ✓ Stability margin > 20% = Safe, predictable operation ✓ Know your COG = Design stable, reliable robots

Quick Design Process:

Calculate required stability margin (20-30%)
Design support polygon large enough
Plan component placement for low, centered COG
Verify through calculation or measurement
Account for payload and dynamic shifts
Test on target terrain before deployment

Center of Gravity and Stability

Lowering the COG

Centralizing the COG

Dynamic COG Management

Multi-Mode Configurations

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