Center of Gravity and Static Tipping Angle Calculator

Calculate a combined center of gravity, its gravity-line intersection on a slope and directional static tipping angles. This model compares rigid assemblies with fixed masses and contacts; it does not calculate sliding, dynamic motion or individual support loads.

Try this example: a 40 kg body centered at (0, 0, 250) mm plus a 10 kg payload at (180, 0, 750) mm gives a combined center of gravity of (36, 0, 350) mm. With a 600 mm × 500 mm support rectangle, the static tipping threshold towards +x is about 37.03°. At a 20° slope in that direction, the gravity line meets the support plane at (163.39, 0) mm, leaving 136.61 mm to the +x edge. These are illustrative calculations, not measurements of a tested machine.

Compare payload positions and support dimensions

Represent the body, battery, fixed arm and payload as 1–8 masses. Enter each component’s mass and its own center-of-gravity coordinates. Count each mass once: if the battery is already included in the body mass, do not add it again. Moving a payload outward shifts the combined center of gravity towards it. Raising it makes the gravity-line intersection move further on a slope. Change one dimension at a time to compare an arrangement with a lower load or a wider support area.

Select Use current conditions as baseline, then change the payload’s height or position. Keep inclination, downhill direction and support dimensions unchanged when comparing only a change in mass position. Selecting an example replaces the full set of inputs, so check the recorded conditions in the comparison table as well as the results. The baseline lasts only while the current tab stays open.

Use the rectangle formed by the contacts that can actually transmit support forces, rather than the outside dimensions of a cover or chassis. For four point contacts, use their support rectangle. A lifted wheel, three-point contact or a trapezoidal footprint needs a different support model. Anchors, guy wires and other external restraints are outside this calculation.

Measure every position from the support center

The origin is the center of the support rectangle. The x axis follows its length, y follows its width, and z is height perpendicular to the support plane. These coordinates stay attached to the rigid assembly as it tilts. Do not enter the component’s vertical height in world coordinates after tilting. Downhill heading φ is 0° towards +x, 90° towards +y, 180° towards −x and 270° towards −y.

Slope θ is measured from horizontal. At 0°, the gravity-line intersection is (x̄, ȳ). On a slope, use the line running downwards with gravity, rather than a perpendicular projection onto the support plane. The Minimum signed edge margin is the smallest signed margin to the four edges: positive inside, zero on the boundary and negative outside. It is not the straight-line distance to an outer corner. The angular margin is the selected heading’s critical angle minus the current slope.

How the center of gravity and tipping angle are calculated

Total mass M = Σmᵢ, with x̄ = Σ(mᵢxᵢ)/M, ȳ = Σ(mᵢyᵢ)/M and z̄ = Σ(mᵢzᵢ)/M. Each component is represented by its mass at its own center of gravity. In a uniform gravitational field, the center of mass and center of gravity coincide. OpenStax’s center-of-mass equations 9.29–9.32 give the mass-weighted calculation.

The gravity line meets the support plane at qₓ = x̄ + z̄ tanθ cosφ and qᵧ = ȳ + z̄ tanθ sinφ. For length L and width W, the intersection must satisfy −L/2 ≤ qₓ ≤ L/2 and −W/2 ≤ qᵧ ≤ W/2 to remain within the rectangle. The minimum signed edge margin is min(L/2 − |qₓ|, W/2 − |qᵧ|); it becomes zero when the intersection reaches the boundary from inside. OpenStax’s conditions for static equilibrium explain force and moment balance and gravity’s turning effect about a support boundary. The intersection equations here are derived from gravity’s components along and normal to the support plane.

When the level-ground projection starts inside the rectangle and z̄ > 0, let d be the distance along the selected downhill heading to the first edge. Then θcrit = atan(d/z̄). In the example, d towards +x is 300 − 36 = 264 mm. The nearest edges on level ground are instead the ±y edges, 250 mm away. The 250 mm minimum signed edge margin and the 264 mm directional distance describe different margins. Changing the heading can change both the first edge reached and the critical angle.

If the level-ground projection starts on or outside a boundary, there is no tipping margin starting from level. A particular slope may move the gravity-line intersection back inside, but it does not change that starting condition. With zero center-of-gravity height and an interior projection, this idealized model has no finite tipping threshold below 90°. That does not establish contact or resistance to sliding at steep slopes.

What the gravity-only model includes

This calculation includes gravity acting on a rigid assembly with fixed mass positions and a fixed support area. It excludes acceleration, braking, turning, impacts, vibration, liquid slosh, arm motion and structural deformation. It does not take a friction coefficient or predict sliding. OpenStax’s discussion of inclined planes treats gravity components and the limit of static friction separately. Insufficient friction can allow sliding before the tipping limit is reached. Do not use the calculated critical angle as a machine’s permitted operating slope.

For an arm held at a particular pose, use the robot arm static torque calculator to examine joint gravity torque, then the motor-sizing guide for duty cycle and thermal checks. The parallel-gripper force calculator addresses the separate problem of holding an object between two jaws. A combined center of gravity also does not uniquely determine each foot’s load: see why four supports can satisfy equilibrium with different load distributions.