Gravity Torque Calculator
Solve the single-link gravity torque relation τ = m · g · L · cos(θ) for any of its five unknowns — torque, mass, gravity, link length, or joint angle — with step-by-step math, a live arm schematic, and companion robotics metrics.
Maximum torque at horizontal
The gravity torque is maximum when the link is horizontal (θ = 0°), because the lever arm — the horizontal distance from the joint to the payload — is then equal to the full link length L. As the link rotates upward, the lever arm shrinks as L·cos(θ), so the torque falls. At θ = 90° (vertical up) the payload is directly above the joint, the lever arm is zero, and the gravity torque vanishes. This is why robot arms can hold heavy payloads aloft with little effort, but struggle to hold them straight out.
Mass and link length multiply torque
Torque is directly proportional to both the payload mass and the link length. Doubling the payload doubles the torque; doubling the link length also doubles the torque. This is the fundamental reason long-reach robots need large, powerful joints at the base and lighter, weaker joints near the wrist. In practice, the mass that matters is not only the payload but everything attached to the link — the gripper, the tooling, and the link's own mass. For a uniform link, the effective mass is the payload plus half the link mass, concentrated at the link's center of gravity.
Static holding torque and dynamic torque
The gravity torque τ = mgL·cos(θ) is the static torque needed just to hold the arm still against gravity at angle θ. To accelerate the arm, the motor must supply additional torque to overcome inertia: τ_total = τ_gravity + I·α + τ_friction. The peak torque requirement often occurs during acceleration from a horizontal position, when both gravity and inertia demand torque in the same direction. Motor sizing must account for both, plus a safety factor of 1.5–2.0.
Counterbalancing reduces required torque
Many robot arms use counterweights, springs, or gas struts to partially cancel gravity torque, reducing the motor's continuous torque requirement and hence its size and cost. A counterweight of mass m_c at distance L_c opposite the link creates a torque m_c·g·L_c·cos(θ) that opposes the payload torque, reducing the net torque by that amount at every angle. A perfectly balanced arm needs zero holding torque at any angle — the motor only provides acceleration and friction torque. This is why industrial arms often include spring or pneumatic counterbalance systems.
How to Use This Calculator
The gravity torque relation links five quantities: τ, m, g, L, and θ. g is fixed at 9.80665 m/s², and the tool solves for whichever of the other four you choose.
1. Choose what to solve for
Pick τ, m, L, or θ from the toggle. The calculator rearranges τ = m·g·L·cos(θ) automatically for whichever you choose.
2. Use realistic presets
Presets cover typical values: torque from 1 to 500 N·m, mass from 0.5 to 200 kg, link length from 0.2 to 4 m, and joint angle from 0° to 90°.
3. Match units carefully
Use SI units throughout: τ in N·m, m in kg, g in m/s², L in m, θ in degrees. The calculator converts degrees to radians internally. If θ is given in radians, multiply by 57.2958 to convert to degrees first.
4. Read the result and verdict
The result card shows the solved value plus equivalent units. A colour-coded verdict classifies the torque from very low (tiny arm) through moderate (robot joint) to very high (large industrial arm).
5. Check the schematic and charts
The schematic shows the arm at the specified angle, the gravity vector, the joint angle arc, and the torque at the joint. The chart plots torque against joint angle for three m·L products.
6. Switch to Advanced for depth
Advanced mode adds a chart of τ vs. m, a sensitivity check on a +1% input change, and companion metrics including static holding torque, dynamic torque, counterbalance mass, and actuator power.
What This Tool Offers
Built around the single-link gravity torque relation τ = m·g·L·cos(θ), the fundamental link between payload, geometry, and joint torque.
Four-way solving
Solve for torque, mass, link length, or joint angle. The calculator rearranges the formula for whichever variable you choose.
Clean arm schematic
A single-link arm at the specified angle with the payload, gravity vector, joint angle arc, and torque at the joint.
Torque-class verdict
Each result is classified from very low (tiny arm) through moderate (robot joint) to very high (large industrial arm), with a coloured badge.
Multi-unit hints
Each field shows the value converted to related units — ft·lbf and kgf·m for torque; lb and g for mass; cm and ft for length; radians for angle.
Dual comparison charts
τ vs. θ for three m·L products, plus an advanced τ vs. m chart for three L·cosθ values.
Reference tables
Typical torques by robot class, cos(θ) values for common angles, and a full variable reference table.
Reference Tables & Conversions
A torque unit converter, a τ lookup grid, and typical robot arm values.
Torque unit converter
Gravity Torque Lookup at g = 9.80665 m/s², L = 0.5 m
Computed from τ = m·g·L·cos(θ). Values in N·m.
| m (kg) | θ = 0° | θ = 30° | θ = 45° | θ = 60° | θ = 90° |
|---|
Typical Gravity Torque by Robot Class
Representative values at θ = 0° (worst case).
| Robot class | Payload m (kg) | Reach L (m) | Max τ (N·m) | Notes |
|---|---|---|---|---|
| Desktop arm (Dobot-class) | 0.5 | 0.3 | 1.5 | Educational, light payloads |
| Small 6-axis arm | 3 | 0.5 | 14.7 | Small assembly tasks |
| Medium 6-axis arm | 10 | 1.0 | 98.1 | General industrial |
| Large 6-axis arm | 50 | 2.0 | 981 | Heavy handling |
| Palletizing robot | 200 | 2.5 | 4,903 | Bag / box palletizing |
| Delta robot (per arm) | 1 | 0.4 | 3.9 | High-speed pick & place |
| SCARA arm | 5 | 0.6 | 29.4 | Horizontal assembly |
| Humanoid arm | 2 | 0.6 | 11.8 | Research humanoid |
| Exoskeleton shoulder | 8 | 0.3 | 23.5 | Assistive device |
cos(θ) Values for Common Angles
| Angle θ | cos(θ) | Torque factor | Notes |
|---|---|---|---|
| 0° | 1.000 | 100% | Horizontal — maximum torque |
| 15° | 0.966 | 96.6% | Near-horizontal |
| 30° | 0.866 | 86.6% | Common working angle |
| 45° | 0.707 | 70.7% | Mid-range |
| 60° | 0.500 | 50.0% | Steep angle |
| 75° | 0.259 | 25.9% | Near-vertical |
| 90° | 0.000 | 0% | Vertical — zero torque |
| −45° | 0.707 | 70.7% | Below horizontal (|cos| same) |
Variable Reference
| Symbol | Meaning | Typical range | Units |
|---|---|---|---|
| τ | Gravity torque | 0.1 – 10,000 | N·m |
| m | Payload mass | 0.1 – 1,000 | kg |
| g | Gravitational acceleration | 1.62 – 9.83 | m/s² |
| L | Link length | 0.05 – 5 | m |
| θ | Joint angle from horizontal | −180 – 180 | degrees |
| I | Moment of inertia of arm | — | kg·m² |
| α | Angular acceleration | — | rad/s² |
What Does τ = m · g · L · cos(θ) Measure?
The gravity torque relation gives the torque required at a robot joint to hold a single-link arm against gravity at a given joint angle. It is the fundamental static loading case for any manipulator and the starting point for actuator sizing, counterbalance design, and trajectory planning. The formula captures the essential physics: the payload's weight creates a moment about the joint, and the moment arm is the horizontal projection of the link, L·cos(θ).
Gravity Torque Formula
The defining relationship for a single-link arm is:
Rearranged to solve for the other unknowns:
L = τ / ( m · g · cos(θ) )
θ = arccos( τ / ( m · g · L ) )
When the link's own mass is significant, the effective mass at the tip becomes:
The total joint torque during motion adds the inertial term:
The counterbalance mass needed to cancel gravity torque at all angles is:
Gravity Torque Variables
| Symbol | Meaning | Notes |
|---|---|---|
| τ | Gravity torque | Torque needed to hold arm against gravity |
| m | Payload mass | Includes payload, gripper, tooling, and part of the link |
| g | Gravitational acceleration | 9.80665 m/s² on Earth; lower on Moon and Mars |
| L | Link length | Distance from joint to payload center of mass |
| θ | Joint angle | From horizontal; 0° = horizontal, 90° = vertical up |
| I | Moment of inertia | Added for dynamic torque calculation |
| α | Angular acceleration | Added for dynamic torque calculation |
How Each Variable Affects Gravity Torque
Gravity torque is directly proportional to mass, gravity, and link length, and follows a cosine dependence on joint angle. The most important design implication is that the torque required to hold a payload is largest when the arm is horizontal and smallest when it is vertical. This is why robot controllers often specify different speed and acceleration limits for horizontal and vertical arm configurations, and why some arms park in a vertical position to minimize holding torque and energy consumption.
In real manipulators, several additional factors modify the simple formula. The link's own mass adds to the effective payload, typically contributing about half the link mass at the tip for a uniform link. Off-axis loads, tooling, and cable harnesses add further torque. The gearbox reduces the torque seen by the motor by its ratio, but introduces friction and backlash. Counterbalance springs or weights can cancel much of the gravity torque, allowing a smaller motor for the same payload. And during acceleration, the inertial torque I·α can exceed the gravity torque, especially for fast moves with light payloads.
Gravity Torque Calculation Examples
τ = 10 × 9.80665 × 0.5 × cos(45°) = 49.03 × 0.7071 = 34.60 N·m
m = 34.60 / (9.80665 × 0.5 × 0.7071) = 34.60 / 3.467 = 9.98 kg
L = 34.60 / (10 × 9.80665 × 0.7071) = 34.60 / 69.34 = 0.499 m
cos θ = 34.60 / (10 × 9.80665 × 0.5) = 34.60 / 49.03 = 0.7057
θ = arccos(0.7057) = 45.1°
τ = 5 × 9.80665 × 0.8 × 1 = 39.23 N·m
τ = 5 × 9.80665 × 0.8 × 0 = 0 N·m
Frequently Asked Questions
Because torque is the product of force and lever arm, and the lever arm is the perpendicular distance from the joint to the line of action of the force. The weight acts vertically downward, so the relevant lever arm is the horizontal distance from the joint to the payload, which is L·cos(θ). When the arm is horizontal (θ = 0°), the lever arm equals the full link length and the torque is maximum. When the arm is vertical (θ = ±90°), the line of action of the weight passes through the joint, the lever arm is zero, and the torque vanishes. The cosine function captures exactly this geometric projection.
Yes, if the link mass is significant. For a uniform link of mass m_link, the center of mass is at L/2 from the joint, so its gravity torque is (m_link/2)·g·L·cos(θ) — exactly as if half the link mass were concentrated at the tip. The total torque is then τ = (m_payload + m_link/2)·g·L·cos(θ). If the link has a non-uniform mass distribution or additional components (motors, sensors, cable carriers), compute the center of mass and effective mass from the actual layout. In many small arms, the link mass is small compared to the payload and can be neglected; in large arms it is often the dominant term.
Static torque is what the actuator must supply to hold the arm motionless against gravity at a given angle: τ_static = m·g·L·cos(θ). Dynamic torque includes the additional torque needed to accelerate the arm and payload: τ_dynamic = τ_static + I·α + τ_friction, where I is the moment of inertia of the arm about the joint, α is the angular acceleration, and τ_friction includes viscous and Coulomb friction in the joint and gearbox. The peak torque requirement usually occurs during acceleration from a horizontal position, when gravity and inertia demand torque in the same direction. Motor sizing must use the peak dynamic torque, with a safety factor of 1.5–2.0.
A counterbalance is a spring, weight, or pneumatic cylinder arranged to produce a torque that opposes the gravity torque at every angle, so the actuator has less to supply. For a counterweight of mass m_c at distance L_c on the opposite side of the joint, the counter-torque is m_c·g·L_c·cos(θ) — the same cosine dependence as the payload torque, so it cancels at every angle if m_c·L_c = m·L. A spring counterbalance uses the fact that a spring's torque depends on its extension, which can be arranged to follow the cosine as well. Counterbalances reduce the continuous torque rating required of the motor, allowing a smaller, cheaper actuator, and they reduce energy consumption during holding.
Motor sizing for a single-link arm proceeds in four steps. First, compute the peak gravity torque from τ_g = m·g·L·cos(θ), using the maximum value over the working range (usually at θ = 0°). Second, compute the peak inertial torque from τ_i = I·α, where I is the moment of inertia of the arm plus payload about the joint (for a point mass at the tip, I = m·L²) and α is the maximum angular acceleration. Third, add friction and any external loads. Fourth, apply a safety factor of 1.5–2.0 and select a motor whose peak torque rating exceeds the total, and whose continuous torque rating exceeds the RMS torque over the duty cycle. If the motor is geared, remember that the load torque is divided by the gear ratio and the load inertia by the gear ratio squared.
For a point mass at the end of a massless link, only the position of the payload's center of mass matters — its orientation does not. For a real payload with finite size, the center of mass position depends on the payload's orientation relative to the link, so rotating the payload about the link axis or about a wrist axis can change the effective lever arm and hence the gravity torque. This is why some robot tasks are easier with the payload in one orientation than another, and why wrist joints also need gravity torque calculations. In a full analysis, each joint's torque depends on the positions and masses of everything distal to it, computed via the Jacobian and the manipulator's dynamic equations.
Gravity torque is directly proportional to g, so a robot designed for Earth (g = 9.80665 m/s²) would experience only 16.5% of the torque on the Moon (g = 1.62 m/s²) and 37.8% on Mars (g = 3.71 m/s²). This means the same arm could lift about 6 times the payload on the Moon and 2.6 times on Mars before the motor reached its Earth-rated torque. Conversely, a motor sized for the Moon would be much smaller and lighter — a significant advantage for space robotics. However, other factors such as thermal environment, radiation, and lubrication differ, so space-rated motors are not simply scaled-down Earth motors.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| tau | Gravitational torque at the joint | N.m |
| m | Mass of the arm (concentrated at its center of mass) | kg |
| g | Gravitational acceleration | 9.81 m/s2 |
| L | Distance from the joint to the arm's center of mass | m |
| theta | Angle of the arm from horizontal | degrees |
What it means
For a single link of mass m, length L, and angle θ from horizontal, the gravitational torque about the joint is τ = m g L cosθ. This torque is caused by the weight acting at the centre of mass (assumed at the link centre). It is used in robot control to implement gravity compensation (feed‑forward) to reduce the torque required from motors, especially in static holding or slow motions. It is also used in the dynamic equations of robotic arms. Understanding gravity torque helps in designing controllers that can compensate for gravity and improve energy efficiency.
Worked example
Gravity Torque (Single‑Link Arm) – Two Detailed Examples
Real‑World| Parameter | Value |
|---|---|
| m (kg) | 1 |
| L (m) | 0.5 |
| θ (deg) | 0 |
| Parameter | Value |
|---|---|
| m | 1 |
| L | 0.5 |
| θ | 90 |
Common mistakes
- Gravity torque on a single‑link arm: τ = m·g·L·cos(θ) – where θ is the angle from the horizontal.
- Angle definition: θ measured from the horizontal (or from vertical) – check the sign convention.
- Units: m in kg, g in m/s², L in m → τ in N·m.
- Direction: The torque acts to rotate the arm toward the vertical downward position.
- Assumes: The mass is concentrated at the end of the link (point mass).
Applications
The gravity torque for a single‑link robot arm, τ = m·g·L·cos(θ), gives the torque due to gravitational load on the joint. This is essential for feedforward control in robots to compensate for the static load caused by gravity. By adding this torque to the control signal, the motor can maintain a position without excessive effort. Engineers use this to improve positioning accuracy and to reduce energy consumption. It is also used in calculating the static balance and in sizing actuators for arms operating in different orientations. This formula is a fundamental part of robot dynamics modelling.
- Feedforward compensation for gravity in robot arms
- Static torque calculation for actuator sizing
- Balancing and counterweight design for manipulators
- Energy‑efficient control of robotic systems
- Simulation of robotic arm dynamics
Related calculators · Kinematics
Recently updated formulas
Suggest an improvement
Found a mistake? Have a better example? Let us know.
You may send up to 3 suggestions per 24 hours.