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Robotics

Pendulum-driven locomotion

1996ActivePublished: 4 September 2026Updated: 4 September 2026Published
Key innovation
Propelling and steering a rolling sphere solely by shifting the internal center of mass — with no wheels, legs, or exposed drive parts outside the shell.
Category
Robotics
Abstraction level
Pattern
Operation level
Robot control
Use cases
Inspection and patrol robots in rough terrainExploration and monitoring in dusty or wet environmentsAmphibious robots with a fully sealed shellEducational and consumer robots (spherical platforms)Research on nonholonomic and underactuated system dynamics

How it works

A motor rotates the axle on which the pendulum weight is suspended. When the pendulum swings forward, the system's center of mass moves ahead of the vertical axis through the sphere's ground-contact point. Gravity acting on the offset mass creates a torque about the contact point, rolling the shell toward the swing direction. Holding a constant offset yields continuous motion, while reversing the offset side reverses travel. In single-axis systems the pendulum moves only in the driving plane (fore/aft). In two-axis systems a second axle shifts the mass sideways: the sphere tilts and changes its rolling path, enabling steering. Attainable speed and acceleration depend on the pendulum's mass and arm length, which set the maximum available torque. The system is nonholonomic and underactuated, so precise control requires sensor feedback (e.g. an IMU).

Problem solved

How to propel and steer a fully sealed spherical robot without protruding wheels, legs, or exposed drive parts, while keeping the shell tight and resistant to dust, water, and impacts.

Components

Pendulum (weight/bob)Carrier of the shifting mass that generates driving torque

A movable mass suspended inside the shell whose deflection shifts the system's center of gravity.

Drive motorActuator controlling the pendulum position

Motor that rotates the pendulum axle and holds its deflection; the source of torque controlling the mass position.

Official

Drive axle / shaftPower transmission and definition of the mass degrees of freedom

Element linking the motor to the pendulum; defines the plane in which the mass moves (one axle for fore/aft travel, a second for steering).

Single-axis configurationOne axle: the pendulum moves only in the driving plane; steering is indirect or limited.
Two-axis configurationA second axle enables sideways mass shifting and direct steering by tilting the sphere.

Official

Spherical shellGround-contact interface and protective enclosure

The outer, usually sealed sphere that rolls on the ground and protects the robot internals.

Control and power unitMotion control and power supply

Controllers, sensors (e.g. IMU), and a battery enclosed inside, providing the feedback needed for the underactuated dynamics.

Official

Implementation

Implementation pitfalls
Limited torque on slopes and soft groundMedium

Driving torque is bounded by the pendulum mass times arm length, so the robot struggles on steep slopes and high-rolling-resistance ground.

Fix:Increase pendulum mass/arm or reduce rolling resistance; plan routes that avoid steep slopes.
Coupling of drive and steeringMedium

In single-axis systems the same mass handles both driving and heading changes, complicating independent control and increasing controller complexity.

Fix:Use a two-axis variant or an additional steering mechanism.
Pendulum oscillation when stoppingLow

Underactuated dynamics cause residual pendulum swing and shell wobble when stopping or changing speed.

Fix:Oscillation-damping control based on IMU feedback.

Evolution

Original paper · 1996 · IEEE AMC 1996 (4th Int. Workshop on Advanced Motion Control) · Aarne Halme
Motion control of a spherical mobile robot
Aarne Halme, Torsten Schönberg, Yan Wang
1996
Motion control of a pendulum-driven spherical robot
Inflection point

Halme, Schönberg, and Wang publish early work on motion control of a spherical robot based on center-of-mass shifting.

2015
Center-of-mass-shift spherical robots reach consumer robotics

Commercial spherical platforms (e.g. Sphero BB-8) popularized locomotion based on internal mass shifting.

Hyperparameters (configurable axes)

Pendulum massHigh

Greater mass increases available driving torque (and climbable slopes) but raises energy use and inertia.

Pendulum arm lengthHigh

A longer arm increases torque for the same deflection but is bounded by the shell's inner diameter.

Number of control axesCritical

One axis enables fore/aft travel; a second axis adds direct steering via sideways mass shifting.

1Single-axis variant (simplified control).
2Two-axis variant (drive + steer).
Maximum deflection angleMedium

The upper range of pendulum deflection bounds the maximum torque and attainable acceleration.