Mobile Robots and Locomotion

Introduction

A mobile robot is a robotic system that changes its position and orientation by generating forces against its environment. Unlike a fixed manipulator, whose workspace is largely determined by its mounting structure, a mobile robot must solve an integrated problem involving mechanical design, terrain interaction, sensing, control, energy, and navigation. Its locomotion system determines not only how fast it can move, but also the surfaces it can cross, the payload it can carry, the accuracy of its motion estimate, and the risks it presents to people and infrastructure.

Mobile robots include wheeled vehicles, tracked platforms, legged machines, aerial vehicles, underwater vehicles, and hybrid systems. The appropriate platform depends on terrain, payload, endurance, required manoeuvrability, environmental hazards, and mission objectives. A warehouse robot may require high efficiency and precise floor navigation, whereas an agricultural robot may need low ground pressure and obstacle tolerance. An inspection robot for a pipeline may need to climb, fly, or swim rather than roll.

This chapter develops models for common mobile platforms, explains locomotion constraints and stability, introduces odometry and its limitations, and presents a framework for platform selection.

Learning Objectives

By the end of this chapter, the reader should be able to:

1. Classify wheeled, tracked, legged, aerial, and underwater robots.
2. Model planar motion using pose, velocity, and nonholonomic constraints.
3. Relate wheel speed, vehicle geometry, and body motion for common drive systems.
4. Explain traction, slip, stability, ground pressure, and energy requirements.
5. Derive and use basic wheel odometry equations.
6. Compare locomotion technologies for terrain, payload, and mission requirements.
7. Identify sources of modelling error and select appropriate sensing and control methods.
8. Evaluate safety, environmental, and professional implications of mobile robot deployment.

1. Locomotion as a Robotic Function

Locomotion is the controlled generation of motion through interaction with a supporting medium. For ground robots the medium is usually soil, pavement, or a manufactured floor. Aerial robots interact primarily with air, while underwater robots interact with water and buoyancy forces.

A mobile robot normally contains:

• a mechanical body and locomotion actuators;
• a power source and power electronics;
• sensors for motion, terrain, and obstacles;
• a controller that converts motion objectives into actuator commands;
• communication and computing systems; and
• safety mechanisms such as emergency stop, collision detection, or return-to-safe-state behaviour.

The robot pose describes its location and orientation. In a planar environment, pose is commonly written as q = [x, y, θ]^T, where x and y are Cartesian coordinates and θ is heading angle. The pose belongs to a configuration space: a mathematical representation of all positions and orientations available to the robot. A robot may have several actuated joints or wheels, but its useful motion can still be described by a smaller number of pose variables.

A central distinction is between holonomic and nonholonomic motion. A holonomic robot can command independent instantaneous motion in all configuration directions. A conventional car-like robot is nonholonomic: it can move forward or backward and turn, but it cannot translate directly sideways without changing its orientation. Nonholonomic constraints strongly influence path planning and control.

2. Wheeled Mobile Robots

Wheeled vehicles are the most energy-efficient ground platforms on firm, reasonably smooth surfaces. Their advantages include mechanical simplicity, high speed, low power consumption, and relatively straightforward control. Their principal limitations are poor performance on steps, loose soil, deep obstacles, and highly irregular terrain.

2.1 Differential-drive robots

A differential-drive robot has two independently driven wheels separated by a track width b. Let the left and right wheel radii be r, and let their angular velocities be ω_L and ω_R. Assuming pure rolling, no lateral slip, and equal wheel radii, the forward velocity v and angular velocity Ω of the robot body are:

Here v is measured in metres per second, Ω in radians per second, wheel angular velocities in radians per second, and b in metres. Equal wheel speeds produce straight motion. Opposite equal wheel speeds produce rotation about the midpoint between the wheels. The instantaneous centre of curvature has radius:

provided Ω ≠ 0. Substituting the wheel equations gives:

These equations assume rigid wheels, a level contact model, negligible wheel deformation, and rolling without slip. On low-friction or uneven surfaces, the commanded and actual velocities may differ substantially.

The planar kinematic model is:

The dot denotes time derivative. The model shows why a differential-drive robot cannot instantaneously move sideways: velocity is constrained to the direction of its longitudinal axis.

2.2 Unicycle and bicycle models

The differential-drive robot is often represented by an equivalent unicycle model with state (x,y,θ) and inputs (v,Ω). This abstraction is useful for path planning and control because it removes wheel-level details while retaining the principal nonholonomic behaviour.

A car-like robot is commonly represented by a bicycle model. If L is the wheelbase and δ is the steering angle, then, under small-slip assumptions:

The turning radius is:

The bicycle model is suitable for front-steered vehicles whose rear wheels approximately follow the vehicle centreline. It is not accurate during severe skidding, high-speed tyre slip, or independent-wheel steering manoeuvres.

2.3 Omnidirectional and mecanum platforms

Omnidirectional platforms use special wheel arrangements, such as Swedish or mecanum wheels, to generate motion in multiple planar directions. In contrast with differential drive, they can often command forward, lateral, and rotational velocity simultaneously. This is valuable in factories, hospitals, and narrow workspaces.

The benefits are manoeuvrability and precise positioning. The disadvantages include reduced traction, greater sensitivity to floor discontinuities, more complex wheel-speed transformations, and increased mechanical cost. The ideal kinematic model assumes each roller or wheel maintains the intended contact direction. Uneven floors and contaminated surfaces violate this assumption and may cause lateral error.

3. Tracks and Articulated Ground Platforms

Tracked vehicles distribute load over a larger contact area than wheeled vehicles. This reduces ground pressure and improves performance on sand, mud, snow, and rubble. Tracks also provide a continuous contact surface, allowing the robot to cross some gaps and obstacles that would stop a small wheeled robot.

A simple skid-steer tracked robot uses independently driven left and right tracks. Its kinematic equations resemble those of a differential-drive robot, but turning normally requires lateral slip between the tracks and ground. Consequently, the ideal no-slip equations are only approximate. The actual turning radius depends on track geometry, soil properties, normal load, speed, and friction.

Track-ground interaction involves shear deformation. When a track turns, parts of the contact patch may move relative to the soil. This produces energy loss and can damage sensitive surfaces. Tracked robots therefore have excellent terrain access but may consume more energy than wheeled platforms on firm floors.

Articulated tracked robots divide the body into linked modules. Articulation allows the platform to conform to steps, rocks, and uneven terrain. The additional joints increase mechanical complexity, control requirements, and maintenance demands. They are common in search-and-rescue, planetary exploration, and hazardous-environment inspection.

4. Legged Locomotion

Legged robots make intermittent or continuous contact with the ground through feet. Their major advantage is terrain adaptability: a foot can be placed on selected support locations rather than requiring a continuous smooth path. Legs can also lift the body over obstacles and vary body height.

A legged robot operates through a sequence of support phases. In a static gait, the centre of mass remains within the support polygon formed by the feet in contact with the ground. If the projection of the centre of mass lies outside this polygon, the robot tends to tip unless dynamic effects, external support, or active acceleration prevent it.

For a quadruped, common gaits include walk, trot, pace, and bound. A walk maintains at least three supporting legs under normal conditions, giving a large support polygon. A trot moves diagonal leg pairs together and can provide efficient dynamic motion but requires balance control. A hexapod can maintain a stable tripod while moving another tripod, which gives useful static stability on irregular terrain.

Legged robots require coordination of body motion, foot placement, joint torques, and contact forces. Their disadvantages include high energy consumption, complex actuators, impact loads, and difficult state estimation. Foot slippage and uncertain contact make purely kinematic models unreliable; force sensing and compliant control are often necessary.

5. Aerial and Underwater Platforms

5.1 Aerial robots

Multirotor aerial vehicles generate lift through rotating propellers. A quadrotor controls its motion by varying the thrust of four rotors. Total upward thrust must approximately balance weight during hover:

where T is total thrust in newtons, m is vehicle mass in kilograms, and g is gravitational acceleration, approximately 9.81 m/s² near Earth’s surface. For manoeuvring, thrust is tilted by changing vehicle attitude, producing horizontal force. Aerial robots have access to three-dimensional space and can inspect roofs, towers, crops, and disaster sites. Their limitations include short endurance, sensitivity to wind, acoustic disturbance, restricted payload, and safety risks from rotating propellers and falling objects.

A fixed-wing aerial robot is more energy-efficient during forward flight and can cover large areas, but it generally cannot hover and requires launch and recovery space. Hybrid vertical-take-off-and-landing aircraft combine both capabilities at the cost of additional complexity.

5.2 Underwater robots

Remotely operated vehicles are connected to a surface vessel by a tether that supplies power and communication. Autonomous underwater vehicles carry their own energy and execute missions with limited communication. Both types operate under pressure, buoyancy, hydrodynamic drag, and uncertain visibility.

The buoyant force is given by Archimedes’ principle:

where F_B is buoyant force in newtons, ρ_w is water density in kilograms per cubic metre, V is displaced volume in cubic metres, and g is gravitational acceleration. If buoyant force exceeds weight, the vehicle accelerates upward; if weight exceeds buoyancy, it accelerates downward, subject to hydrodynamic forces. Many underwater robots are designed to be neutrally buoyant, so small control forces can change depth.

Water produces significant drag. A simplified drag magnitude is:

where C_D is a dimensionless drag coefficient, A is reference area in square metres, and v is speed in metres per second. This relationship applies to steady or quasi-steady flow and is an approximation when complex unsteady flow, appendages, or wake effects are present.

6. Traction, Slip, and Ground Pressure

A mobile robot can move only if the contact forces required by the motion are within the available friction or soil shear strength. On a firm surface, the maximum ideal friction force is approximately:

where μ is the coefficient of friction and N is normal force. This is an idealized Coulomb-friction model. The coefficient varies with material, contamination, speed, temperature, and contact pressure.

Longitudinal slip can be defined differently for driving and braking. For a driven wheel moving at ground speed v and having effective rolling radius r, one common driving-slip definition is:

where s is dimensionless and rω is the peripheral speed. Zero indicates ideal rolling; positive s indicates wheel spin. Definitions must be stated because braking slip is often defined with a different denominator.

Average ground pressure is:

where p is pressure in pascals, W is supported weight in newtons, and A_c is total contact area in square metres. Lower pressure generally reduces sinking in soft ground, although soil behaviour is more complex than this average measure. Tracks often outperform wheels in soft terrain because they enlarge A_c.

7. Stability and Tip-Over

Stability is the ability to resist overturning or loss of controlled contact. For a stationary ground robot, the projection of the centre of mass should remain within the support polygon. During acceleration, braking, turning, or climbing, inertial forces shift the effective load.

For a simple robot on a slope of angle α, the gravitational component parallel to the slope is:

and the normal component is:

These expressions neglect suspension dynamics and other forces. A robot must generate sufficient traction to overcome the parallel component without excessive slip. Tip-over risk also depends on centre-of-mass height, track width, wheelbase, turning acceleration, and terrain irregularity.

Dynamic legged robots cannot be analysed solely by static support polygons. Their momentum, foot contact timing, compliance, and feedback control are important. Stability margins should therefore be evaluated under expected motion rather than only in a stationary configuration.

8. Odometry and State Estimation

Odometry estimates robot motion by integrating measured wheel rotations, joint positions, inertial measurements, or other local motion signals. It is inexpensive and available at high rate, but it accumulates error because even small velocity errors are integrated over time.

For a differential-drive robot over a short sampling interval, let Δs_L and Δs_R be distances travelled by the left and right wheels. The incremental forward displacement and heading change are:

Using the midpoint heading approximation, the pose update is:

The wheel distances are estimated from encoder counts. If an encoder produces n counts per wheel revolution and records Δc counts, then:

for an ideal wheel of radius r. Encoder quantization, incorrect radius, unequal wheel diameters, wheel slip, and an incorrect estimate of b create systematic or random errors.

A practical robot combines odometry with inertial, lidar, camera, satellite, or landmark observations. Sensor fusion methods such as the extended Kalman filter or factor-graph optimization can reduce drift. Odometry remains valuable because it provides local motion prediction between external observations.

9. Platform Selection Framework

Platform selection should begin with mission requirements rather than a preferred technology. Important criteria include terrain, required speed, payload, endurance, manoeuvrability, environmental exposure, localisation conditions, maintenance, cost, and safety.

A useful engineering process is to establish constraints first, reject infeasible platforms, score the remaining candidates against weighted criteria, and then validate the choice through prototype testing. A scoring matrix must not conceal hard constraints: a platform that cannot carry the payload or survive the environment should be rejected regardless of its score elsewhere.

Worked Example: Differential-Drive Motion

An indoor robot has wheel radius r = 0.10 m, wheel separation b = 0.50 m, left-wheel speed ω_L = 8 rad/s, and right-wheel speed ω_R = 10 rad/s. Assuming pure rolling, calculate its body velocity and turning radius.

The forward speed is:

The angular speed is:

Therefore the instantaneous turning radius is:

The robot moves forward while turning toward the left or right according to the wheel convention and sign definition. The result is valid only if wheel radius, wheel separation, and no-slip assumptions are sufficiently accurate.

Applied Case: Illustrative Warehouse Robot Selection

Background

A distribution centre needs a robot to transport 80 kg loads between storage aisles and packing stations. The floor is level concrete, aisles are narrow, and operation is indoors. Human workers and forklifts share parts of the route.

Problem

The design team is considering a differential-drive robot, a tracked robot, and a small quadruped. The robot must operate for an eight-hour shift with scheduled charging, stop safely near people, and reach packing stations within a positional tolerance of 50 mm.

Analysis

The firm floor does not require tracks or legs. Differential drive offers good efficiency, a small turning radius, and mature navigation software. Mecanum wheels could provide lateral manoeuvring, but their lower traction and sensitivity to floor gaps are unnecessary disadvantages. The payload affects motor torque, braking distance, structural strength, and battery capacity; it cannot be treated as a simple software parameter.

The design team estimates wheel torque from the required tractive force. A simplified force balance on level ground is:

where rolling resistance, acceleration demand, and grade resistance are included. For a level floor, F_grade is approximately zero. If total tractive force is distributed equally between two driven wheels, each wheel must provide approximately half the force, subject to traction and drivetrain efficiency.

Approach and solution

The team selects a differential-drive platform with rubber wheels, wheel encoders, an inertial measurement unit, lidar, safety scanners, and independent braking. It uses a local obstacle-avoidance controller but imposes a conservative speed limit near people. A map-based localisation system corrects encoder drift at known environmental features. Charging is scheduled during low-demand periods.

Safety validation includes emergency-stop tests, loss-of-communication tests, sensor-obstruction tests, braking-distance measurements, and tests with maximum payload. The robot is not approved merely because it follows a route in an empty aisle.

Outcome

The differential-drive platform meets the mobility and accuracy requirements at lower energy consumption and maintenance cost than the tracked alternative. The quadruped is rejected because its additional terrain capability does not contribute to this mission and its complexity increases cost and safety exposure.

Learning points

1. Platform capability should match actual environmental demands.
2. Payload changes mechanical, energetic, and safety requirements.
3. Odometry should be corrected by independent observations when accuracy matters.
4. Human-shared operation requires validated protective functions, not only navigation accuracy.
5. A technically impressive platform may be a poor engineering choice if its extra capability is irrelevant.

Control, Safety, and Professional Practice

Mobile robots operate near people, vehicles, animals, and fragile infrastructure. Engineers must identify hazards including collision, crushing, falling payloads, propeller contact, battery fire, loss of communication, unexpected restart, and uncontrolled motion after sensor failure.

Best practice includes speed and force limits, physical emergency stops, protective separation, fault monitoring, redundant or diverse sensing where risk justifies it, controlled software updates, and documented verification. A safety function should move the system to a defined safe state rather than assuming that ordinary mission software will always respond correctly.

Aerial and underwater operations may require licensing, restricted-zone compliance, environmental permissions, and operator qualification. Privacy is relevant when cameras or lidar record people or private property. Underwater robots can disturb habitats, entangle wildlife, or damage archaeological sites. Engineers should assess environmental impact, data governance, cybersecurity, accessibility, and accountability throughout the system life cycle.

Challenges and Limitations

No locomotion model is universally accurate. Wheel equations neglect deformation and slip; track models depend heavily on soil; legged models depend on contact timing and compliance; aerial models are affected by wind and aerodynamic coupling; underwater models require hydrodynamic parameters that are difficult to estimate.

Common practical limitations include:

• battery capacity and charging time;
• degradation of tyres, tracks, joints, and seals;
• poor localisation in repetitive or featureless environments;
• communication loss and electromagnetic interference;
• dust, water, temperature, and vibration;
• uncertainty in payload mass and centre of gravity; and
• mismatch between laboratory tests and field conditions.

Robust design therefore combines analytical models, simulation, hardware-in-the-loop testing, controlled field trials, and operational monitoring. Model uncertainty should be measured and included in safety margins.

Contemporary Developments

Recent mobile robots increasingly combine high-rate inertial sensing, lidar or vision, learned perception, and model-based control. Machine learning can assist terrain classification, visual odometry, foot placement, and obstacle recognition, but learned components should be tested against changing lighting, weather, surfaces, and unusual obstacles. They should not be treated as infallible safety mechanisms without evidence and fallback behaviour.

Soft actuators, variable-stiffness legs, articulated modular vehicles, autonomous docking, and multi-robot coordination are expanding the range of applications. Digital twins and simulation enable earlier testing of terrain and traffic scenarios, although simulation-to-reality differences remain important. Efficient autonomy depends not only on algorithms but also on reliable sensing, maintainable mechanics, and clear operational procedures.

Chapter Summary

Mobile robots generate motion through interaction with ground, air, or water. Wheeled robots are efficient on firm surfaces; tracked systems provide improved terrain access and lower ground pressure; legged robots negotiate discontinuous terrain but require complex contact control; aerial systems provide three-dimensional access; and underwater vehicles operate under buoyancy, drag, pressure, and communication constraints.

Differential-drive and bicycle models provide useful kinematic descriptions, but their assumptions must be stated. Traction, slip, ground pressure, centre-of-mass location, and support polygons determine whether a platform can move safely. Odometry converts encoder or motion measurements into an estimate of pose, but errors accumulate and should be corrected using independent sensors or landmarks.

Platform selection is a systems-engineering decision. Terrain, payload, speed, endurance, accuracy, safety, maintenance, regulation, and environmental effects must be evaluated together. The best platform is not the most sophisticated one; it is the one whose capabilities satisfy the mission with acceptable risk, cost, and lifecycle performance.

Key Terms

Mobile robot: A robot capable of changing its position or orientation.

Locomotion: Generation and control of movement through environmental interaction.

Pose: Position and orientation of a robot.

Configuration space: Mathematical space representing possible robot configurations.

Holonomic: Able to command independent motion in all configuration directions.

Nonholonomic: Subject to motion constraints that cannot be integrated into position constraints.

Differential drive: Drive system using independently controlled left and right wheels or tracks.

Wheelbase: Distance between the front and rear wheel axes of a vehicle.

Skid steering: Turning produced by different speeds of left and right wheels or tracks, involving lateral slip.

Omnidirectional drive: Locomotion allowing controlled motion in multiple planar directions.

Traction: Ability to transmit useful force without unacceptable slip.

Slip: Relative motion between a contact element and the supporting surface.

Ground pressure: Supported normal force divided by contact area.

Support polygon: Polygon formed by the ground contacts supporting a stationary legged robot.

Odometry: Motion estimation obtained by integrating local motion measurements.

Dead reckoning: Estimating current state from a previous state and measured motion without external correction.

Buoyancy: Upward force exerted by a fluid on a displaced volume.

Drag: Resistance force caused by motion through a fluid or medium.

Payload: Useful mass or equipment carried by a robot.

Sensor fusion: Combination of measurements from multiple sensors to estimate state or environment.

Review Questions

Short Answer Questions

1. What is the principal difference between holonomic and nonholonomic motion?
2. State two advantages and two limitations of wheeled robots.
3. What is meant by wheel slip?
4. Why does odometry drift over time?
5. Define the support polygon of a legged robot.

Long Answer Questions

1. Explain the differential-drive kinematic model and discuss its assumptions.
2. Compare wheeled, tracked, and legged locomotion for rough outdoor terrain.
3. Explain how buoyancy and hydrodynamic drag affect underwater robot design.
4. Discuss the factors that should be included in a mobile robot platform-selection process.
5. Describe a safety-engineering approach for a robot operating in a human-shared workspace.

Analytical / Application Questions

1. A differential-drive robot has wheel radius 0.08 m, wheel separation 0.40 m, and wheel speeds 6 rad/s and 9 rad/s. Calculate its forward speed, angular speed, and turning radius, stating your assumptions.
2. Select a locomotion platform for inspecting a damaged building containing stairs, loose rubble, and narrow passages. Justify the choice using terrain, stability, sensing, payload, and safety criteria.
3. An autonomous warehouse robot shows increasing lateral position error despite accurate encoder counts. Diagnose at least three possible causes and propose a validation and sensor-fusion strategy.

Multiple Choice Questions with Answers

Question 1

For a differential-drive robot with equal wheel speeds, what ideal motion results?

A. Pure lateral translation

B. Straight-line translation

C. Rotation about the left wheel

D. Vertical motion

Correct Answer: B. Straight-line translation

Question 2

Which assumption is required by the basic differential-drive equations?

A. The robot is airborne

B. The wheels have infinite mass

C. There is negligible lateral slip

D. The track width is zero

Correct Answer: C. There is negligible lateral slip

Question 3

A tracked robot commonly has lower ground pressure than a similar wheeled robot because it has:

A. A smaller contact area

B. A larger contact area

C. No normal force

D. No friction

Correct Answer: B. A larger contact area

Question 4

For a stationary legged robot, static stability generally requires the centre-of-mass projection to lie:

A. Outside the support polygon

B. At the highest foot

C. Inside the support polygon

D. At the geometric centre of every leg

Correct Answer: C. Inside the support polygon

Question 5

What is the principal weakness of dead-reckoning odometry?

A. It cannot use wheel measurements

B. Its error generally accumulates with distance travelled

C. It requires a global map for every update

D. It measures only battery voltage

Correct Answer: B. Its error generally accumulates with distance travelled

Question 6

A vehicle with total thrust approximately equal to mg is most nearly in which condition?

A. Hover or vertical force equilibrium

B. Free fall with zero gravity

C. Maximum forward acceleration

D. Submerged neutral buoyancy

Correct Answer: A. Hover or vertical force equilibrium

Question 7

Which platform is generally most efficient on a smooth, level warehouse floor?

A. A wheeled robot

B. A tracked excavator

C. A six-legged robot

D. An underwater vehicle

Correct Answer: A. A wheeled robot

Question 8

In the simplified friction model F_max = μN, what does μ represent?

A. Wheel radius

B. Angular velocity

C. Coefficient of friction

D. Robot mass

Correct Answer: C. Coefficient of friction

Question 9

Why are fixed-wing aerial robots less suitable than multirotors for stationary inspection?

A. They cannot normally hover

B. They cannot generate lift

C. They cannot move forward

D. They require underwater propulsion

Correct Answer: A. They cannot normally hover

Question 10

Which is the most appropriate engineering response to odometry drift in a navigation-critical robot?

A. Ignore the drift because encoders are exact

B. Use independent observations such as lidar, vision, or landmarks

C. Reduce the wheel separation to zero

D. Remove all inertial sensors

Correct Answer: B. Use independent observations such as lidar, vision, or landmarks