5.2 Inertial navigation
5 min read•august 20, 2024
Inertial navigation uses accelerometers and gyroscopes to track a vehicle's position and orientation. Unlike , it provides more accurate estimates but requires complex calculations and error compensation.
IMUs, consisting of accelerometers and gyroscopes, measure specific force and angular velocity. These measurements are integrated to determine position, velocity, and orientation, but errors accumulate over time, necessitating with other navigation systems.
Principles of inertial navigation
Dead reckoning vs inertial navigation
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- Dead reckoning estimates position based on previously known position, velocity, and time elapsed
- Inertial navigation uses accelerometers and gyroscopes to measure acceleration and angular velocity
- Inertial navigation provides more accurate position estimates compared to dead reckoning
- Dead reckoning is simpler and less expensive but prone to error accumulation over time
Frames of reference
- Inertial frame is a non-accelerating, non-rotating reference frame (Earth-centered inertial frame)
- Body frame is fixed to the vehicle or robot and moves with it
- Navigation frame is a local-level frame used for position and velocity representation (North-East-Down frame)
- Coordinate transformations between frames are essential for inertial navigation calculations
Equations of motion
- Newton's second law relates force, mass, and acceleration:
- Rotational equations of motion describe angular velocity and acceleration
- Specific force measured by accelerometers includes gravity and motion-induced accelerations
- Integration of acceleration and angular velocity yields position, velocity, and orientation
Inertial measurement units (IMUs)
Accelerometers
- Measure specific force along one or multiple axes
- Common types: mechanical, piezoelectric, and MEMS (microelectromechanical systems)
- output includes motion-induced accelerations and gravity
- Bias, scale factor, and are common error sources in accelerometers
Gyroscopes
- Measure angular velocity about one or multiple axes
- Common types: mechanical, optical, and MEMS
- output is integrated to obtain orientation changes
- Bias, scale factor, and noise are common error sources in gyroscopes
Magnetometers
- Measure the Earth's magnetic field to determine heading
- Provide absolute orientation reference in the horizontal plane
- Can be affected by magnetic disturbances and hard/soft iron effects
- Often used to complement gyroscope measurements for heading estimation
Error sources in inertial navigation
Bias errors
- Constant offset in the sensor output
- Can cause position and orientation errors to grow unbounded over time
- Bias estimation and compensation are crucial for accurate navigation
Scale factor errors
- Linear or nonlinear deviations in sensor sensitivity
- Cause errors proportional to the magnitude of the measured quantity
- Scale factor calibration is necessary to minimize these errors
Misalignment errors
- Angular offsets between sensor axes and the body frame
- Lead to cross-coupling of measurements and navigation errors
- Careful sensor alignment and calibration can reduce these errors
Random noise
- High-frequency fluctuations in sensor output
- Causes short-term navigation errors and limits the achievable
- Noise characteristics (power spectral density) are important for filter design
Inertial navigation algorithms
Strapdown integration
- Integrates body frame measurements to update position, velocity, and orientation
- Requires coordinate transformations between body, navigation, and inertial frames
- Accumulates errors over time due to sensor imperfections and numerical integration
Attitude representation
- Common methods: Euler angles, rotation matrices, and quaternions
- Euler angles (roll, pitch, yaw) are intuitive but suffer from gimbal lock
- Rotation matrices are computationally expensive but avoid singularities
- Quaternions are compact, efficient, and avoid gimbal lock
Position and velocity updates
- Acceleration measurements are transformed to the navigation frame and corrected for gravity
- Velocity is updated by integrating acceleration over time
- Position is updated by integrating velocity over time
- Earth's rotation and Coriolis effects must be accounted for in the navigation equations
Kalman filtering for inertial navigation
State estimation
- combines IMU measurements with a system model to estimate the navigation state
- Navigation state typically includes position, velocity, and orientation (attitude)
- Kalman filter provides optimal estimates in the presence of sensor noise and model uncertainties
Error state vs total state
- Error state Kalman filter estimates the errors in the navigation state
- Total state Kalman filter directly estimates the navigation state
- Error state formulation is more common due to better numerical stability and linearity
- Error state estimates are used to correct the navigation solution from the strapdown integration
Measurement updates
- External measurements (GPS, vision, lidar) are used to update the Kalman filter
- Measurement model relates the observations to the navigation state
- Kalman gain determines the weight given to the measurements based on their uncertainty
- Measurement updates help correct the in the inertial navigation solution
Inertial navigation applications
Autonomous vehicles
- IMUs provide high-frequency motion estimates for control and localization
- Inertial navigation aids GPS during outages and in challenging environments (urban canyons, tunnels)
- Sensor fusion with cameras, lidar, and GPS improves overall navigation performance
Robotics
- IMUs enable attitude estimation and stabilization for mobile robots and drones
- Inertial navigation supports odometry and mapping in GPS-denied environments
- Lightweight and low-cost MEMS IMUs are widely used in robotic applications
Aerospace systems
- High-grade IMUs are essential for aircraft, spacecraft, and missile navigation
- Inertial navigation provides autonomy and robustness in the absence of external aids
- Integration with GPS and other sensors enhances navigation accuracy and reliability
Challenges of inertial navigation
Error accumulation over time
- Inertial navigation errors grow unbounded due to sensor biases and noise
- Long-term accuracy is limited without external aiding or frequent updates
- Error modeling and calibration are crucial for minimizing drift
Cost vs performance tradeoffs
- High-performance IMUs (fiber-optic, ring laser) are expensive and bulky
- MEMS IMUs are low-cost and compact but have lower accuracy and stability
- Selecting the appropriate IMU depends on the application requirements and budget
Integration with other sensors
- Inertial navigation is often combined with GPS, vision, lidar, or other sensors
- Sensor fusion algorithms (Kalman filters, particle filters) are needed to optimally combine measurements
- Time synchronization, coordinate frame alignment, and calibration are important for effective sensor integration
- Robust sensor fusion can compensate for individual sensor limitations and improve overall navigation performance
Key Terms to Review (19)
Accelerometer: An accelerometer is a sensor that measures the acceleration forces acting on an object, providing data on the object's speed and direction of movement. By detecting changes in motion, accelerometers can play a crucial role in determining the position and orientation of a device, which is vital for navigation systems and autonomous robots.
Accuracy: Accuracy refers to the degree to which a measured or calculated value reflects the true value or a reference standard. In various fields, achieving high accuracy is crucial for ensuring reliable results, as it influences the effectiveness of systems that rely on precise data interpretation and decision-making.
Autonomous Vehicles: Autonomous vehicles are machines capable of navigating and operating without human intervention, relying on advanced technologies like sensors, algorithms, and artificial intelligence. These vehicles can understand their environment, make decisions, and adapt to dynamic conditions, which connects closely to various features such as localization, recognition, movement, navigation systems, and complex interactions that emerge during operation.
Complementary filter: A complementary filter is a mathematical algorithm used to combine data from multiple sensors, typically by merging low-pass filtered data from one sensor with high-pass filtered data from another. This technique is often employed to achieve a more accurate estimation of an object's state by effectively leveraging the strengths of each sensor type, especially in scenarios involving noise and drift.
Cumulative Error: Cumulative error refers to the total error that accumulates over time or through a series of measurements in a system, often leading to significant deviations from the true value. In navigation systems, particularly those that use inertial methods, this error can result from small inaccuracies in measurements of position, velocity, or acceleration, which, when combined over time, produce a larger discrepancy. Understanding cumulative error is crucial for improving accuracy and reliability in navigation algorithms.
Dead reckoning: Dead reckoning is a navigation technique used to estimate the position of a moving object by calculating its current location based on a previously determined position, using speed, time, and course. This method relies heavily on accurate measurements and calculations, making it crucial for autonomous robots to understand their trajectory over time, as errors can accumulate and lead to significant deviations from the intended path.
Drift: Drift refers to the gradual deviation of a robot's estimated position from its actual position over time, often caused by accumulated errors in sensor measurements. This phenomenon is particularly significant in applications where precise navigation is crucial, as it can lead to increasing inaccuracies in a robot's location and orientation. Understanding drift is essential for improving the reliability of navigation methods like odometry and inertial navigation, which rely on continuous updates to maintain accurate positioning.
Drone navigation: Drone navigation refers to the methods and technologies used by unmanned aerial vehicles (UAVs) to determine their position and trajectory while flying. This involves a combination of sensors, algorithms, and techniques that allow drones to autonomously navigate through various environments, avoiding obstacles and reaching their intended destinations efficiently. Accurate navigation is crucial for the safe operation of drones, particularly in complex scenarios where GPS signals may be weak or unreliable.
Gimbaled System: A gimbaled system is a mechanism that allows an object, such as a sensor or platform, to remain level and stable while in motion, by utilizing a series of rings or pivots that permit rotation in multiple axes. This technology is crucial for applications like inertial navigation, where it helps maintain the orientation of sensors despite changes in the object's position or movement. By keeping instruments stable, gimbaled systems ensure accurate data collection and enhance the reliability of navigation systems.
GPS integration: GPS integration refers to the process of incorporating Global Positioning System data into navigation and control systems, allowing for precise location tracking and navigation in real-time. This integration enhances various applications by providing accurate positional information, which is crucial for autonomous systems to determine their location and navigate effectively within their environment.
Gyroscope: A gyroscope is a device that utilizes the principles of angular momentum to maintain orientation and provide stability in motion. By spinning rapidly around an axis, gyroscopes can resist changes to their orientation, which makes them crucial in applications like navigation and control systems, especially in autonomous robots where precise movements are essential.
Inertial Measurement Unit: An inertial measurement unit (IMU) is a device that measures and reports an object's specific force, angular rate, and sometimes magnetic field, allowing for the calculation of its velocity, orientation, and gravitational forces. IMUs are crucial for navigation systems, especially in environments where GPS signals may be unavailable or unreliable, enabling accurate positioning and motion tracking.
Kalman filter: A Kalman filter is an algorithm that uses a series of measurements observed over time to produce estimates of unknown variables, effectively minimizing the uncertainty in these estimates. It's particularly useful in the context of integrating different sensor data, helping to improve the accuracy and reliability of positioning and navigation systems by predicting future states based on past information.
Latency: Latency refers to the time delay between a stimulus and the response to that stimulus in a system. This delay can significantly impact the performance of systems, especially in real-time applications where quick responses are crucial. Understanding latency is essential for optimizing the performance of various technologies, ensuring that data from sensors is processed efficiently and communicated promptly.
Magnetic field sensing: Magnetic field sensing refers to the ability of a device or system to detect and measure the strength and direction of magnetic fields. This capability is crucial for various applications, especially in navigation systems where it helps in determining orientation and position relative to the Earth's magnetic field. Accurate magnetic field sensing is vital for enabling autonomous robots to navigate effectively and maintain stability during movement.
Noise: In the context of autonomous robots, noise refers to the random variations and disturbances that can affect sensor measurements and the overall performance of the navigation systems. This unwanted information can arise from various sources, including environmental factors, sensor limitations, and inherent system errors. Understanding and managing noise is crucial for ensuring accurate data collection and reliable operation in both sensor technology and inertial navigation.
Sensor calibration: Sensor calibration is the process of adjusting and fine-tuning a sensor's output to ensure it accurately reflects the true physical measurement it is intended to capture. This process is crucial for enhancing the reliability and precision of sensor data, which is especially important when multiple sensors are integrated or when precise measurements are needed for navigation and positioning tasks.
Sensor Fusion: Sensor fusion is the process of combining data from multiple sensors to produce more accurate, reliable, and comprehensive information about an environment or system. By integrating different sensor inputs, such as visual, auditory, and positional data, it enhances the overall understanding and perception of a robotic system, allowing for improved decision-making and navigation.
Strapdown system: A strapdown system is an inertial navigation configuration where the sensors, such as accelerometers and gyroscopes, are rigidly mounted to the moving platform rather than using a gimbaled system that allows for relative motion. This design enables the direct measurement of motion and orientation without complex mechanical structures, making it simpler and often more reliable for various applications, especially in autonomous robots.