More note for this course check: Advanced-Robotics

1. 3D State Description in Robotics

1.1. Demo Case Study: 3D Navigation of a Drone in an Urban Environment

  • Super Domain: Path & Motion Planning
  • Type of Method: State Representation in 3D Space

1.2. Problem Definition and Variables

  • Objective: To represent and track the state of a drone for 3D navigation in an urban environment.
  • Variables:
    • $q$: State vector including position and orientation in 3D space.
    • $(x, y, z)$: 3D Cartesian coordinates for position.
    • $(\phi, \theta, \psi)$: Roll, pitch, and yaw angles for orientation.
    • Environmental variables: Obstacles’ positions, wind conditions, goal location.

1.3. Assumptions

  • The drone operates in a three-dimensional space.
  • External conditions like wind and air traffic are either known or can be sensed.

1.4. Method Specification and Workflow

  1. State Representation: Define $q = [x, y, z, \phi, \theta, \psi]^T$.
  2. 3D Environment Mapping: Utilize 3D maps or LIDAR data for environmental representation.
  3. Sensor Integration: Use GPS, gyroscopes, accelerometers, and LIDAR for real-time state updates.
  4. Advanced State Estimation: Implement algorithms like Extended Kalman Filter or SLAM (Simultaneous Localization and Mapping) for accurate state estimation in 3D.

1.5. Strengths and Limitations

  • Strengths:
    • Comprehensive representation of the drone’s position and orientation in 3D space.
    • Facilitates complex navigation tasks, including obstacle avoidance and route optimization.
    • Adaptable to a variety of 3D navigation tasks in different environments.
  • Limitations:
    • High computational requirements for real-time state estimation and mapping.
    • Sensor accuracy and reliability can significantly affect state estimation.
    • Complex environmental dynamics can challenge state representation accuracy.

1.6. Common Problems in Application

  • Drift in state estimation due to sensor errors, especially in GPS-denied environments.
  • Difficulty in mapping and navigating in cluttered or dynamically changing environments.
  • Balancing computational load and real-time performance requirements.

1.7. Improvement Recommendations

  • Integrate more robust and diverse sensors (e.g., visual odometry) to enhance state estimation.
  • Employ machine learning techniques for predictive modeling of environmental dynamics.
  • Optimize algorithms for efficient computation, considering the constraints of onboard processing capabilities.


2. Euler Angles and Gimbal Lock in Robotics

2.1. Demo Case Study: Orientation Control of a Robotic Camera System

  • Super Domain: Kinematics and Control Systems
  • Type of Method: Orientation Representation and Control

2.2. Problem Definition and Variables

  • Objective: To control the orientation of a robotic camera system using Euler angles while understanding and mitigating the risks of gimbal lock.
  • Variables:
    • Euler Angles $(\phi, \theta, \psi)$: Roll, pitch, and yaw angles representing the orientation of the camera.
    • Gimbal Lock: A condition where two of the three gimbals align, resulting in a loss of one degree of rotational freedom.

2.3. Assumptions

  • The camera system operates under a gimbal mechanism allowing three degrees of rotational freedom.
  • The motion and orientation of the system can be accurately measured and controlled using Euler angles.

2.4. Method Specification and Workflow

  1. Euler Angle Representation: Define orientation using three rotations about principal axes.
    • Sequence of rotations (e.g., Z-Y-X) is crucial for defining the orientation.
  2. Gimbal Lock Understanding: Recognize that gimbal lock occurs when the pitch angle ($\theta$) is at 90°, aligning two of the three rotation axes.
  3. Orientation Control: Implement control algorithms to adjust roll, pitch, and yaw for desired camera orientation.
  4. Gimbal Lock Mitigation: Monitor and manage the pitch angle to avoid approaching critical values where gimbal lock can occur.

2.5. Strengths and Limitations

  • Strengths:
    • Euler angles provide an intuitive way to describe orientation.
    • Suitable for applications with limited rotational movement where gimbal lock can be avoided.
  • Limitations:
    • Gimbal lock leads to a loss of one rotational degree of freedom, making precise control impossible in certain orientations.
    • Euler angles can be less intuitive for complex, three-dimensional rotational movements.

2.6. Common Problems in Application

  • Unintentionally entering gimbal lock, resulting in control issues.
  • Difficulty in visualizing and programming complex rotations using Euler angles.
  • Accumulation of numerical and computational errors, especially near gimbal lock configurations.

2.7. Improvement Recommendations

  • Implement quaternion-based representation as an alternative to avoid gimbal lock.
  • Use advanced control algorithms that can detect and compensate for approaching gimbal lock conditions.
  • Educate operators and programmers about the limitations and risks of gimbal lock in systems using Euler angles.


3. 3D State Description in Robotics with Quaternion Improvement

3.1. Demo Case Study: Advanced 3D Navigation of an Aerial Robot in Complex Environments

  • Super Domain: Path & Motion Planning
  • Type of Method: Enhanced State Representation in 3D Space

3.2. Problem Definition and Variables

  • Objective: To accurately represent and manage the 3D state of an aerial robot for complex navigation tasks.
  • Variables:
    • $q$: Enhanced state vector including position and orientation in 3D space.
    • $(x, y, z)$: 3D Cartesian coordinates for position.
    • Quaternion $(q_w, q_x, q_y, q_z)$: Representing orientation to overcome Euler angle limitations.

3.3. Assumptions

  • The robot operates in a dynamic, three-dimensional environment.
  • High accuracy in orientation representation is crucial for navigation and task execution.

3.4. Method Specification and Workflow

  1. State Representation: Define $q = [x, y, z, q_w, q_x, q_y, q_z]^T$.
  2. Quaternion Usage: Replace Euler angles with quaternions to avoid issues like gimbal lock and provide more accurate and stable orientation representation.
  3. Sensor Fusion: Integrate IMU data with visual odometry and LIDAR for robust state estimation.
  4. Advanced Algorithms: Use state-of-the-art algorithms like Unscented Kalman Filter or SLAM for precise state tracking in 3D space.

3.5. Strengths and Limitations

  • Strengths:
    • Quaternions provide a more robust and singularity-free representation of orientation in 3D space.
    • Enhanced precision in orientation helps in complex maneuvers and accurate navigation.
    • Suitable for high-dynamic environments like aerial robotics in urban or natural terrains.
  • Limitations:
    • Increased complexity in understanding and implementing quaternion mathematics.
    • Higher computational demands for real-time quaternion-based state estimation.
    • Requires sophisticated sensor fusion techniques to fully leverage quaternion advantages.

3.6. Common Problems in Application

  • Complexity in translating quaternion operations into practical control commands.
  • Challenges in seamlessly integrating quaternion data with traditional position sensors.
  • Ensuring numerical stability and avoiding normalization issues with quaternions.

3.7. Improvement Recommendations

  • Integrate machine learning to predict and compensate for environmental dynamics and sensor inaccuracies.
  • Develop specialized algorithms for efficient quaternion normalization and operation.
  • Utilize hardware acceleration for real-time processing of complex quaternion-based calculations.


4. Special groups for rotations


5. Displacement in Robotics (Translation and Rotation)

5.1. Demo Case Study: Precise Manipulator Movement in Industrial Robotics

  • Super Domain: Kinematics
  • Type of Method: Geometric and Kinematic Analysis

5.2. Problem Definition and Variables

  • Objective: To accurately describe and control the displacement of a robotic manipulator, including both translation and rotation.
  • Variables:
    • $\mathbf{p}$: Position vector representing translation in Cartesian coordinates $(x, y, z)$.
    • Rotation: Represented by Euler angles $(\phi, \theta, \psi)$, rotation matrices, or quaternions.

5.3. Assumptions

  • The manipulator operates within a rigid body framework.
  • Joint movements are precise and can be accurately measured.

5.4. Method Specification and Workflow

  1. Translation: Described by linear displacement along x, y, and z axes.
    • Formula: $\Delta \mathbf{p} = \mathbf{p}{final} - \mathbf{p}{initial}$.
  2. Rotation:
    • Euler Angles: Changes in orientation described by roll ($\phi$), pitch ($\theta$), and yaw ($\psi$).
    • Rotation Matrix: $\mathbf{R} = \mathbf{R}_z(\psi) \mathbf{R}_y(\theta) \mathbf{R}_x(\phi)$.
    • Quaternion: $q = [q_w, q_x, q_y, q_z]$, providing a singularity-free representation.
  3. Homogeneous Transformation Matrix: Combines translation and rotation in a single matrix.
    • Formula: $\mathbf{T} = \begin{bmatrix} \mathbf{R} & \mathbf{p} \ \mathbf{0} & 1 \end{bmatrix}$.

5.5. Strengths and Limitations

  • Strengths:
    • Provides a comprehensive way to describe both position and orientation.
    • Suitable for a wide range of robotic applications, including automation and assembly.
    • Facilitates the calculation of kinematic chains and motion planning.
  • Limitations:
    • Complexity in calculations, especially for rotations in 3D space.
    • Euler angles can suffer from gimbal lock in certain configurations.
    • Precision depends on the accuracy of the measurement systems.

5.6. Common Problems in Application

  • Accumulation of numerical errors in successive transformations.
  • Complexity in inverse kinematics for determining joint angles from desired end-effector position.
  • Difficulty in intuitively understanding and visualizing quaternion-based rotations.

5.7. Improvement Recommendations

  • Implement advanced computational tools for handling complex kinematic calculations.
  • Utilize quaternions for rotation to avoid gimbal lock and improve computational efficiency.
  • Integrate feedback systems for error correction in real-time motion control.


Reference: The University of Edinburgh, Advanced Robotics, course link
Author: YangSier (

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