ARO Robot Geometry

More note for this course check: Advanced-Robotics

1. 3D State Description in Robotics

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

State Representation: Define $q = [x, y, z, \phi, \theta, \psi]^T$.
3D Environment Mapping: Utilize 3D maps or LIDAR data for environmental representation.
Sensor Integration: Use GPS, gyroscopes, accelerometers, and LIDAR for real-time state updates.
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.

s17082312162023

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

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.
Gimbal Lock Understanding: Recognize that gimbal lock occurs when the pitch angle ($\theta$) is at 90°, aligning two of the three rotation axes.
Orientation Control: Implement control algorithms to adjust roll, pitch, and yaw for desired camera orientation.
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.

s17160312162023

3. 3D State Description in Robotics with Quaternion Improvement

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

State Representation: Define $q = [x, y, z, q_w, q_x, q_y, q_z]^T$.
Quaternion Usage: Replace Euler angles with quaternions to avoid issues like gimbal lock and provide more accurate and stable orientation representation.
Sensor Fusion: Integrate IMU data with visual odometry and LIDAR for robust state estimation.
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.

s17235112162023

4. Special groups for rotations

s21423412162023

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

Translation: Described by linear displacement along x, y, and z axes.
- Formula: $\Delta \mathbf{p} = \mathbf{p}{final} - \mathbf{p}{initial}$.
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.
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.

s17174512162023