### 1. Kinematic Tree, Kinematic Chain, and Forward Geometry and Backward Geometry in Robotics

#### 1.1. Demo Case Study: Robotic Arm for Assembly Line Tasks

• Super Domain: Kinematics in Robotics
• Type of Method: Kinematic Analysis and Control

#### 1.2. Kinematic Tree and Kinematic Chain

• Objective: To model the structure and movement of a robotic arm.
• Kinematic Tree:
• Represents the hierarchical structure of a robot.
• Includes all possible movements and connections between different components.
• Kinematic Chain:
• A sequence of links and joints from the base to the end effector.
• Typically, a subset of the full kinematic tree, focusing on a specific task.

#### 1.3. Forward Geometry

• Objective: To determine the position and orientation of the robot’s end effector, given its joint parameters.
• Variables:
• $\mathbf{q}$: Vector of joint parameters (angles for revolute joints, displacements for prismatic joints).
• $\mathbf{T}$: Homogeneous transformation matrix representing end effector position and orientation.
• Method:
• Utilize the kinematic chain to calculate $\mathbf{T}$ as a function of $\mathbf{q}$.

#### 1.4. Inverse Geometry

• Objective: To compute the joint parameters that achieve a desired position and orientation of the end effector.
• Method:
• More complex than forward kinematics, often requiring numerical methods (find inverse of forward kinematics) or iterative algorithms (check inverse kinematics).

#### 1.5. Jacobian Matrix

• Objective: To relate the velocities in the joint space to velocities in the Cartesian space.
• Variables:
• $\mathbf{J}$: Jacobian matrix.
• Method:
• Construct $\mathbf{J}$ such that $\dot{\mathbf{x}} = \mathbf{J} \dot{\mathbf{q}}$, where $\dot{\mathbf{x}}$ is the velocity in Cartesian space and $\dot{\mathbf{q}}$ is the velocity in joint space.

#### 1.6. Strengths and Limitations

• Strengths:
• Provides a systematic approach to model and control robotic mechanisms.
• Facilitates the understanding of complex robotic systems and their motion capabilities.
• Essential for the design and control of robotic systems in various applications.
• Limitations:
• Inverse kinematics can be computationally intensive and may not have a unique solution.
• Jacobian matrices can become singular, leading to control difficulties.

#### 1.7. Common Problems in Application

• Singularities in the kinematic chain, leading to loss of control or undefined behavior.
• Accuracy issues in forward and inverse kinematics due to mechanical tolerances and sensor errors.
• Computational complexity, especially for robots with a high degree of freedom.

#### 1.8. Improvement Recommendations

• Implement redundancy management strategies to handle singularities.
• Use advanced algorithms like pseudo-inverse or optimization-based methods for inverse kinematics.
• Integrate real-time feedback and adaptive control mechanisms to compensate for inaccuracies.

### 2. Spatial Velocity Vector in Robotics

#### 2.1. Demo Case Study: Calculating the Movement of a Robotic Arm’s End Effector

• Super Domain: Kinematics and Dynamics
• Type of Method: Vector Analysis

#### 2.2. Problem Definition and Variables

• Objective: To represent and compute the motion of a robotic arm’s end effector through its spatial velocity, which includes both linear and angular components.
• Variables:
• $\mathbf{v}$: Linear velocity component of the spatial velocity vector.
• $\mathbf{\omega}$: Angular velocity component of the spatial velocity vector.
• ${}^A\mathbf{v}_{BC}$: Spatial velocity of frame C with respect to frame B, expressed in frame A.

#### 2.3. Assumptions

• The frames of reference are rigid, with no deformation under motion.
• The motion can be captured within the Euclidean space $se(3)$ framework.

#### 2.4. Method Specification and Workflow

1. Spatial Velocity Representation: Use the $se(3)$ group to describe the spatial velocity as a six-dimensional vector combining linear and angular velocity.
• Representation: $\mathbf{v} \in se(3) = \begin{bmatrix} \mathbf{v}^A \ \mathbf{\omega}^A \end{bmatrix}$.
2. Velocity Composition: Determine the overall spatial velocity of the end effector (frame C) with respect to the base (frame A) by summing up individual velocities along the kinematic chain.
• Composition formulas:
$${}^A\mathbf{v}{AC} = {}^A\mathbf{v}{AB} + {}^A\mathbf{v}{BC}\ {}^A\mathbf{v} {AC} = {}^A\mathbf{v}{AB} + {}^A X_B ; {}^B\mathbf{v}{BC}$$
3. Transformation Matrix Utilization: Apply the appropriate transformation matrices to express all velocities in a common frame of reference, typically the base frame.

#### 2.5. Strengths and Limitations

• Strengths:
• Provides a complete description of motion, essential for control and analysis.
• Can be used to compute velocities for any point along a robotic arm or system.
• Limitations:
• Requires precise knowledge of the kinematic chain and transformations.
• The computation can become complex, especially for systems with multiple moving parts.

#### 2.6. Common Problems in Application

• Errors in calculating transformation matrices can lead to incorrect velocity composition.
• The intricacy of multi-body systems can introduce challenges in real-time applications.

#### 2.7. Improvement Recommendations

• Implement software frameworks that can automatically handle spatial transformations and velocity computations.
• Use state estimation techniques to accurately measure and update velocities, reducing errors from model inaccuracies.
• Integrate sensor feedback for real-time correction of spatial velocities.

### 3. Forward Kinematics for a Chain of Joints

#### 3.1. Demo Case Study: Movement Analysis of a Robotic Arm

• Super Domain: Kinematics in Robotics
• Type of Method: Analytical Modeling

#### 3.2. Problem Definition and Variables

• Objective: To determine the position and orientation of a robotic arm’s end effector based on joint configurations.
• Variables:
• $q$: Vector of joint parameters (e.g., angles for revolute joints, displacements for prismatic joints).
• $J(q)$: Jacobian matrix at configuration $q$.
• $v(q, \dot{q})$: Velocity of the end effector in Cartesian space.
• $\dot{q}$: Vector of joint velocities.

#### 3.3. Assumptions

• The robotic arm is composed of a series of rigid bodies connected by joints, forming a kinematic chain.
• The kinematic model of the robot is known and can be described by a set of parameters $q$.

#### 3.4. Method Specification and Workflow

1. Jacobian Computation: Calculate the Jacobian matrix $J(q)$, which relates joint velocities to end effector velocities in Cartesian space.
2. Velocity Mapping: Use the Jacobian to map joint velocities $\dot{q}$ to the velocity of the end effector $v$ in Cartesian space using the formula:
• $v(q, \dot{q}) = J(q)\dot{q}$.
3. Local Validity: Understand that the Jacobian is only valid locally, meaning it accurately describes the relationship between joint velocities and end effector velocities only in the vicinity of the configuration $q$.

#### 3.5. Strengths and Limitations

• Strengths:
• Provides a direct and efficient way to calculate the end effector’s position and orientation.
• Essential for real-time control and path planning of robotic manipulators.
• Limitations:
• The Jacobian does not provide global information about the robot’s workspace.
• Singularities in the Jacobian matrix can lead to issues in control algorithms.

#### 3.6. Common Problems in Application

• Computational complexity for robots with a high degree of freedom.
• Inaccuracies in the Jacobian matrix due to modeling errors or physical changes in the robot’s structure.

#### 3.7. Improvement Recommendations

• Implement real-time Jacobian updates to accommodate changes in the robot’s configuration and avoid singularities.
• Utilize redundancy in robotic design to handle singularities and increase the robustness of the kinematic model.
• Integrate advanced sensors and feedback systems to refine the accuracy of the Jacobian matrix continually.

### 4. Inverse Kinematics (IK) and Inverse Geometry (IG) in Robotics

#### 4.1. Demo Case Study: Configuring a Robotic Arm to Reach a Target Position

• Super Domain: Kinematics and Numerical Optimization
• Type of Method: Mathematical Problem-Solving

#### 4.2. Problem Definition and Variables

• Objective: To determine the joint parameters that will position a robotic arm’s end effector at a desired location and orientation.
• Variables:
• IK: Inverse Kinematics, focusing on finding joint velocities $\dot{q}$ to achieve a desired end-effector velocity $\nu^*$.
• IG: Inverse Geometry, solving for joint positions $q$ that reach a specific end-effector pose.
• $J(q)$: Jacobian matrix at joint configuration $q$, relating joint velocities to end-effector velocities.

#### 4.3. Assumptions

• The kinematic structure of the robotic arm is well-defined and the Jacobian matrix can be computed.

#### 4.4. Method Specification and Workflow

1. IK Formulation:
• Solve $\min_{\dot{q}} | \nu(q, \dot{q}) - \nu^* |^2$, where $\nu(q, \dot{q}) = J(q)\dot{q}$.
• Utilize the Moore-Penrose pseudo-inverse $J^+$ when the Jacobian is not invertible.
2. IG Formulation:
• Solve $\min_q \text{dist}(^{0}M_e(q), ^{0}M_*)$ for pose and $\min_{\dot{q}} \text{dist}(^{0}M_e(q_0 \oplus \dot{q}), ^{0}M_*)$ for velocity, with $^{0}M_e$ denoting the end-effector pose in the base frame and $^{0}M_*$ being the target pose.
3. IK with Constraints:
• Consider velocity bounds $\dot{q}^- \leq \dot{q} \leq \dot{q}^+$ and joint limits $q^- \leq q + \Delta t \dot{q} \leq q^+$ to the optimization problem.

#### 4.5. Strengths and Limitations

• Strengths:
• IK is a linear problem and generally easier to solve, providing a precise control mechanism for robotic motion.
• IG offers a more direct approach to reaching a specific target pose.
• Limitations:
• The Jacobian matrix may not always be invertible, necessitating pseudo-inverse solutions.
• IG is a nonlinear problem and can be computationally challenging, often requiring iterative methods.

#### 4.6. Common Problems in Application

• Jacobian singularities can lead to control problems and undefined velocities.
• High computational cost for iterative solutions in IG.
• Difficulty in ensuring convergence to the global minimum for IG problems.

#### 4.7. Improvement Recommendations

• Apply redundancy in the robotic design to mitigate the effects of singularities.
• Use optimization libraries that are robust to numerical issues and can handle constraints effectively.
• For IG, consider gradient descent methods and initialization strategies to improve convergence.