2.4 Velocity kinematics and static forces

The Jacobian matrix is a powerful tool in robot kinematics, mapping joint velocities to end-effector velocities. It's crucial for understanding how a robot's joints affect its overall motion and for solving forward and inverse velocity problems.

Static forces and manipulability are key concepts in robot control. The Jacobian transpose relates end-effector forces to joint torques, while manipulability measures help assess a robot's dexterity and performance in different configurations.

Jacobian Matrix and Robot Kinematics

Jacobian matrix for robot manipulators

• Jacobian matrix linearly maps joint velocities to end-effector velocities representing instantaneous kinematics of the robot
• Geometric approach considers linear and angular velocity contributions of each joint (revolute, prismatic)
• Analytical approach uses partial derivatives of forward kinematics equations with respect to joint variables
• Linear velocity Jacobian (top 3 rows) and angular velocity Jacobian (bottom 3 rows) form complete Jacobian
• Dimensionality depends on number of degrees of freedom (DOF) and task space dimensions (6x6 for 6-DOF robot)

Joint vs end-effector velocities

• Velocity kinematics equation: $v = J(q)\dot{q}$ relates joint and end-effector velocities
• Forward velocity kinematics calculates end-effector velocities from given joint velocities
• Inverse velocity kinematics determines required joint velocities for desired end-effector velocities
• Pseudo-inverse of Jacobian used for redundant manipulators with more DOF than task space dimensions
• Singularities occur when Jacobian loses rank impacting velocity relationships (wrist singularity, elbow singularity)

Static Forces and Manipulability

Static forces via Jacobian transpose

• Force transformation equation: $\tau = J^T(q)F$ relates end-effector forces to joint torques
• Virtual work principle equates work done in joint space to work in task space
• Static force analysis determines joint torques to counteract external forces (gravity compensation)
• Force ellipsoid visualizes force transmission capabilities in different directions (major axis indicates direction of maximum force)

Jacobian and robot manipulability

• Manipulability measure $w = \sqrt{\det(JJ^T)}$ indicates dexterity of manipulator at given configuration
• Condition number of Jacobian (ratio of max to min singular values) shows isotropy of manipulator configuration
• Manipulability ellipsoid graphically represents velocity or force transmission capabilities (shape indicates dexterity)
• Impacts robot performance through workspace analysis, path planning, and singularity avoidance (optimizing trajectories)