🤖Medical Robotics Unit 2 – Robot Kinematics and Dynamics Fundamentals

Key Concepts and Terminology

• Kinematics studies the motion of objects without considering the forces causing the motion
• Dynamics analyzes the forces and torques that cause motion in robotic systems
• Degrees of freedom (DOF) refers to the number of independent parameters defining a robot's configuration
• Joint space represents the set of all possible joint configurations of a robotic manipulator
• Cartesian space (also known as task space) describes the position and orientation of the end-effector
• Forward kinematics determines the end-effector pose given the joint angles or positions
• Inverse kinematics calculates the joint angles or positions required to achieve a desired end-effector pose
• Jacobian matrix relates the joint velocities to the end-effector velocities

Robotic Systems in Medicine

• Medical robots assist surgeons in performing minimally invasive procedures (laparoscopic surgery)
• Robotic systems enhance surgical precision, dexterity, and visualization
• Teleoperated robots allow surgeons to control the robot's movements from a remote console
• Haptic feedback provides tactile sensations to the surgeon, improving situational awareness
• Image-guided robots integrate medical imaging (CT, MRI) for precise targeting and navigation
• Rehabilitation robots help patients regain motor function and improve their quality of life
• Robotic prosthetics and exoskeletons restore mobility and assist in daily activities

Forward Kinematics

• Forward kinematics computes the position and orientation of the end-effector based on the joint angles or positions
• Denavit-Hartenberg (DH) convention standardizes the assignment of coordinate frames to robotic links
  • DH parameters include link length ($a_i$), link twist ($\alpha_i$), joint offset ($d_i$), and joint angle ($\theta_i$)
• Homogeneous transformation matrices represent the spatial relationship between adjacent coordinate frames
  • Transformation matrices combine rotation and translation information
• The product of transformation matrices from the base to the end-effector yields the overall forward kinematics solution
• Forward kinematics is essential for robot control, motion planning, and simulation

Inverse Kinematics

• Inverse kinematics determines the joint angles or positions required to achieve a desired end-effector pose
• Inverse kinematics is crucial for task-level robot programming and motion planning
• Analytical methods solve inverse kinematics equations directly using geometric or algebraic techniques
  • Analytical solutions are fast but may not exist for all robot configurations
• Numerical methods iteratively search for joint angles that minimize the error between the desired and current end-effector pose
  • Numerical methods are more general but computationally expensive
• Redundant robots have more DOF than necessary for a given task, leading to multiple inverse kinematics solutions
• Optimization techniques (pseudo-inverse, null-space projection) handle redundancy and incorporate additional constraints

Robot Dynamics and Control

• Robot dynamics describes the relationship between the forces/torques acting on a robot and its resulting motion
• Dynamic models consider the robot's mass, inertia, and external forces (gravity, friction)
• The equations of motion relate joint torques to joint accelerations, velocities, and positions
• Forward dynamics calculates the robot's motion given the applied joint torques
• Inverse dynamics determines the joint torques required to achieve a desired motion
• Control algorithms ensure that the robot follows a desired trajectory or applies a specific force
  • PID control, computed torque control, and impedance control are common techniques
• Stability analysis guarantees that the robot's motion remains bounded and converges to the desired behavior

Workspace Analysis

• Workspace refers to the set of all reachable positions and orientations of the end-effector
• Reachable workspace includes all points the end-effector can reach with at least one orientation
• Dexterous workspace consists of points the end-effector can reach with any desired orientation
• Workspace analysis helps determine the robot's capabilities and limitations for a given task
• Workspace visualization techniques (point cloud, voxelization) provide insights into the robot's operating range
• Singularities occur when the robot loses one or more DOF, leading to reduced manipulability
• Workspace optimization methods design robots with enhanced reachability and dexterity

Motion Planning and Trajectory Generation

• Motion planning generates collision-free paths for the robot to move from an initial to a goal configuration
• Configuration space (C-space) represents all possible robot configurations, considering obstacles
• Sampling-based methods (RRT, PRM) explore the C-space by randomly sampling configurations and connecting them
• Graph search algorithms (A*, Dijkstra) find optimal paths in the constructed roadmap or tree
• Trajectory generation creates time-parameterized paths that satisfy kinematic and dynamic constraints
• Polynomial interpolation, splines, and minimum-jerk trajectories are common techniques for smooth motion
• Obstacle avoidance ensures the robot maintains a safe distance from obstacles during motion
• Real-time motion planning adapts to dynamic environments and changing objectives

Applications in Medical Procedures

• Robotic-assisted surgery systems (da Vinci) enhance precision, dexterity, and visualization in minimally invasive procedures
• Orthopedic robots (MAKO, ROBODOC) improve the accuracy of joint replacements and bone resections
• Neurosurgical robots (Neuromate, ROSA) assist in electrode placement, biopsy, and tumor resection
• Vascular robots (Magellan, Sensei) enable precise catheter navigation in cardiac and peripheral vascular interventions
• Microsurgery robots (Steady-Hand, MUSA) provide tremor filtration and motion scaling for delicate procedures
• Robotic endoscopes (Medrobotics Flex) offer enhanced flexibility and maneuverability in hard-to-reach anatomical regions
• Robotic needle steering systems improve the accuracy of needle insertions in biopsy and drug delivery
• Rehabilitation robots (Lokomat, InMotion) assist in gait training and upper limb therapy for patients with neurological disorders