The Safety Problem
In minimally invasive surgery, a robot arm guides a needle or tool through soft tissue toward a target. Two physically similar events can occur during the procedure: the tool tip intentionally puncturing tissue layers (skin, muscle, organ), and an accidental impact from medical personnel bumping into the robot arm. Both create external forces on the system.
The control challenge: the robot must behave differently depending on which event is occurring. Tissue puncture requires the robot to maintain its trajectory with high stiffness. An accidental operator impact requires the robot to yield compliantly, absorb the shock, and realign the tool tip, to avoid injuring the patient.
Kinematic Modeling
The robot is a 4-DOF surgical arm: three rotational joints and one translational axis. We modeled the kinematics using the modified Denavit-Hartenberg (D-H) convention, which describes the relative position of each joint frame using only four parameters per link, a significant reduction in model complexity compared to general rigid-body transformations.
The Jacobian matrix, derived analytically from the D-H parameters, maps joint velocities to end-effector Cartesian velocities. This was implemented in MATLAB LiveScript and validated against the geometric model.
Modeling Patient Tissue
Soft tissue (skin, muscle, organ) was modeled using the Kelvin-Voigt viscoelastic model: each tissue layer is represented as a spring and dashpot in parallel, capturing both the elastic return and the viscous energy dissipation of biological tissue.
The tissue interaction force on the tool is F = −kx − α·(dx/dt), where k and α are the stiffness and damping coefficients of the current tissue layer, and x is the penetration depth. Layer boundaries (skin, muscle, bone, organ) are defined as position thresholds along the insertion axis.
Variable Impedance Control
The control architecture implements variable impedance: the robot's apparent mechanical stiffness changes in real time depending on the detected interaction type.
During normal tissue puncture, the controller maintains high stiffness and the robot tracks the reference trajectory despite tissue reaction forces. When an accidental operator impact is detected (modeled as a 30N rectangular impulse applied to the shaft), the controller reduces stiffness and allows the robot to yield, then realigns the tool tip to the reference trajectory once the perturbation is resolved.
Results
The simulation confirmed that the variable impedance architecture successfully distinguishes the two interaction types. Under operator impact, the tool tip displacement remained within acceptable bounds and converged back to the reference trajectory within the simulation window. The tissue puncture profile, with progressive force increase through skin, muscle, and organ layers and characteristic drops at layer boundaries, matched expected biomechanical behavior.