[Part2] Planning for Interaction: The Trajectory Generation of End-Effector

Subtitle: Why Cartesian Impedance Control is the key to safe Human-Robot Interaction (HRI).

4. Necessity of Trajectory Generation (Smoothness is Safety)

One common pitfall in impedance control is feeding a step input or a “Trapezoidal Velocity Profile (TVP)” as the desired trajectory $x_d$.

Why TVP is dangerous in Impedance Control

The Problem: TVP has discontinuous acceleration (infinite jerk) at the corners of the velocity profile.
The Consequence: Since torque is directly related to acceleration ($\tau \propto \ddot{x}$), a jump in acceleration causes a torque spike.
Result: This induces severe vibrations, damages the gearbox, and creates unsafe, jerky motions in HRI scenarios.

The Solution: Minimum Jerk Trajectory

To ensure smooth interaction, we must use trajectories with $C^2$ continuity (continuous acceleration). The Minimum Jerk Trajectory minimizes the change of acceleration, ensuring that the generated torque is smooth and the robot’s behavior remains compliant and safe.

5. Robot Dynamics for Sophisticated Control

A simple PD-like law ($K_p e + K_d \dot{e}$) is insufficient for high-performance manipulation. We must account for the robot’s intrinsic dynamics:

\[\tau_{cmd} = \underbrace{g(q) + C(q,\dot{q})\dot{q}}_{\text{Dynamics Compensation}} + J^T F_{impedance}\]

The Importance of Gravity Compensation

Gravity compensation $g(q)$ is critical. The impedance controller assumes the robot is “weightless.”

Mathematical Insight: If the gravity model is imperfect ($\hat{g}(q) \neq g_{real}(q)$), a residual torque $\Delta \tau_g$ remains.
Steady-State Error: At equilibrium ($\ddot{x}=0, \dot{x}=0$), the spring force must fight this residual gravity: $K_d \cdot e_{ss} = J^{-T} \Delta \tau_g$ This results in a steady-state error ($e_{ss}$), causing the robot to “sag” under its own weight. To minimize this without increasing stiffness (which reduces compliance), precise dynamic identification is required.

6. Conclusion

Impedance control is not just a control algorithm; it is a philosophy of compliant interaction. By shaping the robot’s energy exchange with the environment, we enable safer and more versatile robotic applications.

However, fixed-gain impedance control has limitations. Future research focuses on Variable Impedance Control (VIC) and Learning-based Impedance, where the robot intelligently adapts its stiffness and damping based on the task requirements and environmental contact.