Hardware desiredVelocity + Error PD(e) - dv/dt systemVelocity dt Software Control Sensor (a) desiredVelocity P(vd) F + - dv/dt systemVelocity dt Damping (b) Velocity-Space Response Damping Forcing + Stable Velocity = vy vx (c) Figure 7. A simple example of open-loop velocity control by balancing directional forcing against isotropic damping. (a) Feedback control of velocity requires an accurate measurement of the system velocity. (b) Feed-forward forces combine with damping in the world to close the feedback loop for velocity, with the added benefit that accurate sensing of the ground-truth velocity is no longer necessary, nor is accurate application of force in response to changes in velocity. (c) In velocity space, where the current velocity of a system is represented by a point in that space, forces represent the gradual change in position of those points. The force of damping always acts toward the origin. If an external force is applied, the equilibrium velocity shifts in the direction of that force. Nominal Leg Force Disturbed Disturbed Step Up Time Decaying Nominal Step Up Forcing Step Down Step Down Net Velocity Step Down Gravity (a) (b) Figure 8. The effect of physical damping on the net velocity of the robot. (a) Envelopes around the mass center represent the history of directional impulses on the robot; impulses in the same direction sum, while (b) impulses in different directions push out the envelope in those directions. The envelopes record impulses only from forces other than damping, which is represented by the gradual decay of the envelope. september 2018 * IEEE ROBOTICS & AUTOMATION MAGAZINE * 31