error_floor

Error floor calculation

In this section, we derive the standing pose error ${\tilde{\mathbf{x}}}_{ss}$ that the outer loop must hold. To determine this, examine the closed loop at steady state, reduce it and solve for $\tilde{\mathbf{x}}$. In steady state, the tracking error has stopped changing, $\dot{\tilde{\mathbf{x}}}=0$. The pose-error rate is tied to the reduced velocity error by the Jacobian,

$$\dot{\tilde{\mathbf{x}}}={\mathbf{J}}_{\tilde{x}}(\breve{\mathbf{v}}-{\breve{\mathbf{v}}}_d),$$

so when ${\mathbf{J}}_{\tilde{x}}^{-1}$ exists, the steady-state condition $\dot{\tilde{\mathbf{x}}}=0$ forces the velocity error to vanish,

$$\breve{\mathbf{v}}={\breve{\mathbf{v}}}_d\to \dot{\breve{\mathbf{v}}}={\dot{\breve{\mathbf{v}}}}_d$$

As a result, the damping term $-\breve{\mathbf{D}}(\breve{\mathbf{v}}-{\breve{\mathbf{v}}}_d)$ vanishes.

Differentiating the equality $\breve{\mathbf{v}}={\breve{\mathbf{v}}}_d$ gives $\dot{\breve{\mathbf{v}}}={\dot{\breve{\mathbf{v}}}}_d$. Use this to cancel the inertial terms:

$$0=-\breve{\mathbf{C}}{\breve{\mathbf{v}}}_d-{\mathbf{J}}_{\tilde{x}}^{\top }\breve{\mathbf{K}}{\tilde{\mathbf{x}}}_{ss}-({\mathbf{C}}_c+\breve{\mathbf{D}}{\breve{\mathbf{G}}}_{v_c}){\dot{\tilde{\mathbf{x}}}}_c$$

Then, when ${\mathbf{J}}_{\tilde{x}}^{\top }{\breve{\mathbf{K}}}^{-1}$ exists,

$${\tilde{\mathbf{x}}}_{ss}=-\left({\mathbf{J}}_{\tilde{x}}^{\top }\breve{\mathbf{K}}{)}^{-1}\right[\breve{\mathbf{C}}{\breve{\mathbf{v}}}_d+({\mathbf{C}}_c+\breve{\mathbf{D}}{\breve{\mathbf{G}}}_{v_c}){\dot{\tilde{\mathbf{x}}}}_c]$$

It is tempting to drop the second bracket. The centre-of-mass loop is the decoupled inner system, and in error coordinates it is the damped second-order system

$$m{\ddot{\tilde{\mathbf{x}}}}_c+{\mathbf{D}}_c{\dot{\tilde{\mathbf{x}}}}_c+{\mathbf{K}}_c{\tilde{\mathbf{x}}}_c=0,$$

whose only equilibrium is ${\tilde{\mathbf{x}}}_c=0,{\dot{\tilde{\mathbf{x}}}}_c=0$. Unfortunately, the system does not reach it at this time. The measured centre-of-mass position error settles near $10^{-3}$ m, small but not zero, during unforced flight. However, when the arm and base are enabled, at this time Tikhonov damping is applied homogeneously to all coordinates. This causes spikes in ${\tilde{\mathbf{x}}}_c$ and ${\dot{\tilde{\mathbf{x}}}}_c$ near singularity.

The Coriolis-only form ${\tilde{\mathbf{x}}}_{ss}=-({\mathbf{J}}_{\tilde{x}}^{\top }\breve{\mathbf{K}}{)}^{-1}\breve{\mathbf{C}}{\breve{\mathbf{v}}}_d$ is the idealized limit of perfect centre-of-mass tracking.