Rigid Manipulators--Control控制--Task Space Control任务空间控制

任务空间动力学模型：

H_{x} (q) \ddot{x} + C_{x} (q, \dot{q}) \dot{x} + g_{x} (q) = J_{a}^{- T} (q) u

H_{x} (q) = J_{a}^{- T} (q) H (q) J_{a}^{- 1} (q)

C_{x} (q, \dot{q}) = J_{a}^{- T} (q) C (q, \dot{q}) J_{a}^{- 1} (q) - H_{x} (q) {\dot{J}}_{a} (q) J_{a}^{- 1} (q)

g_{x} (q) = J_{a}^{- T} (q) g (q)

一、Regulation位置控制

1.PD control with gravity compensation

u = J_{a}^{T} (q) (- K_{D} (\dot{x} - {\dot{x}}_{d}) - K_{P} (x - x_{d})) + g (q)

稳态解

S = {(e, \dot{e}) : e = 0, \dot{e} = 0}

1.Inverse dynamics control

Rigid Manipulators--Control控制--Task Space Control任务空间控制

关键：线性化与解耦
目标： $\ddot{x} = u_{0}$

u = J_{a}^{T} (q) (H_{x} (q) u_{0} + C_{x} (q, \dot{q}) \dot{x} + g_{x} (q))

u_{0} = \ddot{x_{d}} - K_{D} (\dot{x} - \dot{x_{d}}) - K_{P} (x - x_{d})

\ddot{e} + K_{D} \dot{e} + K_{P} e = 0

稳态解

S = {(e, \dot{e}) : e = 0, \dot{e} = 0}

2.Passivity-based control

\dot{ζ} = J_{a}^{+} (q) {\dot{ζ}}_{x}

\ddot{ζ} = J_{a}^{+} (q) ({\ddot{ζ}}_{x} - {\dot{J}}_{a} (q) \dot{ζ})

u = H (q) \ddot{ζ} + C (q, \dot{q}) \dot{ζ} + g (q) + J_{a}^{T} (q) K_{D} J_{a} (q) (\dot{ζ} - \dot{q})