Rigid Manipulators--Control控制--Joint Space Control关节空间控制

关节空间动力学模型：

H (q) \ddot{q} + C (q, \dot{q}) \dot{q} + g (q) = u

一、Regulation位置控制

1.PD control

u = - K_{P} e - K_{D} \dot{e}

H (e + q_{d}) \ddot{e} + C (e + q_{d}, \dot{e}) \dot{e} + K_{D} \dot{e} + K_{P} e + g (e + q_{d}) = 0

稳态解

S = {(e, \dot{e}) : K_{P} e + g (e + q_{d}) = 0, \dot{e} = 0}

稳态误差不为0

2.PID control

u = - K_{P} e - K_{D} \dot{e} - K_{I} \int_{0}^{t} e d τ

H (e + q_{d}) \ddot{e} + C (e + q_{d}, \dot{e}) \dot{e} + K_{D} \dot{e} + K_{P} e + K_{I} \int_{0}^{t} e d τ + g (e + q_{d}) = 0

稳态解

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

稳态误差为0，不需要知道外力的准确模型参数

3.PD control with gravity compensation

u = - K_{P} e - K_{D} \dot{e} + g (q)

H (e + q_{d}) \ddot{e} + C (e + q_{d}, \dot{e}) \dot{e} + K_{D} \dot{e} + K_{P} e = 0

稳态解

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

稳态误差为0，但需要知道外力的准确模型参数

1.Inverse dynamics control

Rigid Manipulators--Control控制--Joint Space Control关节空间控制

关键：线性化与解耦
目标： $\ddot{q} = u_{0}$

u = H (q) u_{0} + C (q, \dot{q}) \dot{q} + g (q)

u_{0} = \ddot{q_{d}} - K_{D} \dot{e} - K_{P} e

\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} = {\dot{q}}_{d} - K (\cdot) e

K (s) = \sum_{j = 0}^{n} \frac{K_{j}}{s^{j}}

σ = F^{- 1} (\cdot) e

F^{- 1} (s) = s I + K (s)

u = H (q) \ddot{ζ} + C (q, \dot{q}) \dot{ζ} + g (q) - K_{D} σ

H (q) \dot{σ} + C (q, \dot{q}) σ + K_{D} σ = 0

特别的

K (s) = Λ \Rightarrow L y a p u n o v - b a s e d c o n t r o l

当模型的参数在一定范围内波动时，需要使用鲁棒控制

1.Robust inverse dynamics control

u = H_{0} (q) ({\ddot{q}}_{d} - K_{D} \dot{e} - K_{P} e) + C_{0} (q, \dot{q}) \dot{q} + g_{0} (q) + u_{0}

H (q) (\ddot{e} + K_{D} \dot{e} + K_{P} e) = Y (q, \dot{q}, q_{d}, \dot{q_{d}}, \ddot{q_{d}}) ρ + u_{0}

Y (\cdot) ρ = (H_{0} (q) - H (q)) (\ddot{q_{d}} - K_{D} \dot{e} - K_{P} e) + (C_{0} (q, \dot{q}) - C (q, \dot{q})) \dot{q} + (g_{0} (q) - g (q))

2.Robust passivity-based control

u = H_{0} (q) \ddot{ζ} + C_{0} (q, \dot{q}) \dot{ζ} + g_{0} (q) - K_{D} σ + u_{0}

H (q) \dot{σ} + C (q, \dot{q}) σ + K_{D} σ = Y (q, \dot{q}, q_{d}, \dot{q_{d}}, \ddot{q_{d}}) ρ + u_{0}

Y (\cdot) ρ = (H_{0} (q) - H (q)) \ddot{ζ} + (C_{0} (q, \dot{q}) - C (q, \dot{q})) \dot{ζ} + (g_{0} (q) - g (q))

Rigid Manipulators--Control控制--Joint Space Control关节空间控制

1.Adaptive gravity compensation

u = - K_{P} e - K_{D} \dot{e} + Y_{g} (q) {\hat{ρ}}_{g}

{\dot{\hat{ρ}}}_{g} = - ν Y^{T} (q) (γ \dot{q} + \frac{2 e}{1 + 2 e^{T} e})

2.Adaptive inverse dynamics control

u = \hat{H} (q) ({\ddot{q}}_{d} - K_{D} \dot{e} - K_{P} e) + \hat{C} (q, \dot{q}) \dot{q} + \hat{g} (q)

\hat{H} (q) (\ddot{e} + K_{D} \dot{e} + K_{P} e) = Y (q, \dot{q}, \ddot{q}) ρ

Y (\cdot) ρ = (\hat{H} (q) - H (q)) \ddot{q} + (\hat{C} (q, \dot{q}) - C (q, \dot{q})) \dot{q} + (\hat{g} (q) - g (q))

3.Adaptive passivity-based control

u = \hat{H} (q) \ddot{ζ} + \hat{C} (q, \dot{q}) \dot{ζ} + \hat{g} (q) - K_{D} σ

\hat{H} (q) \dot{σ} + \hat{C} (q, \dot{q}) σ + K_{D} σ = Y (q, \dot{q}, \ddot{q}) ρ

Y (\cdot) ρ = (\hat{H} (q) - H (q)) \ddot{ζ} + (\hat{C} (q, \dot{q}) - C (q, \dot{q})) \dot{ζ} + (\hat{g} (q) - g (q))