E j = 1 2 ( m j j v j T j j + j ω j T j 0 j j ω j + 2 j m s j T ( j v j × j ω j ) ) = e j χ j E_{j}=\frac{1}{2}\left(m_{j}^{j} \mathbf{v}_{j}^{T} j_{j}+^{j} \omega_{j}^{T} j_{0_{j}}^{j} \omega_{j}+2^{j} \mathbf{m s}_{j}^{T}\left(^{j} \mathbf{v}_{j} \times^{j} \omega_{j}\right)\right)=\mathbf{e}_{j} \chi_{j} E j = 2 1 ( m j j v j T j j + j ω j T j 0 j j ω j + 2 j m s j T ( j v j × j ω j ) ) = e j χ j U j = − [ 0 g T 0 ] 0 T j ( q ) [ j j m s m j ] = u j χ j U_{j}=-\left[\begin{array}{cc}
^{0} \mathbf{g}^{T} & 0
\end{array}\right]^{0} \mathbf{T}_{j}(\mathbf{q})\left[\begin{array}{c}
^{j \mathbf{m} \mathbf{s}}_{j} \\
m_{j}
\end{array}\right]=\mathbf{u}_{j} \chi_{j} U j = − [ 0 g T 0 ] 0 T j ( q ) [ j j m s m j ] = u j χ j χ j = [ x x j x y j x z j y y j y z j z z j m x j m y j m z j m j ] T \chi_{j}=\left[\begin{array}{llllllllll}
x x_{j} & x y_{j} & x z_{j} & y y_{j} & y z_{j} & z z_{j} & m x_{j} & m y_{j} & m z_{j} & m_{j}
\end{array}\right]^{T} χ j = [ x x j x y j x z j y y j y z j z z j m x j m y j m z j m j ] T i λ j ^i\lambda_j i λ j
编码器提供离散数据，我们只知道每一时刻的关节的位置信息而不知道速度和加速度，速度要进一步求解。m ˙ ( t ) = m ( t ) − m ( t − T ) T \dot{m}(t)=\frac{m(t)-m(t-T)}{T} m ˙ ( t ) = T m ( t ) − m ( t − T ) m ˙ ( t ) = m ( t + T ) − m ( t − T ) 2 T \dot{m}(t)=\frac{m(t+T)-m(t-T)}{2 T} m ˙ ( t ) = 2 T m ( t + T ) − m ( t − T )
Definition of the standard dynamic parameters χ j χ_j χ j B j B_j B j χ j T = [ x x j , x y j , x z j , y y j , y z j , z z j , m x j , m y j , m z j , m j , I a j , f v j , f s j , o f f j ] χ^T_j =
[xx_j ,xy_j ,xz_j ,yy_j ,yz_j ,zz_j, mx_j, my_j ,mz_j ,m _j, I_{a _j},
f{v_j} ,f{s_j}, off_j] χ j T = [ x x j , x y j , x z j , y y j , y z j , z z j , m x j , m y j , m z j , m j , I a j , f v j , f s j , o f f j ]
Y ( τ ) = W s t ( q ^ a , q ^ a , q ^ a ) χ s t + ρ \mathbf{Y}(\tau)=\mathbf{W}^{s t}\left(\hat{\mathbf{q}}_{a}, \hat{\mathbf{q}}_{a}, \hat{\mathbf{q}}_{a}\right) \chi_{s t}+\rho Y ( τ ) = W s t ( q ^ a , q ^ a , q ^ a ) χ s t + ρ
两种转矩控制策略 ：PID，Computed torque control ，我们很熟悉的是PID，在这里我们主要考虑的是线性关系，寻找增益系数。我们假设机器人模型如下：τ i = a i q ¨ i + F v i q ˙ i + γ i \tau_{i}=a_{i} \ddot{q}_{i}+F v_{i} \dot{q}_{i}+\gamma_{i} τ i = a i q ¨ i + F v i q ˙ i + γ i
在关节空间还是在工作空间里设计控制系统唯一的区别就在于参考值给我们的是各个关节位置速度加速度还是在世界坐标系下的宏观变量。J T J^T J T
Computed torque control的应用场景 ：τ = M ^ ( q ) w + c ^ ( q , q ˙ ) \boldsymbol{\tau}=\widehat{\mathbf{M}}(\mathbf{q}) \mathbf{w}+\widehat{\mathbf{c}}(\mathbf{q}, \dot{\mathbf{q}}) τ = M ( q ) w + c ( q , q ˙ ) q ¨ ≡ w = q ¨ d + K p ( q d − q ) + K d ( q ˙ d − q ˙ ) \ddot{\mathbf{q}} \equiv \mathbf{w}=\ddot{\mathbf{q}}^{d}+\mathbf{K}_{p}\left(\mathbf{q}^{d}-\mathbf{q}\right)+\mathbf{K}_{d}\left(\dot{\mathbf{q}}^{d}-\dot{\mathbf{q}}\right) q ¨ ≡ w = q ¨ d + K p ( q d − q ) + K d ( q ˙ d − q ˙ ) 0 = ( q ¨ d − q ¨ ) + K p ( q d − q ) + K d ( q ˙ d − q ˙ ) 0=(\ddot{\mathbf{q}}^{d}-\ddot{\mathbf{q}}) +\mathbf{K}_{p}\left(\mathbf{q}^{d}-\mathbf{q}\right)+\mathbf{K}_{d}\left(\dot{\mathbf{q}}^{d}-\dot{\mathbf{q}}\right) 0 = ( q ¨ d − q ¨ ) + K p ( q d − q ) + K d ( q ˙ d − q ˙ )