The dynamic equations governing the behavior of a single link robot with flexible joint are traditionally obtained from Lagrangian dynamics considerations. The simple robot system under study is shown in the attached image. Let $x_1=\theta_m$ be the motor angular position, the corresponding angular velocity $x_2 = d\theta/dt$, the elastic force $x_3 = k_s(\theta_t - \theta_m)$ and $x_4 = \{ d\theta_l/dt - d\theta_m/dt\}/\rho$, where $\rho^2=1/k_s$. Then the state variable representation is:

$$\begin{align*} \dot{x}_1(t) &= x_2(t)\\ \dot{x}_2(t) &= -a_5x_2(t)+a_1x_3(t)+a_1u(t)\\ \dot{x}_3(t) &= x_4(t)/\rho\\ \dot{x}_4(t) &= \{ -a_2a_3\sin{[\rho^2x_3(t)+x_1(t)]}-a_4x_3(t)-a_7x_2(t)-a_6\rho x_4(t) - a_1u(t)\}/\rho \end{align*}$$

with $a_1=1/J_m$, $a_2=1/J_l$, $a_3=mgl$,$a_4=a_1+a_2$,$a_5=B_m/J_m;a_6=B_l/J_l$,$a_7=a_6-a_5$ and $u(t)=\tau(t)$.

Consider the VTOL example. The following shortcuts ${\rm c}(\cdot)=\cos(\cdot),\quad {\rm s}(\cdot)=\sin(\cdot)$ are used. The $\Delta_0$ is given as

$$\begin{align*} \omega_{0}^{1} &: {\mathrm d}x-v_{x}{\mathrm d}t \\ \omega_{0}^{2} &: {\mathrm d}v_{x}-(u^{2}\epsilon {\mathrm c}(\theta)-u^{1}{\mathrm s}(\theta)){\mathrm d}t \\ \omega_{0}^{3} &: {\mathrm d}z-v_{z}{\mathrm d}t \\ \omega_{0}^{4} &: {\mathrm d}v_{z}-(u^{1}{\mathrm c}(\theta)+u^{2}\epsilon {\mathrm s}(\theta)-1){\mathrm d}t \\ \omega_{0}^{5} &: {\mathrm d}\theta-\omega {\mathrm d}t \\ \omega_{0}^{6} &: {\mathrm d}\omega-u^{2}{\mathrm d}t, \end{align*}$$

where $\epsilon$ is a constant parameter. It can be shown that $\Delta_{0,\mathrm{d}t}^{\perp} = \mathrm{span}\{\delta_{u^1},\delta_{u^2}\}$. Construct $\Delta_1 \in \Delta_0$ such that $v_0(\Delta_1)\in \Delta_1$ holds with $v_0=\Delta_{0,{\mathrm d}t}^{\perp}$, i.e. $\Delta_1$ is given as:

$\begin{align*} \omega_{1}^{1} &: {\mathrm d}x-v_{x}{\mathrm d}t \\ \omega_{1}^{2} &: {\mathrm c}(\theta){\mathrm d}v_{x}+{\mathrm s}(\theta){\mathrm d}v_{z}-\epsilon {\mathrm d}\omega+{\mathrm s}(\theta){\mathrm d}t \\ \omega_{1}^{3} &: {\mathrm d}z-v_{z}{\mathrm d}t \\ \omega_{1}^{4} &: {\mathrm d}\theta-\omega {\mathrm d}t \end{align*}$

Consider a two-link rigid robot manipulator moving a horizontal plane. The dynamic equations of this MIMO system are

$$\left[\matrix{ \ddot{q}_{1}\cr \ddot{q}_{2} }\right]=\left[\matrix{ M_{11} & M_{12}\cr M_{21} & M_{22} }\right]^{-1} \left\{\left[\matrix{ u_{1}\cr u_{2} }\right]-\left[\matrix{ -h\dot{q}_{2} & -h(\dot{q}_{1}+\dot{q}_{2})\cr h\dot{q}_{1} & 0 }\right]\left[\matrix{ \dot{q}_{1}\cr \dot{q}_{2} }\right]\right\},$$

where

$\begin{align*} M_{11}&=a_{1}+2a_{3}\cos(q_{2})+2a_{4} \sin (q_{2}),\ M_{22}=a_{2} \\ M_{12}&=M_{21}=a_{2}+\alpha_{3}\cos(q_{2})+a_{4}\sin(q_{2}) \\ h&=a_{3}\sin(q_{2})-a_{4}\cos(q_{2}) \end{align*}$

with

$\begin{align*} a_{1}&=I_{1}+m_{1}l_{c1}^{2}+I_{e}+m_{e}l_{ce}^{2}+m_{e}l_{1}^{2} \\ a_{2}&=I_{e}+m_{e}l_{ce}^{2} \\ a_{3}&=m_{e}l_{1}l_{ce}\cos(\delta_{e}) \\ a_{4}&=m_{e}l_{1}l_{ce}\sin(\delta_{e}). \end{align*}$

The state model of the knee-quadriceps can be expressed as

$$\begin{cases} \begin{align*} \dot{x}_1 &= \left[ s_0 \alpha K_m + s_v q\dfrac{s_0\alpha F_mx_1 - s_ux_2x_1}{1 + px_1 - s_vqx_2}\right] u_{ch} - s_ux_1u_{ch} - \dfrac{s_v ax_1 r_p x_4}{L_0 (1+px_1-s_vqx_2)}\\ \dot{x}_2 &= \left[ \dfrac{s_0\alpha F_m - s_ux_2}{1 + px_1 - s_vqx_2} \right]u_{ch} + \dfrac{bx_1r_px_4 - s_vax_2r_px_4}{L_0(1+px_1-s_vqx_2)}\\ \dot{x}_3 &= x_4\\ \dot{x}_4 &= \dfrac{1}{I}[x_2r_p - \lambda x_3 - \mu x_4 - mgl_c \cos{x_3}], \end{align*} \end{cases}$$

where $\textbf{x}=[x_1, \ldots, x_4]^{\mathrm T} = [K_c, F_c, \theta, \dot{\theta}]^{\mathrm T}$ is the state vector and $\textbf{u}=[u_{ch},\alpha ]^{\mathrm T}$ the control vector. The variable $\theta$ represents the knee joint angle and the variables $K_c, F_c, u_{ch}, \alpha$ represent the state variables of the quadriceps muscle model.

The present study of tension leg platform is the first commercial application of a revolutionary design of offshore production platform developed by well-known oil company. Intended for oil and gas production in water depths beyond the reach of traditional fixed structures, the tension leg platform was designed as a rectangular shaped floating platform which was connected to the ocean floor by 16 vertical steel tethers or legs, four per corner. The legs were kept in tension so that vertical movement was suppressed, while limited horizontal movement may occur.

The estimation model is:

$$\begin{align*} y(k) &= 0.590y(k − 3)+1.0598y(k − 1) − 1.0931y(k − 2) + 121.13u(k − 1)u(k− 1)u(k − 9) \\ &− 116.54u(k − 6)u(k − 6)u(k− 6) − 19.797u(k − 4)u(k − 8)u(k − 8) \\ &+ 214.04u(k − 5)u(k − 9) − 34.877u(k− 1)u(k − 1)u(k − 1) − 3.7983u(k − 1)u(k− 2)u(k − 7) \\ &− 25.04u(k − 4)u(k − 8)u(k− 11) + 165.93u(k − 2)u(k − 3)u(k − 4) \\ &− 173.85u(k − 6)u(k − 7) − 69.693u(k− 4)u(k − 12) + 203.12u(k − 5)u(k − 6)u(k − 6) \\ &+ 727.86u(k − 2)u(k − 3)u(k − 5) − 11.107u(k− 3)u(k − 10)u(k − 11) \\ &+ 11.506u(k − 6)u(k− 6)u(k − 12) − 68.607u(k − 2)u(k − 4)u(k− 6) \\ &− 366.75u(k − 3)u(k − 5)u(k − 6)− 25.696u(k − 4)u(k − 8)u(k − 12) \\ &+ 137.86u(k − 1)u(k − 2)u(k − 5)− 142.24u(k − 2)u(k − 2)u(k − 9) \\ &+ 101.44u(k− 1)u(k − 6)u(k − 9) − 9.0283u(k − 3)u(k− 3)u(k − 12) \\ &− 168.30u(k − 2)u(k − 5)u(k − 6)+ 30.295u(k − 5)u(k − 6)u(k − 8) \\ &− 0.158u(k− 1)u(k − 2)u(k − 2) − 433.21u(k − 2)u(k− 2)u(k − 4) \\ &+ 39.88u(k − 3)u(k − 8)u(k − 11)− 162.26u(k − 1)u(k − 4)u(k − 11) \\ &− 212.08u(k− 1)u(k − 1)u(k − 5) − 438.7u(k − 3)u(k− 3)u(k − 5) \\ &+ 162.15u(k − 2)u(k − 4)u(k − 11)− 3.607u(k − 4)u(k − 4)u(k − 11) \\ &+ 13.262u(k− 6)u(k − 9)u(k − 9)+448.4u(k − 3)u(k − 4)u(k− 6) \\ &− 46.475u(k − 4)u(k − 4)u(k − 9) + 119.95u(k − 1)u(k − 1)u(k − 2) + noise \:terms. \end{align*}$$

Here, the input is the wave, and the output is the pitch. Sample rate is 2.2473 Hz