Data-driven identification and control based on optic tracking feedback for robotic systems

Abstract

This paper presents the control of a robotic system based on a data-driven model. The components of the robotic system are a redundant robot and a motion capture system, both considered as a class of nonlinear discrete Multi-Input and Multi-Output system. The strong tracking Kalman filter algorithm approximates the Jacobian matrix of an equivalent model for the robotic system considering only the input and output on-line data. Moreover, a type of proportional controller based on the estimated Jacobian matrix for the robot’s end-effector is designed. The Lyapunov stability analysis guarantees the convergence of the equivalent model and the control law. The estimation and control approach are validated with thorough experimental results.

Appendices

Appendix 1: Proof of the Theorem 2.1

The Lyapunov function in terms of the updated estimation error in the modelling is:

$$ \begin{array}{@{}rcl@{}} V_{\epsilon}(k+1)= \frac{1}{2} \epsilon(k+1)\epsilon^{T}(k+1) \end{array} $$

(48)

the change in the Lyapunov function is:

$$ \begin{array}{@{}rcl@{}} {\varDelta} V_{\epsilon}(k+1)= V_{\epsilon}(k+1)- V_{\epsilon}(k) \end{array} $$

(49)

the change in the Lyapunov function in terms of the change of estimation error Δ𝜖(k + 1) from Eq. 23 is:

$$ \begin{array}{@{}rcl@{}} {\varDelta} V_{\epsilon}(k+1) ={\varDelta} \epsilon(k+1) \left[\epsilon(k)+ \frac{1}{2} {\varDelta} \epsilon(k+1) \right]^{T} \end{array} $$

(50)

substituting the estimation error in the change in Lyapunov function:

$$ \begin{array}{@{}rcl@{}} {\varDelta} V_{\epsilon}(k + 1) \!& =&\! - \mathit{H}(\mathit{k}) \mathit{K}_{\mathit{F}}(\mathit{k}) \epsilon(k) \left[\!\epsilon(k) - \frac{1}{2} \mathit{H}(\mathit{k}) \mathit{K}_{F}(\mathit{k}) \epsilon(k)\! \right]^{T} \\ \!& =&\!-H(k) K_{F}(k) \epsilon (k)\epsilon^{T}(k) \left[\!I - \frac{1}{2} H(k) K_{F}(k)\! \right] \\ \end{array} $$

(51)

according to Corollary I $H(k) K_{F}(k)\triangleq I$; therefore, Eq. 51 becomes:

$$ \begin{array}{@{}rcl@{}} {\varDelta} V_{\epsilon}(k+1) & =& - \frac{1}{2} \epsilon^{T}(k) \epsilon(k) \\ & \leq&- \frac{1}{2} \parallel \epsilon(k) \parallel^{2} \end{array} $$

(52)

Equations 48 and 52 satisfy that the Lyapunov conditions V_𝜖(k + 1) > 0 and ΔV_𝜖(k + 1) < 0; hence, we can conclude that the estimation error 𝜖(k) converges to zero while $k \rightarrow \infty$.

Appendix 2: Proof of the Theorem 3.1

The Lyapunov function is defined:

$$ \begin{array}{@{}rcl@{}} V(k+1)=\frac{1}{2} e(k+1) e^{T}(k+1) \end{array} $$

(53)

the change in the Lyapunov function is:

$$ \begin{array}{@{}rcl@{}} {\varDelta} V(k+1)=V(k+1)- V(k) \end{array} $$

(54)

the change in the Lyapunov function in terms of the change in the error:

$$ \begin{array}{@{}rcl@{}} {\varDelta} V(k+1)= {\varDelta} e(k+1) \left[ e(k) + \frac{1}{2} {\varDelta} e(k+1) \right] \end{array} $$

(55)

from the position error definition in Eq. 30, and χ_d(k + 1) = χ_d(k) + Δχ_d(k + 1), it is obtained the next equation:

$$ \begin{array}{@{}rcl@{}} e(k+1)&= & \chi(k)+T_{s} \hat{J}_{A}(k) \omega(k)+T_{s}\epsilon(k) \omega(k) \\ & -&\chi_{d}(k) - {\varDelta} \chi_{d}(k+1) \end{array} $$

(56)

where $J_{A}^{*}(k)=\hat {J}_{A}(k)+\epsilon (k)$. The change in the error Δe(k + 1) includes the estimated Jacobian matrix $\hat {J}_{A}(k)$ and the residual error of the model 𝜖(k) in Eq. 4:

$$ \begin{array}{@{}rcl@{}} {\varDelta} e(k+1)&= & T_{s} \hat{J}_{A}(k) \omega(k)+T_{s}\epsilon(k) \omega(k)- {\varDelta} \chi_{d}(k+1) \\ {\varDelta} e(k+1)&= & T_{s} \hat{J}_{A}(k) \frac{{\varDelta} q(k)}{T_{s}}+T_{s}\epsilon(k) \frac{{\varDelta} q(k)}{T_{s}}- {\varDelta} \chi_{d}(k+1) \\ {\varDelta} e(k+1)&= & \hat{J}_{A}(k) {\varDelta} q(k)+\epsilon(k) {\varDelta} q(k)- {\varDelta} \chi_{d}(k+1) \\ &= &-J_{A}^{*}(k) \hat{J}_{A}^{\dagger}(k)C_{k} \left[ \alpha e(k) -{\varDelta} \chi_{d}(k+1)\right] \\ & -& {\varDelta} \chi_{d}(k+1) \end{array} $$

(57)

According to [21], the generalized pseudo-inverse matrix $\hat {J}_{A}^{\dagger }(k)$ of a full row rank matrix $\hat {J}_{A}(k) \in \mathbb {R}^{m \times n}$ with m < n satisfies the next criterion $P_{k}=J_{A}^{*}(k) \hat {J}_{A}^{\dagger }(k)$, which is a positive definite matrix considering (42).

$$ \begin{array}{@{}rcl@{}} {\varDelta} e(k+1)&= &-P_{k} C_{k}\alpha e(k)+\left[ P_{k}C_{k} -I\right]{\varDelta}\chi_{d}(k+1) \\ &= &-P_{k} C_{k}\alpha e(k)+{\varOmega}_{k} \end{array} $$

(58)

where ${\varOmega }_{k}=\left [P_{k}C_{k} -I\right ]{\varDelta }\chi _{d}(k+1)\leq {\varGamma }_{1}$ if ${\mathscr{B}}_{1} \triangleq P_{k}C_{k} -I$, ${\varGamma }_{1} \triangleq \lambda _{max}({\mathscr{B}}_{1}) \parallel {\varDelta }\chi _{d}(k+1) \parallel$. Substituting the change in control error into the change in the Lyapunov function:

$$ \begin{array}{@{}rcl@{}} {\varDelta} V(k+1)&= & {\varDelta} e(k+1) \left[ e(k) +\frac{1}{2} {\varDelta} e(k+1) \right]^{T} \\ &=&-P_{k}C_{k} \alpha e(k) e^{T}(k) \left[I- \frac{1}{2} P_{k} C_{k} \alpha \right] \\ & +& {\varOmega}_{k}e^{T}(k) \left[I - P_{k}C_{k} \alpha \right] \\ &+& \frac{1}{2} {\varOmega}_{k}{{\varOmega}_{k}^{T}} \end{array} $$

(59)

where the terms in function of Ω_ke(k) are undefined in sign, to cancel them, we need to fulfill the following condition:

$$ \begin{array}{@{}rcl@{}} I-P_{k}C_{k} \alpha=0 \end{array} $$

(60)

according to Corollary II P_k ≈ I when 𝜖(k) ≈ 0, by design it is chosen that $\alpha =I C_{k}^{-1}$. Therefore, $\parallel {\varOmega }_{k} {{\varOmega }_{k}^{T}}\parallel ^{2} \leq {\varGamma }_{2}$ if ${\mathscr{B}}_{2}=P_{k}\alpha ^{-1}-I$, ${\varGamma }_{2} \triangleq \left [ \lambda _{\max \limits }({\mathscr{B}}_{2}) \parallel {\varDelta }\chi _{d}(k+1) \parallel ^{2} \right ]$. Then, the stability condition should satisfy:

$$ \begin{array}{@{}rcl@{}} {\varDelta} V(k+1)&= &-C_{k} \alpha e(k) e^{T}(k) \left[I- \frac{1}{2} C_{k} \alpha \right] \\ & +&\frac{1}{2} {\varOmega}_{k}{{\varOmega}_{k}^{T}} \end{array} $$

(61)

the requirement for the stability condition is ΔV (k + 1) < 0. Considering that $C_{k}\in \mathbb {R}^{3 \times 3}$ is a diagonal matrix which contains varying time parameters C_kx, C_ky, and C_kz and $\alpha \in \mathbb {R}^{3 \times 3}$ is a diagonal matrix which contains constant parameters α_x, α_y, and α_z, we can define the selection of α as a constant regarding the performance of the parameters C_kx, C_ky, and C_kz in Fig. 17.

Fig. 17

Adaptive parameters C_kx, C_ky, and C_kz.

Therefore, $\alpha =I C_{k}^{-1}$ when 𝜖(k) ≈ 0 and $e(k) \rightarrow 0$ and it is possible to obtain the lower and the upper bounded constants for the values in the diagonal matrix α as follows:

$$ \begin{array}{@{}rcl@{}} 0 < \alpha_{x} <1.8362 \\ 0 < \alpha_{y} <1.6641 \\ 0 < \alpha_{z} <0.6100 \end{array} $$

(62)

The selected values used for α_x, α_y, and α_z in Tables 1 and 2 are in agreement for the condition in Eq. 62, and α_x, α_y, and α_z have lower values for the experimental conditions to prevent damages in the robot actuators. Once the parameters in the diagonal matrix α are established, we can find the value of ρ_x, ρ_y, and ρ_z by the next expressions:

$$ \begin{array}{@{}rcl@{}} \rho_{x} =\frac{ 1-C_{kx} \parallel \hat{J}_{A_{x}}(k) \parallel^{2}}{C_{kx}} \\ \rho_{y} =\frac{ 1-C_{ky} \parallel \hat{J}_{A_{y}}(k) \parallel^{2}}{C_{ky}} \\ \rho_{z} =\frac{ 1-C_{kz} \parallel \hat{J}_{A_{z}}(k) \parallel^{2}}{C_{kz}} \end{array} $$

(63)

now, it is possible to select the values for ρ_x, ρ_y, and ρ_z considering that C_k fulfills the conditions $\alpha =I C_{k}^{-1},$ $e(k) \rightarrow 0$, and 𝜖(k) ≈ 0 and also that the values of α satisfy the condition in Eq. 62:

$$ \begin{array}{@{}rcl@{}} 0 < \rho_{x} <1.8350 \\ 0 < \rho_{y} <1.6631 \\ 0 < \rho_{z} <0.6099 \end{array} $$

(64)

the selected values for ρ_x, ρ_y, and ρ_z in Tables 1 and 2 are in agreement for the condition in Eq. 64. Considering $\alpha =IC_{k}^{-1}$ and the term $I- \frac {1}{2} C_{k} \alpha$ must be positive to fulfill the Lyapunov stability condition ΔV (k + 1) < 0 in Eq. 61:

$$ \begin{array}{@{}rcl@{}} 0<I- \frac{1}{2} C_{k} \alpha \end{array} $$

(65)

therefore, the next term satisfies the stability condition:

$$ \begin{array}{@{}rcl@{}} \alpha< 2 I C_{k}^{-1} \end{array} $$

(66)

and substituting $\alpha =IC_{k}^{-1}$ in the condition (66), it is obtained:

$$ \begin{array}{@{}rcl@{}} IC_{k}^{-1}< 2 I C_{k}^{-1} \end{array} $$

(67)

Then, the Lyapunov condition ΔV (k + 1) is satisfied. It is possible to fulfill according to Eq. 61, where the term $\frac {1}{2} {\varOmega }_{k}{{\varOmega }_{k}^{T}}$ is bounded, the term − C_kαe(k)e^T(k) remains negative, by choosing $\alpha \leq IC_{k}^{-1}$, then the result is the positive definite matrix $\frac {1}{2}I$. Therefore, Eq. 61 becomes:

$$ \begin{array}{@{}rcl@{}} {\varDelta} V(k+1) & \leq -\frac{1}{2}\parallel e(k) \parallel^{2} +\frac{1}{2} {\varGamma}_{2} \end{array} $$

(68)

By construction V (k + 1) > 0, see Eq. 53; moreover, according to Eq. 68, ΔV (k + 1) < 0 in a vicinity of the origin. Therefore, the e(k) approaches to a compact set also near to a vicinity of the origin. With this, it is conclude that the control (42) stabilizes the robot system in Eq. 6.