Position-Based Visual Servoing Control for Multi-Joint Hydraulic Manipulator

Abstract

Manipulators are widely used in various fields of industrial automation, and hydraulic manipulators have more extensive application in future by virtue of high power to weight ratio. However, compared with the traditional motor manipulators, the dynamic model of hydraulic manipulator is more complex such as higher model order characteristics, which makes the design of precision motion controller of hydraulic manipulators more challenging. In addition, the application demand of hydraulic manipulator is gradually increasing even under various harsh conditions or extreme environments, which puts forward higher requirements for the autonomous environmental perception ability of hydraulic manipulators. Therefore, the visual servoing control for hydraulic manipulator is of great research value. In this paper, a position-based visual servoing control method for multi-joint hydraulic manipulator is proposed. To be specific, based on the obtained target position, the discrete desired path points can be got by velocity-limited path interpolation, which will be transformed into a discrete sequence of desired joint angles according to well-constrained inverse kinematics. Then the continuous desired angle trajectory of manipulator joint is generated by B-spline trajectory planner. Finally, the ideal angle trajectory is input into backstepping controller to contrive the angle tracking of each joint, so as to realize the visual target following control of multi-joint hydraulic manipulator.

Appendix : A: Proof for Theorem

Proof Proof for Theorem I

The Lyapunov function is defined as:

$$ \begin{array}{@{}rcl@{}} & {{V}_{2}}=\frac{1}{2}{{z}_{2}}^{T}{{M}_{j}}{{z}_{2}} \end{array} $$

(A1)

$$ \begin{array}{@{}rcl@{}} &{{V}_{3}}={{V}_{2}}+\frac{1}{2}{{z}_{3}}^{T}{{z}_{3}} \end{array} $$

(A2)

Combine (??),(??), (??), (??) and Eq. A1 the differential of V₂ can be simplified as:

$$ \begin{array}{@{}rcl@{}} {{\dot{V}}_{2}}&=&{{J}_{j}}^{T}{{z}_{2}}{{z}_{3}}-{{z}_{2}}^{T}{{k}_{2}}{{z}_{2}}+{{z}_{2}}^{T}\\&&\left[ {{J}_{j}}{{u}_{js}}-{{\varphi }_{c}}\left( q,\dot{q},{{{\dot{q}}}_{eq}},{{{\ddot{q}}}_{eq}} \right)\tilde{\theta }+{\varDelta} \left( q,\dot{q},t \right) \right] \end{array} $$

(A3)

Combine (??), (??), (??) and (A2), the differential of V₃ can be simplified as:

$$ \begin{array}{@{}rcl@{}} {{\dot{V}}_{3}}\!&=&\!{{ {{{\dot{V}}}_{3}} \rvert}_{{{z}_{3}}=0}}+{{z}_{3}}^{T}\left[ {{J}_{j}}^{T}{{z}_{2}}+{{\beta }_{e}}\left[ {{u}_{Qs}}-{{k}_{3}}{{z}_{3}}\vphantom{\tilde{\theta }} \right.\right.\\&&\left.\left.\!+{{\varphi }_{Q}}\left( q,\dot{q},{{{\dot{q}}}_{eq}},{{{\ddot{q}}}_{eq}} \right)\tilde{\theta } + {{\Delta }_{Q}}\left( q,\dot{q},t \right) \right] - {{{\tilde{P}}}_{Ld}} \right] \end{array} $$

(A4)

According to the Condition III, (A4) can be convert to inequality form as follow:

$$ {{\dot{V}}_{3}}\le {{ {{{\dot{V}}}_{2}} \rvert}_{{{z}_{3}}=0}}-{{k}_{3}}{{z}_{3}}+{{\zeta }_{3}} $$

(A5)

${{ {{{\dot {V}}}_{2}} \rvert }_{{{z}_{3}}=0}}$can be written as:

$$ \begin{array}{@{}rcl@{}} {{ {{{\dot{V}}}_{2}} \rvert}_{{{z}_{3}}=0}}&=&{{z}_{2}}^{T}\left[ -{{k}_{2}}{{z}_{2}}+{{J}_{j}}{{u}_{js}}-{{\varphi }_{j}}\left( q,\dot{q},{{{\dot{q}}}_{eq}},{{{\ddot{q}}}_{eq}} \right)\tilde{\theta }\right.\\&&\left.+{{{\varDelta} }_{Q}}\left( q,\dot{q},t \right)\vphantom{\tilde{\theta }} \right] \end{array} $$

(A6)

According to the Condition I, (A6) can be convert to inequality form as follow:

$$ \begin{array}{@{}rcl@{}} & {{ {{{\dot{V}}}_{2}} \rvert}_{{{z}_{3}}=0}}\le -{{z}_{2}}^{T}{{k}_{2}}{{z}_{2}}+{{\zeta }_{2}} \end{array} $$

(A7)

$$ \begin{array}{@{}rcl@{}} & \text{ }\le -{{\eta }_{2}}{{V}_{2}}+{{\zeta }_{2}} \end{array} $$

(A8)

where ${{\eta }_{2}}=2{{M}_{j}}^{-1}{{k}_{2}}$, the following equation can be obtained by solving (A8) in time domain:

$$ {{ {{V}_{2}} \rvert}_{{{z}_{3}}=0}}(t)\le {{ {{V}_{2}} \rvert}_{{{z}_{3}}=0}}(0){{e}^{-{{\eta }_{2}}t}}+\frac{{{\zeta }_{2}}}{{{\eta }_{2}}}\left( 1-{{e}^{-{{\eta }_{2}}t}} \right) $$

(A9)

According to Eq. A9, ${{ {{V}_{2}} \rvert }_{{{z}_{3}}=0}}\le \frac {{{\zeta }_{2}}}{{{\eta }_{2}}}$, and ${{ {{V}_{2}} \rvert }_{{{z}_{3}}=0}}\to 0$ as $t\to \infty$.

Combined (A6) and Eq. A9, the following inequality of ${{\dot {V}}_{3}}$ can be obtained the as:

$$ {{V}_{3}}\le -{{\eta }_{3}}{{V}_{3}}+{{\zeta }_{3}}+\frac{{{\eta }_{3}}}{{{\eta }_{2}}}{{\zeta }_{2}} $$

(A10)

where η₃ = 2k₃, the following equation can be obtained by solving (A8) in time domain:

$$ {{V}_{3}}(t)\le {{V}_{3}}(0){{e}^{-{{\eta }_{3}}t}}+\frac{{{\zeta }_{c}}}{{{\eta }_{c}}}\left( 1-{{e}^{-{{\eta }_{3}}t}} \right) $$

(A11)

where $\frac {{{\zeta }_{c}}}{{{\eta }_{c}}}=\frac {{{\zeta }_{2}}}{{{\eta }_{2}}}+\frac {{{\zeta }_{3}}}{{{\eta }_{3}}}$.

According to Eq. A11, ${{V}_{3}}\le \frac {{{\zeta }_{c}}}{{{\eta }_{c}}}$, and V₃ → 0 as $t\to \infty$.

It can be proved that V3 is ultimately bounded to $\frac {{{\zeta }_{c}}}{{{\eta }_{c}}}$ and z₃ is bounded. On the premise that z₃ is bounded, according to eq.49, V₂ is ultimately bounded to $\frac {{{\zeta }_{2}}}{{{\eta }_{2}}}$ and z₂ is bounded.

In addition, by frequency domain transformation of Eq. ??, the conversion function L(s) between z₁ and z₂ can be obtained as follows:

$$ L(s)=\frac{{{z}_{1}}}{{{z}_{2}}}=\frac{1}{s+{{k}_{1}}} $$

(A12)

According to Eq. ??, z₁ is bounded on the premise that z₂ is bounded. The proof for Theorem I is complete. □

Proof Proof for Theorem II

When disturbance error, Δ = 0, gets zero and the dynamic model parameters are accurate ($\tilde {\theta }=0$), the differential of Lyapunov function such as Eqs. A2 and A4 can be rewritten as:

$$ \begin{array}{@{}rcl@{}} {{\dot{V}}_{2}}&=&{{J}_{j}}^{T}{{z}_{2}}{{z}_{3}}-{{z}_{2}}^{T}{{k}_{2}}{{z}_{2}}+{{z}_{2}}^{T}{{J}_{j}}{{u}_{js}} \end{array} $$

(13)

$$ \begin{array}{@{}rcl@{}} {{\dot{V}}_{3}}&=&\!{{ {{{\dot{V}}}_{3}} \rvert}_{{{z}_{3}}=0}} + {{z}_{3}}^{T}\left[ {{J}_{j}}^{T}{{z}_{2}} + {{\upbeta }_{e}}\left[ {{u}_{Qs}}-{{k}_{3}}{{z}_{3}} \right]-{{{\tilde{P}}}_{Ld}} \right] \end{array} $$

(14)

According to Condition II, the upper bound of Eq. 13 can be expressed as:

$$ {{ {{{\dot{V}}}_{2}} \rvert}_{{{z}_{3}}=0}}\le -{{z}_{2}}^{T}{{k}_{2}}{{z}_{2}}\le 0 $$

(15)

According to Condition IV, (14) can be rewritten as:

$$ {{V}_{3}}\le -{{z}_{3}}^{T}{{k}_{3}}{{z}_{3}}\le 0 $$

(16)

Thus, the asymptotic output tracking of Step II in the process of control law design is achieved, i.e., z₃ → 0 as $t\to \infty$. And on this premise, the asymptotic output tracking of Step I is also achieved.

$$ {{ {{{\dot{V}}}_{2}}={{{\dot{V}}}_{2}} \rvert}_{{{z}_{3}}=0}}\le -{{z}_{2}}^{T}{{k}_{2}}{{z}_{2}}\le 0 $$

(17)

That is,

$$ \begin{array}{@{}rcl@{}} {{z}_{2}}\to \text{0} \end{array} $$

as $t\to \infty$.

And according to Eq. ??, z₁ → 0 can be achieved on the premise of that z₂ → 0 as $t\to \infty$. The proof for Theorem 2 is complete. □