Unified Passivity-Based Visual Control for Moving Object Tracking

Abstract

In this chapter, a unified passivity-based visual servoing control structure considering a vision system mounted on the robot is presented. This controller is suitable to be applied for robotic arms, mobile robots, as well as mobile manipulators. The proposed control law makes the robot able to perform a moving target tracking in its workspace. Taking advantage of the passivity properties of the control system and considering exact knowledge of the target velocity, the asymptotic convergence of the control errors to zero is proved. Later, a robustness analysis is carried out based on L ₂-gain performance, thus proving that control errors are ultimately bounded even when bounded errors exist in the estimation of the target velocity. Both numerical simulation and experimental results illustrate the performance of the algorithm in a robotic manipulator, a mobile robot, and also a mobile manipulator.

Appendices

Appendix 1

1.1 Mobile Robot Model

This chapter considers a unicycle-like robot, consisting of two self-driven wheels located on the same axle and a castor wheel, as Fig. 12.37 shows. Therefore, if the robot is considered as a concentrated mass placed in the point C at the middle of the wheel’s axle, the kinematic model that represents the pose of the robot in the plane are

$$ {\displaystyle \begin{array}{c}\dot{x}=u\cos \phi \\ {}\dot{y}=u\sin \phi \\ {}\dot{\phi}=\omega \end{array}} $$

(12.41)

where (x, y) represents the position of the robot, ϕ represents the robot’s orientation, u is the linear velocity of the robot, and ω is the angular velocity of the robot.

Fig. 12.37

Geometric description of the mobile robot

This type of robots has a non-holonomic constraint given by

$$ \dot{y}\cos \phi -\dot{x}\sin \phi =0 $$

(12.42)

This restriction states that the robot is not able to move laterally but it is able only to navigate in the perpendicular direction to the wheels’ axle.

1.2 Feature Selection

Without loss of generality for the control laws proposed in this chapter, a cylindrical object is selected, defining the vector of image features as $\boldsymbol{\upxi} ={\left[\begin{array}{cc}{\xi}_1& {\xi}_2\end{array}\right]}^{\mathrm{T}}={\left[\begin{array}{cc}{x}_{\mathrm{m}}& {d}_{\mathrm{m}}\end{array}\right]}^{\mathrm{T}}$, where x _m is the projection of the cylinder middle point x-coordinate on the image plane; and d _m represents the projection of the actual width D of the cylinder on the image plane [25]. This feature definition is depicted in Fig. 12.38. According to the camera projection model (12.4), the image features are straightforwardly obtained as

$$ {x}_{\mathrm{m}}=f\frac{x_{\mathrm{Tmc}}}{z_{\mathrm{Tmc}}};\kern1em {d}_{\mathrm{m}}=f\frac{D}{z_{\mathrm{Tmc}}}\vspace*{-10pt} $$

(12.43)

Fig. 12.38

Image features

Now, the problem consists of obtaining the vision system model. This model has to describe the time variation of the image characteristics $\dot{\boldsymbol{\upxi}}$ as a function of both the robot motion ${\left[\begin{array}{cc}u& \omega \end{array}\right]}^{\mathrm{T}}$ and the target velocity v _T. For this purpose, the pose of the target on the X _mc − Z _mc plane with respect to the vision system can be written as a function of the distance d and the angle φ, defined as depicted in Fig. 12.39, as follows:

$$ {\displaystyle \begin{array}{c}{x}_{\mathrm{Tmc}}=d\sin \varphi \\ {}{z}_{\mathrm{Tmc}}=d\cos \varphi \end{array}} $$

(12.44)

Fig. 12.39

Relative posture between the target and the robot

Replacing (12.44) with (12.43), the vector of image characteristics is expressed as a function of the relative pose between the target and the camera

$$ \boldsymbol{\upxi} ={\left[\begin{array}{cc}f\tan \varphi & f\frac{D}{d\cos \varphi}\end{array}\right]}^{\mathrm{T}} $$

(12.45)

and the time derivative of (12.45) is

$$ \dot{\boldsymbol{\upxi}}=\frac{\partial \left({\xi}_1,{\xi}_2\right)}{\partial \left(\varphi, d\right)}{\left[\begin{array}{cc}\dot{\varphi}& \dot{d}\end{array}\right]}^{\mathrm{T}}={\mathbf{J}}_1{\left[\begin{array}{cc}\dot{\varphi}& \dot{d}\end{array}\right]}^{\mathrm{T}} $$

(12.46)

with

$$ {\mathbf{J}}_1=\left[\begin{array}{cc}f{\sec}^2\varphi & 0\\ {}\frac{fD}{d}\sec \left(\varphi \right)\tan \left(\varphi \right)& -\frac{fD}{d^2}\sec \left(\varphi \right)\end{array}\right] $$

(12.47)

The variation in relative position between the robot and the target is due to both the robot motion and the target motion: ${\left[\begin{array}{cc}\dot{\varphi}& \dot{d}\end{array}\right]}^{\mathrm{T}}={\left[\begin{array}{cc}\dot{\varphi}& \dot{d}\end{array}\right]}_{\mathrm{R}}^{\mathrm{T}}+{\left[\begin{array}{cc}\dot{\varphi}& \dot{d}\end{array}\right]}_{\mathrm{T}}^{\mathrm{T}}$. Now, from the kinematic model of the mobile robot in polar coordinates (see Fig. 12.39) and first considering a static object, the time variation of the relative posture between the target and the robot (time variation of d and φ) is obtained as a function of the linear and angular velocities of the robot $\boldsymbol{\upmu} ={\left[\begin{array}{cc}u& \omega \end{array}\right]}^{\mathrm{T}}$ as follows:

$$ {\left[\begin{array}{c}\dot{\varphi}\\ {}\dot{d}\end{array}\right]}_{\mathrm{R}}=\left[\begin{array}{cc}\frac{\sin \left(\varphi \right)}{d}& 1\\ {}-\cos \left(\varphi \right)& 0\end{array}\right]\left[\begin{array}{c}u\\ {}\omega \end{array}\right]={\mathbf{J}}_2\boldsymbol{\upmu} $$

(12.48)

Now consider a static position for the robot, that is, constant values for x, y, ϕ; obtaining the time derivative of (12.44), the target velocity ${\mathbf{v}}_{\mathrm{T}}={\left[\begin{array}{cc}{\dot{x}}_{\mathrm{T}\mathrm{mc}}& {\dot{z}}_{\mathrm{T}\mathrm{mc}}\end{array}\right]}^{\mathrm{T}}=\mathbf{A}{\left[\begin{array}{cc}\dot{\varphi}& \dot{d}\end{array}\right]}_{\mathrm{T}}^{\mathrm{T}}$ is expressed as

$$ {\mathbf{v}}_{\mathrm{T}}=\left[\begin{array}{cc}d\cos \left(\varphi \right)& \sin \left(\varphi \right)\\ {}-d\sin \left(\varphi \right)& \cos \left(\varphi \right)\end{array}\right]{\left[\begin{array}{c}\dot{\varphi}\\ {}\dot{d}\end{array}\right]}_{\mathrm{T}} $$

(12.49)

Then, as A is invertible, the following expression can be obtained

$$ {\displaystyle \begin{array}{c}{\left[\begin{array}{c}\dot{\varphi}\\ {}\dot{d}\end{array}\right]}_{\mathrm{T}}=\left[\begin{array}{cc}\frac{1}{d}\cos \left(\varphi \right)& -\frac{1}{d}\sin \left(\varphi \right)\\ {}\sin \left(\varphi \right)& \cos \left(\varphi \right)\end{array}\right]{\mathbf{v}}_{\mathrm{T}}\\ {}{\left[\begin{array}{c}\dot{\varphi}\\ {}\dot{d}\end{array}\right]}_{\mathrm{T}}={\mathbf{J}}_0{\mathbf{v}}_{\mathrm{T}}\end{array}} $$

(12.50)

Finally, by introducing the motions of both the robot (12.48) and the target (12.50) into (12.46), the model of the vision system is obtained

$$ \dot{\boldsymbol{\upxi}}={\mathbf{J}}_1\left({\mathbf{J}}_2\boldsymbol{\upmu} +{\mathbf{J}}_0{\mathbf{v}}_{\mathrm{t}}\right) $$

(12.51)

Defining

$$ {\displaystyle \begin{array}{c}\mathbf{J}={\mathbf{J}}_1{\mathbf{J}}_2\\ {}{\mathbf{J}}_{\mathrm{T}}={\mathbf{J}}_1{\mathbf{J}}_0\end{array}} $$

(12.52)

a compact form for the vision system model is obtained

$$ \dot{\boldsymbol{\upxi}}=\mathbf{J}\boldsymbol{\upmu } +{\mathbf{J}}_{\mathrm{T}}{\mathbf{v}}_{\mathrm{t}} $$

(12.53)

where

$$ \mathbf{J}=\left[\begin{array}{cc}\frac{x_{\mathrm{m}}{d}_{\mathrm{m}}}{fD}& \frac{f^2+{x}_{\mathrm{m}}^2}{f}\\ {}\frac{d_{\mathrm{m}}^2}{fD}& \frac{x_{\mathrm{m}}{d}_{\mathrm{m}}}{f}\end{array}\right] $$

(12.54)

$$ {\mathbf{J}}_{\mathrm{T}}=\left[\begin{array}{cc}\frac{d_{\mathrm{m}}}{D}& -\frac{x_{\mathrm{m}}{d}_{\mathrm{m}}}{fD}\\ {}0& -\frac{d_{\mathrm{m}}^2}{fD}\end{array}\right] $$

(12.55)

Appendix 2

Formal definitions associated with passivity of operators relating functional spaces, used in this work, follow [44].

Definition 1

L _p signal spaces.

For all p ∈ [1, ∞), L _p signal spaces are defined as

$$ {L}_{\mathrm{p}}=\left\{f:{\mathrm{\mathbb{R}}}_{+}\to \mathrm{\mathbb{R}}/{\int}_0^{\infty }{\left|f(t)\right|}^p dt<\infty \right\}. $$

L _p are the Banach spaces with respect to the norm

$$ {\left\Vert f\right\Vert}_p={\left({\int}_0^{\infty }{\left|f(t)\right|}^p dt\right)}^{\frac{1}{p}}. $$

Definition 2

L _∞ signal spaces.

L _∞ signal spaces are defined as

$$ {L}_{\infty }=\left\{f:{\mathrm{\mathbb{R}}}_{+}\to \mathrm{\mathbb{R}}/\underset{t\in {\mathrm{\mathbb{R}}}_{+}}{\sup}\kern0.5em \left|f(t)\right|<\infty \right\}. $$

L _∞ are the Banach spaces with respect to the norm

$$ {\left\Vert f\right\Vert}_{\infty }=\underset{t\in {\mathrm{\mathbb{R}}}_{+}}{\sup}\kern0.5em \left|f(t)\right|. $$

Definition 3

Truncated function.

Let f : ℝ₊ → ℝ. Then, for each T ∈ ℝ₊, the function f _T : ℝ₊ → ℝ is defined by

$$ {f}_{\mathrm{T}}(t)=\left\{\begin{array}{ll}f(t)& 0\le t<T\\ {}0& \kern1.75em t\ge T\end{array}\right. $$

Definition 4

Extended L _p signal spaces.

Extended L _p spaces are defined as

$$ {L}_{\mathrm{p}\mathrm{e}}=\left\{f/{f}_{\mathrm{T}}\in {L}_{\mathrm{p}}\kern1em \forall T<\infty \right\}. $$

Definition 5

Given g, h ∈ L _2e, the inner product and the norm ‖•‖_2e in the set L _2e are defined as

$$ {\displaystyle \begin{array}{c}{\left\langle g,h\right\rangle}_{\mathrm{T}}={\int}_0^{\mathrm{T}}g(t)h(t)\ dt\kern1em \forall T\in \left[0,\infty \right)\\ {}{\left\Vert g\right\Vert}_{2,\mathrm{T}}={\left\langle g,g\right\rangle}_{\mathrm{T}}^{\frac{1}{2}}={\left({\int}_0^{\mathrm{T}}g(t)g(t)\ dt\right)}^{\frac{1}{2}}\end{array}} $$

Definition 6

Let G : L _2e → L _2e be an input-output mapping. Then, G is passive if there exists some constant β such that

$$ {\left\langle \mathbf{Gx},\mathbf{x}\right\rangle}_{\mathrm{T}}\ge \beta \kern1em \forall \mathbf{x}\in {L}_{2e}\kern1em \forall T\in \left[0,\infty \right) $$

Definition 7

Let G : L _2e → L _2e be an input-output mapping. Then, G is strictly input passive if there exists some constants β ∈ ℝ and δ > 0 such that

$$ {\left\langle \mathbf{Gx},\mathbf{x}\right\rangle}_{\mathrm{T}}\ge \beta +\delta\ {\left\Vert \mathbf{x}\right\Vert}_{2,\mathrm{T}}^2\kern1em \forall \mathbf{x}\in {L}_{2e} $$

Definition 8

Let G : L _2e → L _2e be an input-output mapping. Then, G is strictly output passive if there exists some constants β ∈ ℝ and δ > 0such that

$$ {\left\langle \mathbf{Gx},\mathbf{x}\right\rangle}_{\mathrm{T}}\ge \beta +\delta\ {\left\Vert \mathbf{Gx}\right\Vert}_{2,\mathrm{T}}^2\kern1em \forall \mathbf{x}\in {L}_{2e} $$