>>Time Delay Analysis in Machine Vision Systems

### Time Delay Analysis in Machine Vision Systems

In this work, an approach to control a 6-DoF stereo camera for the purpose of Human-Robot Interaction (HRI) is proposed. The main objective in the presented work is to cope with the critical time-delay introduced by computer vision algorithms used to acquire the feedback variable within a visual controlled system. In the studied HRI architecture, the feedback variable is represented by the 3D position of a human subject. We proposed two methods of control. The fist one uses the generalized form of the Hermite-Biehler theorem applied to quasi-polynomials, while the second one use a predictive control method.

##### The plant

Fig. 1. Block diagram of the pan control system within the proposed active vision architecture.

The control system, with the block diagram of a single DoF illustrated in Fig. 1, consists of two PTZ video cameras. The main blocks are:

• $C(s)$ represents the overall system controller, used to maintain the stability of the control structure;
• $M(s)$ represents a conversion block which converts pixels to degrees;
• $R(s)$ is the inner loop controller used to control the camera\'s DC (Direct Current) motors;
• $P(s)$ represents the plant, that is, the stereo camera;
• $I(s)$ is an integral block for velocity to position transformation;
• $V(s)$ represents the image processing component which includes all 2D and 3D computations.
##### Controller design using the Hermite-Biehler theorem

In order to determine the $C(s)$ controller, the open-loop transfer function is considered, without taking into account the time-delay inserted by the image processing component. Using this function, the maximum delay margin that can be inserted into the system, without destabilizing it, is determined and used to compute the controller's gain. A Proportional (P) controller for a second order transfer function is proposed:

$G(s)=\frac{k}{s^2+a_1s+a_0} e^{-\tau s}$

For $a_1^2 < 2a_0$, the controller's stability interval is computed using the following expression:

$\max_{j=m,\;m+2} \left\{ \frac{a_1v_j}{k \tau \sin v_j} \right\} < k_c < \min_{j=n,\;n+2} \left\{ \frac{a_1v_j}{k \tau \sin v_j} \right\}$

where, $v_j$ represents a solution in the $(0, \pi)$ interval of:

$\text{ctg} v = \frac{v^2-\tau^2 a_0}{\tau a_1v}$

and $\tau$ is the dead-time. Using the above approach, a a stability interval is obtained.

##### Predictive control

The dead time presented in the system cannot be actually separated from the process. In order to stabilize the plant, a prediction control structure can be implemented, such as the Smith predictor technique illustrated in Fig. 2.

Fig. 2. Basic block diagram of a predictive control structure.

The core concept of the Smith predictor is to move the process's dead time outside the feedback loop of the control system and to determine a controller of a time-delay free system. In order to design a controller $C_r(s)$ capable of stabilizing the plant having its dead time outside the control loop, a Smith predictor compensator has been used:

$C_r(s) = \frac{C_r^*(s)}{1+C_r^*(s) G_p(s)(1-e^{-s\\tau})}$

The $C_r^*(s)$ controller can be determined using the pole placement method. The idea is to construct a Proportional-Integral-Derivative (PID) controller and to obtain a characteristic polynomial using an imposed criteria. The obtained controller is implemented as a general expression of the overall time-delay system.

##### References

G. Macesanu, F. Moldoveanu and S.M. Grigorescu "A Time-Delay Control Approach for a Stereo Vision Based Human-Machine Interaction System", Journal of Intelligent and Robotic Systems, Springer, 2013.

G. Macesanu, S.M. Grigorescu, J.F. Ferreira, J. Dias and F. Moldoveanu, "Real Time Facial Features Tracking using an Active Vision System," 13th International Conference on Optimization of Electrical and Electronic Equipment, Brasov, Romania, 24-26 May 2012, pp. 1493-1498.