**5.1 Safe tracking control of an uncertain EL-dynamics with full-state constraints using BF**

In this section, the controller and adaptive laws developed in (37) and (43) are simulated for a two-link robot planar manipulator, with dynamics shown in (46), where c1, c2, c12 denote cos *q*<sup>1</sup> � �, cos *<sup>q</sup>*<sup>2</sup> � ), and cos *<sup>q</sup>*<sup>1</sup> <sup>þ</sup> *<sup>q</sup>*<sup>2</sup> � � respectively, sin <sup>2</sup> denotes sin *q*<sup>2</sup> � �, and *g* is the gravitational constant.

$$
\underbrace{\begin{bmatrix}
\theta\_{1}+2\theta\_{2}c\_{2} & \theta\_{3}+\theta\_{2}c\_{2} \\
\theta\_{3}+\theta\_{2}c\_{2} & \theta\_{3}
\end{bmatrix}}\_{\mathcal{M}(q)}\begin{bmatrix}
\ddot{\boldsymbol{q}}\_{1} \\
\ddot{\boldsymbol{q}}\_{2}
\end{bmatrix} + \underbrace{\begin{bmatrix}
\theta\_{2}\sin\_{2}\dot{q}\_{1} & 0
\end{bmatrix}}\_{\mathcal{C}(q,\dot{q})}\begin{bmatrix}
\dot{\boldsymbol{q}}\_{1} \\
\dot{\boldsymbol{q}}\_{2}
\end{bmatrix} + \underbrace{\begin{bmatrix}
\theta\_{4\xi\mathsf{c}\_{1}+\theta\_{3}\mathsf{c}\_{12}} \\
\theta\_{6\mathsf{g}\mathsf{c}\_{12}}
\end{bmatrix}}\_{\mathcal{G}\_{7}(q)} = \begin{bmatrix}
\boldsymbol{\tau}\_{1} \\
\boldsymbol{\tau}\_{2}
\end{bmatrix} \tag{46}
$$

The nominal values of the parameter vector *θ* ¼ ½ � *θ*1, *θ*2, *θ*3, *θ*4, *θ*<sup>5</sup> *T* are

$$\begin{aligned} \theta\_1 &= 0.325 \text{ kg} \cdot \text{m}^2 & \quad \theta\_3 = 0.217 \text{ kg} \cdot \text{m}^2\\ \theta\_2 &= 0.240 \text{ kg} \cdot \text{m}^2 & \quad \theta\_4 = 2.4 \text{ kg} \cdot \text{m} & \quad \theta\_5 = 1.0 \text{ kg} \cdot \text{m} \end{aligned} \tag{47}$$

The desired trajectory is selected as

$$q\_{d\_1} = \left(-4 - \mathfrak{G}e^{-2t}\right)\sin\left(t\right), q\_{d\_2} = \left(-4 - \mathfrak{G}e^{-t}\right)\cos\left(t\right). \tag{48}$$

The objective is to track the desired joint trajectory provided that the model parameters are unknown while the state *<sup>Q</sup>* <sup>¼</sup> ½ � *<sup>q</sup>*, *<sup>q</sup>*\_ *<sup>T</sup>* satisfies the following constraints,

$$\begin{array}{ll} q\_1 \in (-4.4, 4.1) & \dot{q}\_1 \in (-10.2, 4.2) \\ q\_2 \in (-7.1, 4.2) & \dot{q}\_2 \in (-4.2, 4.93) \end{array} \tag{49}$$

To this end, the barrier function formulation presented in Section 3 is used along with the adaptive control developed in Section 4. The feedback and adaptation gains for the proposed controller are selected as *β* ¼ 14, Λ ¼ diag 2ð Þ *:*01, 2*:*01 , and Γ ¼ diag 30, 30 ð Þ. The results of the simulation are shown in **Figures 2**–**4**. The joints

**Figure 2.** *Evolution of the joint angles for the planar robot simulation using an adaptive law with and without BF.*

with a gradient based adaptive parameter estimation law on the transformed system that tracks the desired trajectories of the original system. The controller guarantees that the robot trajectories remain inside a pre-specified safe region, tracking the desired trajectories and the parameter estimation errors remain bounded. The method can be utilized for applications wherein robots must operate in a confined space to reach an object for grasping or other manipulation tasks such as pick and

*Safe Adaptive Trajectory Tracking Control of Robot for Human-Robot Interaction Using…*

In future, the usefulness of barrier transformation to design a visual servo controller will be shown. Constrained VS approach can guarantee target features to remain within the camera field of view for the duration of the task. Some recent efforts in that direction can be found in [73]. Utilizing CBF for developing safe robot controllers by utilizing human actions and workspaces can be another avenue

The authors would like to thank Daniel Trombetta for discussions related to

of future research for safe human-robot interaction.

*DOI: http://dx.doi.org/10.5772/intechopen.97255*

human-robot interaction application and control design.

, Ghananeel Rotithor<sup>2</sup> and Ashwin Dani<sup>1</sup>

\*Address all correspondence to: ashwin.dani@uconn.edu

2 Electrical and Computer Engineering, University of Connecticut, Storrs, CT, USA

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

1 University of Connecticut, Storrs, USA

provided the original work is properly cited.

\*

place.

**Acknowledgements**

**Author details**

Iman Salehi<sup>1</sup>

**141**

**Figure 3.**

*Evolution of the joint angle errors and joint velocity errors for the planar robot simulation using an adaptive law with BF.*

**Figure 4.**

*Evolution of the parameter estimation error for the planar robot simulation.*

position evolution *q*1ð Þ*t* and *q*2ð Þ*t* of a two degrees-of-freedom planar robot using an adaptive law with and without BF are shown in **Figure 2**. It can be observed from **Figure 2** that when the adaptive law with BF is used, the estimated trajectories are blocked from crossing over the boundaries that are set for each of the joints. The position and velocity estimation errors are depicted in **Figure 3**. From **Figures 2** and **3**, it is clear that the tracking error asymptotically converges to zero, and, because the Lyapunov candidate does not contain any terms that are negative definite in ~*θ*, the parameter estimation does not converge but it does remain bounded. Boundedness of the parameter estimation errors can be seen in **Figure 4**.

#### **6. Conclusions and future directions**

This chapter provides a perspective on problems wherein humans and robots work collaboratively with one another. Research in this field aims to relax the current workplace constraints, such as fences, virtual curtains often seen in manufacturing settings between humans and robots or velocity limits on collaborative robots. This chapter develops an efficient robot control methodology to create a safe working environment without sacrificing the efficiency of the robots. In the context of the chapter, safety is defined as a constrained behavior of a system, and robot effectiveness, as driving the actual behavior of the robot to the desired behavior. To this end, an online safe tracking controller for an uncertain Euler– Lagrange robotic system with is developed where the constraints are placed on all the states. A barrier function transform is used to transform the full-state constrained EL-dynamics into an equivalent unconstrained system with no prior knowledge of the system parameters. An adaptive controller is developed along

*Safe Adaptive Trajectory Tracking Control of Robot for Human-Robot Interaction Using… DOI: http://dx.doi.org/10.5772/intechopen.97255*

with a gradient based adaptive parameter estimation law on the transformed system that tracks the desired trajectories of the original system. The controller guarantees that the robot trajectories remain inside a pre-specified safe region, tracking the desired trajectories and the parameter estimation errors remain bounded. The method can be utilized for applications wherein robots must operate in a confined space to reach an object for grasping or other manipulation tasks such as pick and place.

In future, the usefulness of barrier transformation to design a visual servo controller will be shown. Constrained VS approach can guarantee target features to remain within the camera field of view for the duration of the task. Some recent efforts in that direction can be found in [73]. Utilizing CBF for developing safe robot controllers by utilizing human actions and workspaces can be another avenue of future research for safe human-robot interaction.
