**Author details**

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. Bounded-

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

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

ness of the parameter estimation errors can be seen in **Figure 4**.

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

**6. Conclusions and future directions**

**Figure 4.**

**140**

**Figure 3.**

*Collaborative and Humanoid Robots*

*with BF.*

Iman Salehi<sup>1</sup> , Ghananeel Rotithor<sup>2</sup> and Ashwin Dani<sup>1</sup> \*


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

© 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, provided the original work is properly cited.
