# Plant-Inspired Soft Growing Robots: A Control Approach Using Nonlinear Model Predictive Techniques
## 1 Introduction
In the literature, few studies have aimed to control the growth of robots, whether in the joint or task space. As an example, a stimulus-based feedback controller has been employed in [10] to control the locomotion of a root-like growing robot. This was based on the tactile measurements perceived from the sensors embedded in the robot's tip. In [16], minimizing the energy spent during the penetration of soil environments was achieved by controlling the tip of a plant-inspired root robot through an optimal control technique. To guarantee the performance of the robot when it comes to trajectory following, research in [17] applied a joint-space Proportional-Derivative (PD) controller with gravity compensation to the obtained dynamics model of vine robots to guarantee the robot's performance. A visual servoing Jacobian-based controller is proposed in [18] to maneuver a vine robot into the environment until reaching the desired target defined in image space. However, two key challenges were encountered in the mentioned control schemes, namely, the lack of incorporating the process constraints inhibited by the growing robot, and dealing with the nonlinearity of the coupled dynamics of the robot. Handling the process constraint is particularly critical since the extension of growing robots is managed by an irreversible addition of materials to the robot's tip, and once the robot has grown to a certain length, it can not be retracted back to smaller values. Moreover, such conventional controllers assume complete knowledge of the robot state; however, in reality, measuring all the states could be cumbersome in lengthy growing robots, or it could not be advisable from an economic perspective. In this paper, a Nonlinear Model Predictive Control (NMPC) combined with Moving Horizon Estimation (MHE) is proposed for controlling the growth of vine-like growing robots in terms of point stabilization, trajectory tracking, and obstacle avoidance. The con tribution of this paper is twofold: (1) a nonlinear MPC control scheme is proposed to control the movement of the vine growing robot while considering the process constraints utilizing an adaptive nonlinear kinematic model of the vine robot, and (2) a precise online estimation of the vine robot's state, including its Cartesian coordinates, is introduced based on the MHE scheme considering that no such practical methods exist to deploy sensors to measure the spatial location of the vine robot's tip. Instead of applying the well-known Extended Kalman Filters (EKF) as in [19], an optimization-based nonlinear MHE framework has proposed an online state estimation technique in this research. In this paper, a Nonlinear Model Predictive Control (NMPC) combined with Moving Horizon Estimation (MHE) is proposed for controlling the growth of vine-like growing robots in terms of point stabilization, trajectory tracking, and obstacle avoidance. The contribution of this paper is twofold: (1) a nonlinear MPC control scheme is proposed to control the movement of the vine-growing robot while considering the process constraints utilizing an adaptive nonlinear kinematic model of the vine robot, and (2) a precise online estimation of the vine robot's state, including its Cartesian coordinates, is introduced based on the MHE scheme considering that no such practical methods exist to deploy sensors to measure the spatial location of the vine robot's tip. Instead of applying the well-known Extended Kalman Filters (EKF) as in [19], an optimization-based nonlinear MHE framework has proposed an online state estimation technique in this research. Section 2 describes the model of the vine robot utilized in this study. The moving horizon state estimation is explained in detail in Section 3 while Section 4 describes the nonlinear model predictive control problem. The numerical simulations are presented in Section 5 and the results and their discussion. Finally, the conclusions from this research are summarized in Section 6.