# Plant-Inspired Soft Growing Robots: A Control Approach Using Nonlinear Model Predictive Techniques
## 5 Results and Discussion
### Point Stabilization Results  
As previously discussed, one of the purposes of vine-like growing robots is to act as a channel to transport supplies to those in dangerous situations. In this first simulation experiment, based on the state estimation achieved by the proposed MHE framework, the vine robot began at an initial state \(x_{0}=[0,0,0.4,0.4,0,0]^{T}\), and the suggested NMPCbased growth controller is involved in maintaining the robot's tip at predefined desired states in the space, i.e., \(\mathbf{x}_{d}\in\mathds{R}^{6}\). These goals could describe potential places for the robot to attend within the working environment. The sampling time is decided to be \(T_{s}=0.1\)s with a prediction horizon \(N=10\). Diagonal state and input weighting matrices in (10) are taken, where \(\mathbf{Q}=\text{diag}(1,1,1,0,0,0)\), while \(\mathbf{R}=\text{diag}(0.5,0.5,0.5)\). The working environment is designed to be constrained between \([-4,4]\) m amongst the three Cartesian coordinates. The extremes of the robot's shape variables were selected based on the robot's configuration limits discussed beforehand in the kinematics section.  
\[-4 \leq\begin{bmatrix}x(m)\\ y(m)\\ z(m)\end{bmatrix}\leq 4, \tag{13}\] \[\begin{bmatrix}0\\ -\pi\\ -\pi\end{bmatrix} \leq\begin{bmatrix}s(m)\\ \theta(rad)\\ \phi(rad)\end{bmatrix}\leq\begin{bmatrix}10\\ \pi\\ \pi\end{bmatrix}\]  
On the other side, we chose the input inequality constraints that consider the irreversible characteristics of the growing vine-like robot and the actuator boundaries as follows:  
\[\begin{bmatrix}0\\ -\frac{\pi}{10}\\ -\frac{\pi}{10}\end{bmatrix}\leq\begin{bmatrix}s(m/s)\\ \theta(rad/s)\\ \phi(rad/s)\end{bmatrix}\leq\begin{bmatrix}0.1\\ \frac{\pi}{10}\\ \frac{\pi}{10}\end{bmatrix} \tag{14}\]  
Figure 5 shows the proposed NMPC-based growth controller performance in the point stabilization simulation scenario. Throughout the experiment's initial 25 s, the controller maintains the vine robot's tip at \((1,1,1)\) m distance from its base. Satisfying rising time results (20 s) are achieved to attain the objective using the highlighted actuators' inputs. The \(z\) coordinate of the goal is doubled, as illustrated, during the next 25 s, where the length of the robot \(s\) is required to be increased. Based on the vine robot's kinematics, extending the robot's length to arrive at the new \(z\) value would have an effect on the other \(x\) and \(y\) coordinates. As a result, when reaching this new target, both \(x\) and \(y\) positions have been marginally affected; this is demonstrated in Figure 5. To address this problem and restore the tip of the robot, the NMPC increased the robot length \(s\) while actuating the curvature angle _theta_ in the positive direction. After some time, only the robot length is increased to accommodate for the adjusted \(x\) and \(y\) locations. Subsequently, following the passing of 50 s of simulation time, the robot is intended to attend a new \(z\) target lower than the preceding one. This task necessitates the robot shortening its length. However, because of the irreversible nature of the growing mechanism, the robot is bound and unable to shorten its grown length. As a result, the NMPC reduced the robot curvature in the hope of achieving the current specified goal. While this helped achieve a minimal error in the \(z\) coordinate, it substantially impacted the other two coordinates. The NMPC meets the vine-like robot's state and input saturation limits at all phases.  
The NMPC primarily relies on incorrect state computations by solving the forward kinematics in (1) to find the corresponding robot's Cartesian positions, since the MHE only estimates the robot's state after passing at least \(N_{e}\) samples. As seen in Figure 5, the approximate states and, as a result, the control behaviors are very noisy during the initial samples of \(k<N_{e}\) relative to the remaining samples.