Adaptive neural PD controllers for mobile manipulator trajectory tracking

P, PI, PD and PID controllers brief summary

Adaptive neuron PD controller

The main disadvantage of conventional PD controllers is that they are not suitable for nonlinear, time-variant systems. adaptive neural controllers are an alternative to overcome this issue.

The proposed adaptive single neuron PD (SNPD) controller is illustrated in Fig. 2. The value e represents the error (1) between the reference y_r and the system output y. The inputs x₁ and x₂ are defined as the proportional (2) and the derivative (3) errors (Moradi, Katebi & Johnson, 2001). The weights ω₁ and ω₂, are adapted online using the EKF algorithm. The weight ω₁ and ω₂ represents the proportional gain, and derivative gain, respectively. The value v is computed as the weighted sum of the inputs of the neuron (4). Finally, the output of the neuron $\hat{y}$ is computed with (5), where as activation function is selected tanh(·). The activation function reacts in the range (−1, 1). However, the parameter α can be selected to adjust the control action, since the output of the neuron is directly the control signal $u\left(k\right)=\hat{y}\left(k\right)$.

The proposed EKF-based training method is described in a section bellow. EKF provides faster learning rates and convergence time than backpropagation, which is crucial for online training.

Adaptive multilayer PD controller

The multilayer network PD (MNPD) scheme is shown in Fig. 3; it consists of a fully connected neural network with one hidden layer with multiples nodes and one node at the output layer. The network input is the error and the derivative between a reference value and the system output. The neural network is trained online using an extended Kalman filter-based algorithm; the objective is to reduce the tracking error by adapting online the output of the network, which is the control signal to the system, it is $u\left(k\right)=\hat{y}\left(k\right)$.

Consider a neural network as shown in Fig. 3 with 2 input signals and q nodes in the hidden layer.

The output of the network is given by (6) $${\sigma }_i(k)=\tanh \left(n_i(k)\right),i=1\dots q,$$ (7) $$n_i(k)=\sum _{j=0}^2{\omega }_{ij}^{\left(1\right)}(k)x_j(k),x_0(k)=+1,$$ (8) $$v_1(k)=\sum _{k=0}^q{\omega }_{1j}^{\left(2\right)}(k)u_k(k),u_0(k)=+1,$$ (9) $$\hat{y}(k)=v_1(k).$$

Extended Kalman filter based training algorithm for neural networks

For training of neural networks, the weights of the network become the state to be estimated by the EKF, with the objective of reducing the neural network error, which in this case, since the output is the reference minus the system output e(k) = y_r(k) − y(k), this is because the neural network output is considered directly as the control signal u(k). The fact that e is minimized means that the neural PD controller output u is working in achieving the control obective of tracking the desired reference y_r. The neural network is trained online using an extended Kalman filter-based algorithm (10–12).

(10) $$\mathbf{K}\left(k\right)=\mathbf{P}\left(k\right)\mathbf{H}\left(k\right){\left[\mathbf{R}\left(k\right)+{\mathbf{H}}^{\mathrm{T}}\left(k\right)\mathbf{P}\left(k\right)\mathbf{H}\left(k\right)\right]}^{-1},$$ (11) $$\omega \left(k+1\right)=\omega \left(k\right)+\eta \mathbf{K}\left(k\right)\mathbf{e}\left(k\right),$$ (12) $$\mathbf{P}\left(k+1\right)=\mathbf{P}\left(k\right)-\mathbf{K}\left(k\right){\mathbf{H}}^{\mathrm{T}}\left(k\right)\mathbf{P}\left(k\right)+\mathbf{Q}\left(k\right),$$ (13) $${\mathbf{h}}_{ij}\left(k\right)=\left[{\displaystyle \frac{\mathrm{\partial }{\mathrm{y}}_i\left(k\right)}{\mathrm{\partial }{\omega }_j\left(k\right)}}\right].$$where $\omega \in {\mathbb{R}}^n$ is the weight vector, $\mathbf{K}\in {\mathbb{R}}^{n\times m}$ is the Kalman gain vector with n as the number of weights, and m the number of outputs of the neural network; $\mathbf{P}\in {\mathbb{R}}^{n\times n}$, $\mathbf{Q}\in {\mathbb{R}}^{n\times n}$, and $\mathbf{R}\in {\mathbb{R}}^{m\times m}$ are covariance matrices of weight estimation error, estimation noise, and error noise, respectively; $\eta \in \mathbb{R}$ is the Kalman filter learning rate, and $\mathbf{H}\in {\mathbb{R}}^{n\times m}$ is a matrix whose entries h_ij are the derivative of the neural network output with respect to each weight Eq. (13), ${\mathrm{y}}_i\in \mathbb{R}$ is the i-th output of the neural network and j = 1 ⋯ n, the error $\mathbf{e}\in {\mathbb{R}}^m$ is defined as the difference between the desired output and the neural network output (Sanchez & Alanis, 2006).

Neural network weights are initialized randomly. The Kalman filter learning rate η is selected heuristically to minimized e. It should be considered that if η is sufficiently large, the network could not converge. Conversely, if a lower η is selected, it would take longer to converge. Moreover, matrices $\mathbf{P}$, $\mathbf{Q}$ and $\mathbf{R}$ are initialized as diagonal matrices with initial values chosen heuristically. The $\mathbf{Q}$ matrix is set to deal with process noise, while the $\mathbf{R}$ matrix is set to deal with measurement noise. Metaheuristic algorithms can be used to optimize the initial values of the Kalman settings (Villaseñor et al., 2018).

Let us remark that matrices H, K and P are bounded (Song & Grizzle, 1992).

Mobile manipulator kinematics

Mobile manipulators are composed of one or more manipulators attached to a mobile platform. Conventional mobile robots such as unicycles, differential drives, and car-like robots increase the workspace of manipulators. However, these platforms have limited movement capabilities due to their nonholonomic kinematics constraints, Li et al. (2016). On the other hand, omnidirectional mobile platforms improved the movement capabilities, allowing moving towards any position and desired orientation (Zhang et al., 2016; Wu et al., 2017; Kundu et al., 2017). This section introduces a kinematic model of a mobile manipulator, which consists of a robotic manipulator of n Degrees of Freedom (DOF) attached to an omnidirectional mobile platform.

The Kinematics chain of mobile manipulators is described in Fig. 4. The homogeneous matrix ^wT_b defines the position and orientation of the mobile platform. The transformation ^bT_m is a constant homogeneous matrix between the mobile platform frame and the manipulator base. The matrix ^mT_e can be computed based on the Denavit-Hartenberg (DH) model of the manipulator (Spong & Vidyasagar, 2008; Lopez-Franco et al., 2018).

Considering an omnidirectional mobile platform, the pose of the robot with respect to the world frame w is given by 3 DOF, which are the positions x_b and y_b, and the orientation θ_b. Then, the matrix ^wT_b can be defined as Eq. (18).

The matrix ^bT_m is constant, and it adjusts the distance from the mobile platform base frame b to the manipulator base frame m. The values t_x, t_y and t_z are used to adjust the distance in the direction of the x-axis, y-axis and z-axis, respectively. If it does not need to adjust the frame orientation, then matrix ^bT_m can be described by Eq. (19).

Let consider a joint variable $\mathbf{q}$ to represent the platform configuration ${\mathbf{q}}_m={\left[\begin{array}{ccc}{x}_b& y_b& {\theta }_b\end{array}\right]}^T$ and the manipulator configuration ${\mathbf{q}}_{<}sf>m</sf>={\left[\begin{array}{ccccc}{q}_1& q_2& q_3& \cdots & q_n\end{array}\right]}^T$, where q_i is a joint value for the articulation i. The joint variable for the mobile manipulator is given by $\mathbf{q}={\left[\begin{array}{cc}{{\mathbf{q}}_b}^T& {{\mathbf{q}}_m}^T\end{array}\right]}^T$.

Given the joint variable $\mathbf{q}$, the computation of ${}^w{\mathbf{T}}_e\left(\mathbf{q}\right)$ which is the forward kinematics of the mobile manipulator can be obtained as (20) $${}^w{\mathbf{T}}_e\left(\mathbf{q}\right){=}^w{\mathbf{T}}_b\left({\mathbf{q}}_b\right){}^b{\mathbf{T}}_m{}^m{\mathbf{T}}_e\left({\mathbf{q}}_m\right),$$where ${}^w{\mathbf{T}}_e\left(\mathbf{q}\right)$ represents the end-effector pose respect to the world frame w. The matrix ^wT_e is expressed as (21) $${}^w{\mathbf{T}}_e(\mathbf{q})=\left[\begin{array}{cccc}{r}_{11}& r_{12}& r_{13}& t_x\\ r_{21}& r_{22}& r_{23}& t_y\\ r_{31}& r_{32}& r_{33}& t_z\\ 0& 0& 0& 1\end{array}\right]=\left[\begin{array}{cc}\mathbf{R}& \mathbf{t}\\ \mathbf{0}& 1\end{array}\right],$$where the orientation of the end-effector is represented by the matrix R, and its Cartesian position is given by the vector $\mathbf{t}$. More information about homogeneous matrices, manipulators kinematics, and forward kinematics can be found in Spong & Vidyasagar (2008), Craig (2005) and Sciavicco & Siciliano (2008).

Differential kinematics

The inverse kinematics consists in the computation of the joint variables $\mathbf{q}$ given the end-effector pose ${}^0{\mathbf{T}}_n$. This computation can be solved by minimizing an error function using an iterative process based on the differential kinematics (Sciavicco & Siciliano, 2008). Differential kinematics aims to find the relationship between the joint velocities $\dot{\mathbf{q}}$ and the end-effector velocity $\dot{\mathbf{t}}$. The following differential kinematics Eq. (22) gives this relationship (22) $$\dot{\mathbf{t}}=\mathbf{J}\left(\mathbf{q}\right)\dot{\mathbf{q}},$$where $\mathbf{J}$ is the matrix that relates the contribution of the joint velocities $\dot{\mathbf{q}}$ to the end-effector velocity $\dot{\mathbf{t}}$. The matrix $\mathbf{J}$ is called the geometric Jacobian. This Jacobian matrix can be computed as Eq. (23).

(23) $$\mathbf{J}\left(\mathbf{q}\right)=\left[\begin{array}{cccc}{\displaystyle \frac{\mathrm{\partial }{t}_x}{\mathrm{\partial }{q}_1}}& {\displaystyle \frac{\mathrm{\partial }{t}_x}{\mathrm{\partial }{q}_2}}& \cdots & {\displaystyle \frac{\mathrm{\partial }{t}_x}{\mathrm{\partial }{q}_n}}\\ {\displaystyle \frac{\mathrm{\partial }{t}_y}{\mathrm{\partial }{q}_1}}& {\displaystyle \frac{\mathrm{\partial }{t}_y}{\mathrm{\partial }{q}_2}}& \cdots & {\displaystyle \frac{\mathrm{\partial }{t}_y}{\mathrm{\partial }{q}_n}}\\ {\displaystyle \frac{\mathrm{\partial }{t}_z}{\mathrm{\partial }{q}_1}}& {\displaystyle \frac{\mathrm{\partial }{t}_z}{\mathrm{\partial }{q}_2}}& \cdots & {\displaystyle \frac{\mathrm{\partial }{t}_z}{\mathrm{\partial }{q}_n}}\end{array}\right],$$where $\mathbf{t}={\left[\begin{array}{ccc}{t}_x& t_y& t_z\end{array}\right]}^T$ is the end-effector position related to the joint variable $\mathbf{q}={\left[\begin{array}{cccc}{q}_1& q_2& \cdots & q_n\end{array}\right]}^T$.

An inverse kinematics approach consists in minimizing the error between an actual end-effector position $\mathbf{t}$ and the desired position ${\mathbf{t}}^{\ast }$. This error is defined as $\mathbf{e}={\mathbf{t}}^{\ast }-\mathbf{t}$. The error $\mathbf{e}$ can be mapped to the joint velocities $\dot{\mathbf{q}}$ based on the differential kinematics equation. Eq. (22) is rewritten to compute $\dot{\mathbf{q}}$ given $\mathbf{e}$ as Eq. (24).

(24) $$\dot{\mathbf{q}}=\mathbf{J}{\left(\mathbf{q}\right)}^{\text{†}}\dot{\mathbf{t}}=\mathbf{J}{\left(\mathbf{q}\right)}^{\text{†}}\mathbf{e},$$where ${\mathbf{J}}^{\text{†}}$ is the pseudo-inverse of $\mathbf{J}$. The mentioned inverse kinematics approach can be defined as a first-order algorithm that allows the inversion of a motion trajectory, specified at the end-effector position into equivalent joint position and velocities (Sciavicco & Siciliano, 2008).

A robot system with a Jacobian matrix $\mathbf{J}\in {\mathbb{R}}^{3\times n}$ where n>3, is considered redundant; there are more n DOF than necessary to perform a task with 3 DOF. Commonly, the combination of DOF of the mobile platform and the manipulator represent a redundant robot. In the case of a redundant robot, the solution Eq. (24) can be generalized into Eq. (25).

(25) $$\dot{\mathbf{q}}=\mathbf{J}{\left(\mathbf{q}\right)}^{\text{†}}\mathbf{e}+\left(\mathbf{I}-\mathbf{J}{\left(\mathbf{q}\right)}^{\text{†}}\mathbf{J}\left(\mathbf{q}\right)\right){\dot{\mathbf{q}}}_0,$$where the first term minimizes the error e, the matrix $\left(\mathbf{I}-{\mathbf{J}}^{\text{†}}\mathbf{J}\right)$ allows the protection of vector ${\dot{\mathbf{q}}}_0$ in the null space of $\mathbf{J}$, and $\mathbf{I}$ is the identity matrix. In the case that $\mathbf{e}=\mathbf{0}$, the result of the second term $\left(\mathbf{I}-{\mathbf{J}}^{\text{†}}\mathbf{J}\right){\dot{\mathbf{q}}}_0$ can reconfigure the joint variable $\mathbf{q}$ without changing the end-effector position $\mathbf{t}$.

In this work, it is proposed to design the vector ${\dot{\mathbf{q}}}_0$ to avoid singularities based on the manipulability measure $m\left(\mathbf{q}\right)$, which is defined as Eq. (26).

Then, vector ${\dot{\mathbf{q}}}_0$ can be computed as Eq. (27).

(27) $${\dot{\mathbf{q}}}_0=k_0\left({\displaystyle \frac{\mathrm{\partial }m\left(\mathbf{q}\right)}{\mathrm{\partial }\mathbf{q}}}\right),$$where k₀ > 0. By maximizing the manipulability measure, redundancy is exploited to move away from singularities. More detailed information about differential kinematics can be found in Spong & Vidyasagar (2008), Craig (2005) and Sciavicco & Siciliano (2008).

PID control design

To solve a position tracking for the mobile manipulator, the controller has to compute the joint velocities $\dot{\mathbf{q}}\left(k\right)$ at step time k, to control the motion of the mobile manipulator from the actual end-effector position $\mathbf{t}\left(k\right)$ to the desired position $\mathbf{t}{\left(k\right)}^{\ast }$. This section introduces the use of a discrete PID to control the mobile manipulator motion based on the error $\mathbf{e}\left(k\right)=\mathbf{t}{\left(k\right)}^{\ast }-\mathbf{t}\left(k\right)$, which is described as $\mathbf{e}\left(k\right)={\left[\begin{array}{ccc}{e}_x\left(k\right)& e_y\left(k\right)& e_z\left(k\right)\end{array}\right]}^T$.

A discrete PID control (Moradi, Katebi & Johnson, 2001) can be used for each error $e_x\left(k\right)$, $e_y\left(k\right)$ and $e_z\left(k\right)$ as follows (28) $$u_x\left(k\right)=K_P^x{e}_x\left(k\right)+K_I^x\sum _{j=1}^k{e}_x\left(j\right)+K_D^x\left[e_x\left(k\right)-e_x\left(k-1\right)\right],$$ (29) $$u_y\left(k\right)=K_P^y{e}_y\left(k\right)+K_I^y\sum _{j=1}^k{e}_y\left(j\right)+K_D^y\left[e_y\left(k\right)-e_y\left(k-1\right)\right],$$ (30) $$u_z\left(k\right)=K_P^z{e}_z\left(k\right)+K_I^z\sum _{j=1}^k{e}_z\left(j\right)+K_D^z\left[e_z\left(k\right)-e_z\left(k-1\right)\right],$$where $K_P^x$, $K_I^x$ and $K_D^x$ are the proportional, integrative and derivative gains for error e_x, respectively. Similarly, the parameters $K_P^y$, $K_I^y$ and $K_D^y$ are the gains for error e_y, and $K_P^z$, $K_I^z$ and $K_D^z$ are the gains for error e_z. The control output $\mathbf{u}\left(k\right)={\left[\begin{array}{ccc}{u}_x\left(k\right)& u_y\left(k\right)& u_z\left(k\right)\end{array}\right]}^T$ can be mapped to the joint velocities $\dot{\mathbf{q}}\left(k\right)$ based on Eq. (25) to control the system. This is (31) $$\dot{\mathbf{q}}\left(k\right)=\mathbf{J}{\left(\mathbf{q}\left(k\right)\right)}^{\text{†}}\mathbf{u}\left(k\right)+\left(\mathbf{I}-\mathbf{J}{\left(\mathbf{q}\left(k\right)\right)}^{\text{†}}\mathbf{J}\left(\mathbf{q}\left(k\right)\right)\right){\dot{\mathbf{q}}}_0.$$

Neural PD controllers implementation

Both proposed neural PD presented in previous sections are implemented on the above-described mobile manipulator. Figure 5 shows the general control scheme for both implementations.

An adaptive neural PD control module is designed to minimize the error e_x, e_y and θ_z. Each control signal (neural PD output) u_x, u_y and u_z, are compute for each control module. These control signals $\mathbf{u}\left(k\right)={\left[\begin{array}{ccc}{u}_x\left(k\right)& u_y\left(k\right)& u_z\left(k\right)\end{array}\right]}^T$ are mapped to the joint velocities $\dot{\mathbf{q}}\left(k\right)$ using Eq. (31) to control the system.

If expression Eq. (31) is multiplied by the Jacobian matrix $\mathbf{J}\left(\mathbf{q}\left(k\right)\right)$, then we have (32) $$\mathbf{J}\left(\mathbf{q}\left(k\right)\right)\left[\dot{\mathbf{q}}\left(k\right)\right]=\mathbf{J}\left(\mathbf{q}\left(k\right)\right)\left[\mathbf{J}{\left(\mathbf{q}\left(k\right)\right)}^{\text{†}}\mathbf{u}\left(k\right)+\left(\mathbf{I}-\mathbf{J}{\left(\mathbf{q}\left(k\right)\right)}^{\text{†}}\mathbf{J}\left(\mathbf{q}\left(k\right)\right)\right){\dot{\mathbf{q}}}_0\right],$$ (33) $$=\mathbf{J}\left(\mathbf{q}\left(k\right)\right)\mathbf{J}{\left(\mathbf{q}\left(k\right)\right)}^{\text{†}}\mathbf{u}\left(k\right)+\mathbf{J}\left(\mathbf{q}\left(k\right)\right)\left(\mathbf{I}-\mathbf{J}{\left(\mathbf{q}\left(k\right)\right)}^{\text{†}}\mathbf{J}\left(\mathbf{q}\left(k\right)\right)\right){\dot{\mathbf{q}}}_0,$$ (34) $$=\mathbf{u}\left(k\right)+\left(\mathbf{J}\left(\mathbf{q}\left(k\right)\right)-\mathbf{J}\left(\mathbf{q}\left(k\right)\right)\mathbf{J}{\left(\mathbf{q}\left(k\right)\right)}^{\text{†}}\mathbf{J}\left(\mathbf{q}\left(k\right)\right)\right){\dot{\mathbf{q}}}_0,$$ (35) $$=\mathbf{u}\left(k\right)+\left(\mathbf{J}\left(\mathbf{q}\left(k\right)\right)-\mathbf{J}\left(\mathbf{q}\left(k\right)\right)\right){\dot{\mathbf{q}}}_0,$$ (36) $$=\mathbf{u}\left(k\right),$$which indicates that $\mathbf{u}\left(k\right)=\mathbf{J}\left(\mathbf{q}\left(k\right)\right)\dot{\mathbf{q}}\left(k\right)$ is a solution of the differential kinematics of the system, where the matrix $\mathbf{J}\left(\mathbf{q}\left(k\right)\right)$ is bounded (Hernandez, Nuño & Alanis, 2016). Moreover, the computed control signal of the neural controller Eq. (5) is also bounded. Considering that neural control for system Eq. (31) is a feed forward network control law, it composed a stable system (Khalil, 2002).

Simulations

The first trajectory for simulation is the circular. Although conventional PID controller presents a good response, their gains remain constant, and they cannot adapt to changes in the system operating conditions. On the other hand, the MNPD, SNA-PID, and SNPD approaches can correctly follow the reference once the weights are adjusted.

The system response results for the circular trajectory are given in Fig. 7. The settling time is almost the same for all the approaches. The conventional PID presents overshoot and steady-state errors. The SNA-PID controller suppresses overshoot, but it has small steady-state errors. Although MNPD presents oscillations while the weights are adapting, the neural algorithms can follow the sinusoidal trajectory better than the PID and SNA-PID. Moreover, the SNPD controller does not suffer from overshoot.

The root mean square (RMS) and median absolute deviation (MAD) for the circular trajectory are presented in Table 2. The adaptive approaches present the best results, which are highlighted in bold. In this case, the SNPD control scheme reportesd the smallest RMS results in general.

In Fig. 8, the trajectory tracking and velocity control signals for the circular trajectory are presented. From Figs. 8A to 8D, the PID control passes over the reference caused by the integral part, and MNPD presents oscillations as expected from the system response. However, MNPD, SNA-PID, and SNPD controllers follow the trajectory correctly. As seen from Figs 8E to 8H, at the first steps adaptive weights compute bigger control signals than PID. However, it is necessary to reach the reference with a small tracking error.The adaptation ability of both MNPD and SNPD is also shown.

The same gains and parameters for the four approaches are set for a new desired trajectory to be tested, and the results are shown in Fig. 9. Similar results can be seen in the system response. The settling time is similar, and the MNPD present oscillations during the adaptations of its weights. In this case, PID and SNA-PID controllers present steady-state errors. However, SNA-PID performs better than PID. Moreover, The PID controller also has overshoot. The best results are given by SNPD and MNPD.

Table 3 shown the RMS and MAD results for the rose curve trajectory. The adaptive scheme has been demonstrated to have better results than the conventional PID controller. In this case, MNPD controller shows the smallest RMS results in general. In contrast, SNPD shows the best MAD results.

In Fig. 10, the trajectory following, and the velocity control signals for the rose curve trajectory are reported. From Figs. 8A to 8D can be noticed that the adapting approaches outperform the conventional PID controller. The SNA-PID controller shows a small steady-state error, while MNPD and SNPD report the smallest. The SNA-PID control improved the performance of PID, but it is needed to tune the proportional coefficient K to improve the performance. As can be seen from Figs. 8E to 8H, at the beginning of the trajectory, the weights adaptation of the MNPD and the SNPD compute bigger control signals than SNA-PID and PID; however, this is necessary to reach the reference with a small tracking error.

Experiments

Real experiments are carried out in two parts. In the first part, the proposed SNPD controller is compared against the classical PID. In this case, two trajectories are tested for comparison purposes. In the second part, the robustness of the SNPD controller is tested under the presence of disturbances and non-modeled dynamics.

In the first part of the experiments, the adaptive SNPD and MNPD controllers performed similarly in simulations. However, MNPD shows oscillations during the adaptations of its weights at the beginning. These oscillations can be eliminated if pre-trained weights are used instead of initializing them randomly every time. Thus, it is considered to compare the SNPD controller to the PID controller since PID performed better than PD. Moreover, the same gains and parameters used for simulation were used for real-time experiments. The weights in the SNPD were randomly initialized.

In Fig. 11, the system response for both approaches is shown. The real system is not the same in simulation, and the gains of the conventional PID must be tuned again. Otherwise, it will not be able to follow the trajectory correctly and present a longer settling time. In contrast, using the same parameters as in simulation, the SNPD was able to adapt and showed a shorter settling time.

Table 4 reports the RMS and MAD results for the rose curve trajectory in real experiments. The SNPD scheme has been demonstrated to have better results than conventional PID with the smallest RMS and MAD results in general.

The trajectory following and velocity control signals for the rose curve trajectory are illustrated in Fig. 12. In Figs. 12A and 12B, the response for the rose curve trajectory is shown. As can be seen, PID cannot follow the trajectory correctly, and it is confirmed in Table 4. In Figs. 12C and 12D, adaptive SNPD computes bigger control signals than PID. However, this demonstrates that SNPD is adjusting itself to reject perturbation and changes during experimental tests.

A new trajectory is tested, and the system response results are shown in Fig. 13. The SNPD control performed better than PID for the results for the trapezoidal trajectory. The results exhibited the adaptation ability of the SNPD, while PID control requires the tune of its gains.

The trajectory following and velocity control signals results for the trapezoidal trajectory are given in Fig. 14. In Figs. 14A and 14B, the PID scheme reported bigger tracking error than SNPD controller. From Figs. 14C and 14D, it is clear that bigger control action is required to be able to follow the trajectory with minimum error tracking. This is achieved with the online adaptation of SNPD controller.

Finally, Table 5 reported the RMS and MAD results for the trapezoidal trajectory in real experiments. The SNPD scheme outperformed the PID controller with the smallest RMS and MAD results in general.

In the second part of the experiments, three tests were performed. In Test 1, a trajectory tracking on a smooth floor is considered. This test represents an ideal environment. In Test 2, the same trajectory tracking is performed on a rough floor. This test represents the presence of disturbances. Finally, in Test 3, the same trajectory tracking is performed on the uneven floor with a load on board of 1.5 kg. This last test represents non-modeled dynamics. Moreover, since the experiments were carried out on a real robot, there was measurement noise.

The SNPD controller is considered for this test under the sinusoidal trajectory. The results for the system response are provided in Fig. 15. The system response results are very similar for the three tests. This indicates that no adjustment to the controller parameters is required to handle uncertainties in unmodeled changes in system dynamics. Moreover, there is no presence of steady-state errors, and the controller reaches the references within an adequate settling time.

The trajectory following and velocity control signals for these tests are illustrated in Fig. 16. In Fig. 16A, the results show the correct trajectory following as expected from the system response results. As can be seen in Figs. 16C and 16D, the SNPD computes bigger control signals than the results in Fig. 16B. This shows that the weights are dynamically adapted to reject disturbances and changes in dynamics during the experimental test

Finally, the Table 6 shows the RMS and MAD results for the sinusoidal trajectory. The SNPD scheme has been demonstrated to have better results than conventional PID with the smallest RMS and MAD results in general. There is not a big difference between the RMS and MAD results, which indicates that the proposed SNPD controller is robust to unmodeled dynamics and disturbances.