Robust Nonlinear Model Predictive Control for the Trajectory Tracking of Skid-Steer Mobile Manipulators with Wheel–Ground Interactions

Abstract

This paper presents a robust control strategy for trajectory-tracking control of Skid-Steer Mobile Manipulators (SSMMs) using a Robust Nonlinear Model Predictive Control (R-NMPC) approach that minimises trajectory-tracking errors while overcoming model uncertainties and terra-mechanical disturbances. The proposed strategy is aimed at counteracting the effects of disturbances caused by the slip phenomena through the wheel–terrain contact and bidirectional interactions propagated by mechanical coupling between the SSMM base and arm. These interactions are modelled using a coupled nonlinear dynamic framework that integrates bounded uncertainties for the mobile base and arm joints. The model is developed based on principles of full-body energy balance and link torques. Then, a centralized control architecture integrates a nominal NMPC (disturbance-free) and ancillary controller based on Active Disturbance-Rejection Control (ADRC) to strengthen control robustness, operating the full system dynamics as a single robotic body. While the NMPC strategy is responsible for the trajectory-tracking control task, the ADRC leverages an Extended State Observer (ESO) to quantify the impact of external disturbances. Then, the ADRC is devoted to compensating for external disturbances and uncertainties stemming from the model mismatch between the nominal representation and the actual system response. Simulation and field experiments conducted on an assembled Pioneer 3P-AT base and Katana 6M180 robotic arm under terrain constraints demonstrate the effectiveness of the proposed method. Compared to non-robust controllers, the R-NMPC approach significantly reduced trajectory-tracking errors by 79.5% for mobile bases and 42.3% for robot arms. These results highlight the potential to enhance robust performance and resource efficiency in complex navigation conditions.

1. Introduction

Recent advances in robotics technology have demonstrated that robots are capable of assisting people or even labour in certain tasks that are repetitive and physically demanding for human workers, such as material handling, assembly, quality control in manufacturing settings, or picking and transporting fruit in agricultural applications [1]. Autonomous mobile robots often face issues from external or uncontrollable factors, including unexpected dynamic obstacles and variations in uneven terrain layout, which is typical in outdoor applications such as agriculture and forestry. Indeed, if one focuses on outdoor applications, navigating uneven terrain conditions usually represents challenges such as dealing with slopes, structural rigidness, flexibility, and other terrain characteristics [2]. The slippage phenomenon may arise in such situations, leading to reduced tractive force exertion, motion inaccuracies, and potential damage when the robotic system operates in confined spaces. Nevertheless, earlier motion control designs have not accounted for the navigation terrain, which could be considered as the primary terra-mechanical constraint that may impact the control performance of ground vehicles [3].

A robotic arm mechanically attached to a mobile base is often referred to as a mobile manipulator [4]. This coupling stands for interactions between forces generated by the object manipulation, the manipulator–base integration, and the wheel–terrain contact. These interactions usually result in bidirectional disturbances in the robotic system. In addition to manoeuvrability, obstruction with dynamic obstacles appearing unexpectedly around mobile robots, tractive force losses caused by low-friction terrain, or non-contact conditions become a critical disturbance source as they lead to performance degradation and, consequently, potential risk of slippage or wheel entrapment. Modelling mobile robot dynamics under ideal conditions without slip could yield insufficient motion accuracy due to the lack of robust control [5]. For instance, in the analyses of autonomous mobile robots and motion controllers, some authors assume representations that fulfil nonholonomic constraints [6,7,8], which could lead mobile robots to navigate without experiencing slippage [9]. However, wheel slippage frequently occurs in real-world scenarios, particularly on low-friction surfaces, uneven terrain, or in relative high-speed manoeuvrers. Slip conditions can be regarded as model uncertainties within the motion models of the vehicle, as they are often unknown and difficult to predict due to the heterogeneous nature of the terrain. Therefore, it is essential to investigate mobile robot control strategies that account for wheel slip, modelling errors, and external disturbances.

Combined control strategies for manipulation that integrate the dynamics of both the mobile base and robotic arm have been extensively studied in the literature [7,10,11,12,13]. However, further research is needed to address, in practice, disturbance rejection in wheel–terrain interactions and the resulting propagation of these disturbances through motion coordination of the robot’s arms and joints. Traditional techniques such as PID, back-stepping, and adaptive control approaches have been widely studied to coordinate mobile manipulators and mitigate uncertain dynamics caused by the wheel slip and nonholonomic constraints [7]. For instance, works on feedback linearisation or Lyapunov techniques with parametrized robot models can be found in [14]. However, the approaches are limited to the search of applicable specific Lyapunov functions or nonlinear simplifications of the robot model. The application of these techniques may result in the loss of the system fidelity from the original system. Control approaches and algorithms such as fuzzy logic-based or neural network control [15,16,17,18] can restrict model accuracy, specially in terra-mechanical models. In particular, Model Predictive Control (MPC) is a model-based control scheme designed to deal with constrained linear systems [19]. MPC approaches such as Hybrid MPC, Explicit MPC, and its nonlinear counterpart Nonlinear MPC (NMPC) arise as optimal control frameworks capable of directly handling the complex nonlinear and uncertain dynamics of mobile manipulator robots subject to constraints and disturbances [20]. The NMPC strategy is able to account for complex kinematic and dynamic relationships in multi-degree-of-freedom robotic manipulators. The design of NMPC has evolved to incorporate passivity constraints, which enhance system stability by ensuring energy dissipation during interactions. This adaptation leverages passivity-based control principles within the NMPC framework, enabling the effective handling of nonlinearities and system dynamics. Passivity is based on the principle that a network cannot deliver more energy to its environment than what was initially supplied [21]. This property is particularly beneficial in control systems as it limits energy dissipation, which naturally supports stability within a closed-loop control system when posed as an energy minimisation problem. Then, passivity ensures effective trajectory tracking, even in the presence of nonlinearity and external disturbances [22]. Additionally, the conservation property inherent to passive systems allows stable feedback and the integration of new passive components without compromising overall system stability. The passivity constraint is usually introduced into the optimisation problem to enhance the robustness and stability of the control system, particularly in environments with unpredictable disturbances and system dynamics in which models are uncertain [23]. By ensuring that the control actions do not introduce excessive energy, the passivity constraint contributes to the overall stability and performance of the dynamic system under various operational conditions. By integrating passivity as an additional condition, NMPC could achieve more robust performance in applications where interaction forces are significant, such as in robotic articulations, vehicle actuators, and robot manoeuvrability [21,22]. Then, for this purpose, an analysis of the robotic system is performed to identify a called storage function that can be introduced to set an additional constraint for the underlying optimisation problem of NMPC. Such a constraint ensures that the system maintains stability and performance properties.

Motion control strategies can be addressed considering two main control objectives: trajectory tracking and disturbance regulation. However, specific controllers may not fully ensure those control objectives due to certain conditions of uncertainties and disturbances on the system dynamics. Such conditions in mobile manipulators can be attributed to unpredictable navigability conditions such as unanticipated obstacles, uneven terrain, slopes, confined spaces, and workload variations. In such cases, robust control strategies are required to be incorporated into control systems. Robustness enables the controller to maintain its performance within a specified range of unfavourable conditions, thus ensuring that the control system meets its objectives. Active Disturbance-Rejection Control (ADRC) is a strategy that could ensure robust performance of the motion controller. The ADRC strategy has been designed to minimise the impact of uncertainties in both Single-Input Single-Output (SISO) and Multiple-Input Multiple-Output (MIMO) systems [24]. The ADRC scheme relies on real-time estimation and compensation of internal and external disturbances without requiring a system model representation. Furthermore, the nonlinear ADRC structure incorporates nonlinear formulations into the design of both the observer and the control law. This nonlinear design is able to enhance the effectiveness of the control strategy by enabling it to withstand uncertainties and disturbances, both internal and external [25,26]. The ADRC has been successfully applied to several robotic systems, including manipulators [27,28] and mobile robots [25,29]. However, its compatibility with SSMM remains to be fully exploited.

The purpose of this work is to design a Robust NMPC strategy (R-NMPC) for the trajectory tracking of SSMM subject to the slip phenomena arising from the wheel–terrain interaction. A novel approach that combines NMPC and ADRC strategies is proposed for the control of nonlinear systems. The NMPC generates a response based on the prior knowledge of the robotic system, thereby enabling continuous position and orientation compensation for the mobile base and robotic arm. The main contribution of this work is to enhance the robust control performance of NMPC strategies, enabling accurate tracking of the 5-DoF trajectory while mitigating the effects of unmeasurable terrain disturbances. Furthermore, incorporating passivity constraints into the NMPC optimisation problem contributes to maintaining robust stability and enhances the overall performance of the control system. The ADRC is used as a compensator within the integrated control scheme with the aim of enhancing the robustness of the SSMM, even under unpredictable conditions of terrain. It is used an Extended State Observer (ESO) of the ADRC to predict modelling errors and uncertainties in order to generate compensation control actions additional to the nominal ones against possible slippage and bidirectional disturbances. Due to mechanical coupling between the mobile base and the robotic arm, bidirectional disturbances in the SSMM are produced by the motion of the robotic arm while reaching tracking points, and the mobile base experiences slip. Control performance of the proposed controller is assessed in simulations and field experimentation on a Pioneer 3P-AT mobile base, manufactured by Omron Adept MobileRobots, headquartered in Amherst, New Hampshire, USA, and a Katana 6M180 robotic arm, produced by Neuronics AG, Zurich, Switzerland, using metrics associated with the cumulative control input effort, cumulative tracking error, and cumulative total cost throughout several trajectory-tracking and disturbance-rejection tests [30,31]. For the trajectory-tracking tests, three reference trajectories are proposed to assess the continuous variation of position and orientation. Moreover, linear and angular velocity disturbances are applied to the mobile base to simulate slipping scenarios resulting from the ground–wheel interaction. Additionally, intentional variations in the SSMM model are introduced to assess the robustness of the motion control system. These variations target model parameters such as mass, inertia, and geometry, allowing for a comprehensive evaluation of the system’s performance under model uncertainties.

The paper is organized as follows: Section 2 provides a review of works related to the proposed control methods. In Section 3, the dynamical model of the mobile manipulator is described, which is used as a prediction model for the NMPC strategy in Section 4. In addition, the nonlinear approach that considers motion constraints for the SSMM is detailed. In Section 5, the proposed robust control approach is designed as a combination of nonlinear control and a compensation control action based on the ADRC formulation. Section 6 presents the experimental tests and a detailed analysis of the results. In addition, it includes the simulation and field results of trajectory-tracking and terrain disturbance-rejection tests, as well as robust tests with model parameter variations. Finally, Section 7 presents the findings and conclusions drawn from this work.

2. Related Work

Skid-steer mobile manipulators have drawn increased attention lately due to their potential benefits of flexibility and dexterity to integrate simultaneous multi-tasks under a constrained operability workspace [4]. Developments of mobile manipulators were mainly motivated by the imitation and handling of operator skills in complementary tasks. For example, in the existing literature, several robotic arms were integrated into mobile bases for agriculture tasks that would require autonomous motion, such as soil preparation, nutrient management, agro-chemical application, irrigation, product extraction, harvesting, and pruning, among others [10,32,33,34]. Although mobile manipulators have broad applications, this research focuses specifically on the reachability of the manipulator to a target point for locating agricultural products, rather than encompassing object handling or completing full agricultural tasks.

The implementation of mobile manipulators in industrial contexts needs the development of control strategies for autonomous navigation and manipulation [35]. For instance, due to high payload characteristics and even distribution, Skid-Steer Mobile Manipulators (SSMMs) are suitable for operating autonomously and safely in dynamic environments of the industry. The robot in [35] had to face obstacles such as service objects and markers on walls and floors. On the other hand, an SSMM can perform specific tasks such as reaching objects from a source to a target location, avoiding unexpected objects appearing from above, and facing unexpected motion reactions while moving over terrain. Such challenges showed that is is required to develop mobile manipulation systems that can autonomously navigate and perform manipulation tasks in variable and challenging environments.

Motion controllers have been developed for the trajectory tracking of mobile manipulators [36,37,38,39]. For example, in [37] the authors designed a motion controller based on back-stepping techniques, assisting the control allocation for a four-wheel steering system to improve coordination of steerability with traction while tracking headline trajectories within crop rows. In [38], a sliding-mode control scheme was presented for an articulated manipulator to avoid side skidding effects and external disturbances exerted by impact forces on the end-effector. In [40], such a technique was combined with a fuzzy approach to obtain speed-tracking control gains that best adjust the center of gravity of the agricultural machinery with relative heavy loads. In [41], the work proposed a filtering approach for an adaptive control strategy in a two-link robotic mobile manipulator. The aforementioned techniques relied on solving complex algorithms, including filtering and sub-steps of control compensation, but most of them had an overloaded computation burden. To reduce computational requirements, several control strategies have implemented model simplification strategies, separate analysis of the mobile base and robotic arm, and limitations of the number of joints. Thus, it is necessary to implement a control strategy that includes most motion dynamics of the robotic system without limiting the number of DoF, which could compromise the dexterity of the mobile manipulator robot.

Model-based Predictive Control (MPC) has been selected as a primary tool to applications involving mobile robotic systems [42,43,44]. In [45], a dexterity motion control framework was introduced to tackle manipulation, balancing, and interaction as a unified optimisation approach for unstable mobile actuators. The approach used an MPC strategy to optimise end-effector tasks, encompassing position, orientation, and contact forces. The linear model and soft constraints applied to the MPC did not fully allow fast responses to disturbances. Instead, the control system presented continuous oscillations that had to be compensated for. In [46], a combination of an instantaneous dynamical model and an adaptive MPC method was applied to a mobile manipulator for transportation tasks. In [11], an MPC scheme was used to introduce a set of constraints to encode a motion sequence for base or end-effector pose tracking to face heavy resistive load. The MPC was modified to compensate for the modelling error that was caused by the linear approximation. In [47], an MPC was used for position and speed control in a mobile manipulator considering motion constraints. The controller stabilised the robotic system, taking into account the kinematics of the manipulator and a predefined reference trajectory. In the presence of disturbances or a deviated operating point, the output system could be unable to respond suitably, potentially leading to unstable control characteristics. Alternative methodologies have been developed to address robustness and the nonlinear responses of complex industrial problems [48,49].

Model Predictive Control (MPC) can be effectively adapted for complex dynamic systems through the integration of a nonlinear prediction model. This enhanced approach, known as Nonlinear Model Predictive Control (NMPC) is able to addresses the trajectory-tracking problem in robotic systems. For example, in [50], a Nonlinear Model Predictive Control (NMPC) approach for a mobile manipulator achieved a reduction in computational time by using parameterized control inputs and eliminating terminal constraints. However, the removal of terminal constraints raises concerns regarding the system’s ability to converge to the desired state and maintain stability over time. In [51], NMPC was applied to a tunnel-following task for a 7-DoF manipulator. The approach allowed the exploitation of acceptable position deviations around a reference path. The NMPC used convex-over-nonlinear functions and constraints to solve the Optimal Control Problem (OCP). In [52], an NMPC-based Reinforcement Learning (NMPC-based RL) technique for 6-DoF robot manipulators was developed to address trajectory-tracking control in relation to the motion planning of manipulators under the presence of obstacles. A Q-Learning algorithm fine-tuned the parameters of an NMPC strategy to enhance the closed-loop performance. However, both approaches introduced significant computational complexity, which can extend the response time of the controlled system. Therefore, it was relevant to prioritise the acquisition of accurate data rather than solely relying on models, as this can further increase the robustness and effectiveness of the control strategy. In [53], a motion control strategy was devoted to minimising the lateral slip. The controller was implemented by combining the advantages of a model-based predictive scheme and fuzzy control methods. The problem related to the fuzzy strategy was the prior knowledge about the control requirements to state fuzzy rules, which led the system to become complex. In [54], a Stochastic NMPC (SNMPC) algorithm was used for active target tracking. The controller penalised tracking errors by predicting uncertainty for target position estimation and robot pose. In [55], a robust NMPC approach included the slip parameter, which was the source of uncertainty in the constrained mobile robot. The robot was modelled with bounded additive disturbances. The performance of the robotic systems was affected by external disturbances and model uncertainties. Therefore, robust strategies have been introduced to take advantage of the NMPC performance and improved its responses in mobile robots. However, to the best knowledge of the authors, the use of model predictive controllers has not been fully extended to coupled mobile manipulators.

Some research has enhanced the robustness of NMPC strategies through several modifications. These include robust model prediction using the virtual decomposition control method combined with a time-varying state feedback control law [56]. A robust predictive control law with a weight matrix was applied to different stages of dynamic prediction time in [57], and an event-triggered robust tracking control approach was applied for motion control in [58]. However, the existing literature indicates a lack of effective strategies for rejecting uncertainties in dynamic robotic systems with limited knowledge of their dynamics. Active Disturbances Rejection Control (ADRC) was proposed by [24] as an alternative to traditional industrial control strategies that aid in further enhancing robust performance. The ADRC strategy was designed to work without relying on the system dynamic model, incorporating mechanisms for disturbance estimation and rejection approaches [59]. For instance, mobile robots are sensitive to disturbances such as wheel friction and drifting, and the ADRC is a suitable controller for mobile robots, effectively addressing such challenges [25]. In [60], the robot motion control used the ADRC methodology to enable the robot to navigate through a bumpy road without experiencing destabilization. In [61], the ADRC scheme was designed for the stabilization problem of wheeled mobile robots with uncertainties. In addition, applying a decouple strategy ensures convergence. In [29], ADRC was used to control disturbances occurring in a four-wheeled robot by adjusting the motor output torque and tracking the optimal wheel speed associated with the optimal slip rate. The control strategy was effective in preventing wheel slip when driving on roads with low adhesion coefficients. In [62], the work suggested combining a state feedback controller with an ADRC approach to deal with trajectory tracking subject to slip and external environmental disturbances. The Extended State Observer (ESO) was designed to estimate the robotic system states and the extended states, which represent the total effects of slip and the external environmental disturbances. The ADRC strategy was used as a complement to other controllers to strengthen the controlled system; however, a principal controller used linearisation strategies to deal with complex dynamic systems.

The ADRC has been widely used in robotic applications due to its transversal design, which can work as an additional control loop as a compensation strategy. For instance, in [63] an ADRC strategy incorporating a dual-stage disturbance observer was implemented to enhance the backward trajectory-tracking performance of generalized N-trailers under non-ideal operational conditions. In [64], a robust MPC framework was developed in conjunction with ADRC approaches and applied to a robotic autonomous underwater vehicle. Such integration allowed the ADRC to incorporate disturbances and uncertainties into a total disturbance, which was estimated using a discrete Extended State Observer (ESO) and effectively rejected through feedback control. Furthermore, the ADRC strategy has been extended to robotic manipulator applications by incorporating a linear ESO [25]. In [27], linear ADRC was applied to a 2-degree-of-freedom (DoF) rigid link manipulator for trajectory tracking, addressing high nonlinearity and substantial uncertainties. Additionally, in [28] the ADRC framework was used to tackle the system nonlinearity, strong mechanical coupling, and uncertainty associated with a two-link manipulator system. This research presented a robust feedback control mechanism designed to achieve trajectory tracking despite unknown disturbances.  

3. Dynamic Model of the Skid-Steer Mobile Manipulator

To obtain forward equations of the overall SSMM dynamics, it was considered that there is bi-directional force propagation, encompassing transmission forces from the manipulator towards the mobile base, and vice-versa (see Figure 1). This bi-directional force exchange is essential to account for the propagation of uncertainties and external disturbances, such as the navigation terrain throughout the mobile robot. The Newton–Euler formulation [65] was used in this study to set the momentum and forces acting on each arm joint as a function of the state variables, external forces, and constraints.

The robot base was considered as a joint that can simultaneously possess both rotational and translational motion dynamics. To apply the recursive N–E method, it was necessary to obtain the Denavit–Hartenberg (DH) parameters for a 5-DoF mobile manipulator [65]. The DH convention allowed a homogeneous transformation matrix that described the position and orientation of a i-th link with reference to the previous one, $i-1$. Such a matrix is formed by the production of two translation and two rotational transformations [65]. In this paper, the transformation matrix is denoted by $R_i^j$, where j represents the actual link in respect to the i link for a fixed coordinate axis. The DH convention designated values of rotation and translation between local coordinate axis are detailed in

Table 1. Denavit–Hartenberg parameters for the SSMM robot.

i	Articulation	${\mathit{\theta }}_{\mathit{i}}$	${\mathit{d}}_{\mathit{i}}$	${\mathit{a}}_{\mathit{i}}$	${\mathit{\alpha }}_{\mathit{i}}$
1	Mobile base	0	0	0	0
2	Auxiliary joint	0	0	0	0
3	Manipulator base	${\theta }_{m1}$	$d_1$	0	${\displaystyle \frac{\pi }{2}}$
4	Shoulder joint	${\theta }_{m2}$	0	$d_2$	0
5	Arm joint	${\theta }_{m3}$	0	$d_3$	0

and Figure 1.

Then, the position vector for the SSMM is defined by:

$$q={[{\theta }_{b1}\int v_b{\theta }_{m1}{\theta }_{m2}{\theta }_{m3}]}^T$$

(1)

where the orientation and translation of the mobile base are described by ${\theta }_{b1}$ and $\int v_b$. The motion of the robot arm joints is given by the angular variables ${\theta }_{m1}$, ${\theta }_{m2}$, and ${\theta }_{m3}$. The distances from each link are $d_1,d_2$, and $d_3$, as shown in Figure 1. The arm positions are $x_a,y_a$, and $z_a$ in the $x,y$, and z coordinates. Then, the variables q and $\dot{q}$ denote states of position and speed for the robotic arm and base, respectively. Such variables are used to define the nonlinear state-space model for SSMM.

In the SSMM, the rotational and translational motion of the robot base are considered as two additional DoFs. As a result, frames $\left\{S_1\right\}$ and $\left\{S_2\right\}$ overlap at the same coordinate origin (see Figure 1). To apply the N–E procedure, it was necessary to obtain the rotation matrix and its inverse. Since the base reference frame, $\left\{S_1\right\}$, had the same orientation as the global reference frame, $\left\{S_0\right\}$, the rotation matrix and its inverse are the identity matrix. Moreover, the system of the basis, $\left\{S_1\right\}$, is located at the centre of gravity, which is the same as for the mobile base. Both coordinate systems were aligned with the reference frame of the robot base and did not required rotation. Therefore, the vector $p_1^b$ was equal to zero.

The proposed methodology used force and angular momentum equations to describe the motion dynamics of the SSMR. Thus, it was necessary to obtain the angular speed, ${\omega }_1^b$, and accelerations, ${\dot{\omega }}_1^b$, of the robotic mobile base. These speed vectors were obtained as follows:

$${\omega }_1^b=R_1^0\left({\omega }_0^b+\dot{{\theta }_b}{\varrho }_0\right)=\left[\begin{array}{c}0\\ 0\\ \dot{{\theta }_b}\end{array}\right]$$

(2)

$${\dot{\omega }}_1^b=R_1^0\left[\dot{{\omega }_0^b}+\ddot{{\theta }_b}{\varrho }_0+{\omega }_0^b\times \left(\dot{{\theta }_b}{\varrho }_0\right)\right]=\left[\begin{array}{c}0\\ 0\\ \ddot{{\theta }_b}\end{array}\right]$$

(3)

where R is the rotation matrix and ${\varrho }_0={[0,0,1]}^T$ is the auxiliary transformation vector. To calculate the linear acceleration of the robotic system, it was considered that the mobile base exhibits translational motion, speed, and linear acceleration along the x-axis. Consequently, the $\varrho$-vector in this case has to be ${\varrho }_b={[1,0,0]}^T$. Thus, the linear acceleration of the robot base could be expressed by Equation (4).

$${\dot{v}}_1^b=R_0^1\left(\dot{{\theta }_b}{\varrho }_b+{\dot{v}}_0^b\right)+{\dot{\omega }}_1^b\times p_1^b+2{\omega }_1^b\times \left(\dot{{\theta }_b{R}_0^1{\varrho }_b}\right)+\left({\omega }_1^b\times p_1^b\right)=\left[\begin{array}{c}{\dot{v}}_b\\ 2\dot{{\theta }_b}{v}_b\\ g\end{array}\right]$$

(4)

To obtain joint torques at the arm and base of the mobile manipulator, it was assumed that the mobile base performed both rotational and translational motion concurrently. Then, the linear force in the new coordinate frame is as follows:

$$f_{lin\left(base\right)}=f_1^{b}T\left(R_0^1{\varrho }_b\right)$$

(5)

where f denotes forces and T is transfer function. The forces and torques transferred from the manipulator to the base included forces in both directions, from the manipulator to the base, and vice versa. Considering both the joints and the mobile base, the resultant matrix becomes:

$$M\left(q\right)\ddot{q}+C(q,\dot{q})\dot{q}+G\left(q\right)=\tau$$

(6)

where M, C, G, and $\tau$ are mass, inertia matrix, Coriolis matrix, the gravitational forces vector, and the input torque vector of each joint, respectively. The nonlinear state space, $\dot{\zeta }\left(t\right)=f(\zeta \left(t\right),u\left(t\right),\eta )$, used in the proposed robust motion controller is then as follows:

$$\begin{array}{cc}\hfill \left[\begin{array}{c}\dot{q}\\ \ddot{q}\end{array}\right]& =\left[\begin{array}{c}\dot{q}\\ -M{\left(q\right)}^{-1}\left[G\left(q\right)\right]\end{array}\right]+\left[\begin{array}{c}{0}_{5\times 5}\\ M{\left(q\right)}^{-1}\end{array}\right]\tau +\eta \hfill \\ \hfill \Omega & =[1_{1\times 5},0_{5\times 5}]\left[\begin{array}{c}q\\ \dot{q}\end{array}\right]\hfill \end{array}$$

(7)

where $\zeta ={[q,\dot{q}]}^T$ is the state vector of the nonlinear system represented in state space, $u=[{\tau }_b,F_b,{\tau }_1,{\tau }_2,{\tau }_3]$ is the input control vector, $\eta$ stands for a vector of the model uncertainties, and $\Omega$ is the output vector of the SSMM.

The dynamic model disregards motion constraints such as workspace limitations, environmental interactions, and payload capacity. Instead, dynamic and kinematic constraints are accounted for in the proposed control strategy. As a result, it is assumed that joints cannot achieve a full 360° range of motion control. The arm’s workspace is restricted by the lengths of its links and the geometric constraints of the mobile base. Additionally, the maximum motor speeds on each wheel and joints are given by the specifications of the mechanical actuators, while the maximum payload is also set to prevent structural damage.

4. Nonlinear Model Predictive Control Strategy

This section describes the control architecture layout of the robust NMPC (R-NMPC) approach, encompassing the nominal controller (disturbance-free) and the ancillary control strategy designed to complement the nominal one and mitigate uncertainties and terrain disturbances, as illustrated in Figure 2. This section also presents the dynamic model of motion for the SSMM and details the wheel–terrain interaction between the mobile base and the attached manipulator, which is also integrated into the prediction model for the proposed controller. A 3D reference trajectory is proposed for the manipulator and a 2D reference trajectory is proposed for the mobile base. Inverse kinematics is then applied to generate rotational and translational speed, which completes the references of the robotic arm and the wheels of the mobile base. Constraints are imposed taking into account the range of motion of the joint coordinates, as well as the maximum speed achievable by the physical elements of the robot. The NMPC controller uses the reference trajectory, constraints, and system outputs to optimise nominal inputs, $u_n$. The NMPC strategy is applied to the control position and orientation of the mobile base and robotic arm. Furthermore, an additional strategy based on ADRC is developed to obtain the complementary control action, $u_c$, which incorporates the error model to compensate for potential uncertainties and modelling errors. The combination of the predictive controller and its compensator results in a robust control signal, u, according to Equation (8).

$$u=u_n+u_c$$

(8)

The NMPC strategy uses the nonlinear dynamic model of the SSMM, as set by Equation (7), to predict the future system response in a finite prediction horizon, N. A control sequence is optimised in accordance with the minimisation of the cost function, whereby the tracking error is subject to state and control input constraints. Moreover, the control system uses prescribed references for the position and orientation trajectories of the SSMM, constraints according to model boundaries, and the actuators of the robotic system (see Figure 2). The NMPC uses the sliding horizon technique, which solves an optimisation problem along the N horizon that is continuously updated at each time instant. The solution of the optimisation problem provides an optimised sequence of control actions, $u_n\left(t\right),u_n(t+1),\dots ,u_n(t+N)$. However, only the first control action of this sequence is applied to the robotic system. The optimisation problem is given by:

$$\begin{array}{cc}\hfill \underset{u_n\left(·\right)}{\min }{\int }_{t_k}^{t_{k+N-1}}& J(\tau ,\zeta \left(\tau \right),u_n\left(\tau \right))d\tau +J_N(t_{k+N},\zeta \left(t_{k+N}\right))\hfill \\ \hfill \mathrm{subject}\mathrm{to}:& \dot{\zeta }\left(t\right)=f(\zeta \left(t\right),u_n\left(t\right))\hfill \\ \hfill & \Omega \left(t\right)=g\left(\zeta \right(t\left)\right)\hfill \\ \hfill & {\zeta }_N\left(t_{k+N}\right)\in {\mathbb{Z}}_N\hfill \\ \hfill & \zeta \left(t\right)\in \mathbb{Z}\left(t\right)\hfill \\ \hfill & u_n\left(t\right)\in \mathbb{U}\left(t\right)\hfill \\ \hfill & \dot{V}\left(\zeta \right)\le {\Omega }^T\left(t\right)u_n\left(t\right)\hfill \\ \hfill & \forall t\in \left[t_k,t_{k+N-1}\right]\hfill \end{array}$$

(9)

with:

$$\begin{array}{cc}\hfill & J(t,\zeta ,u_n)={∥{\zeta }^{\mathrm{ref}}\left(t\right)-\zeta \left(t\right)∥}_{Q_{\zeta }}^2+{∥u_n^{\mathrm{ref}}\left(t\right)-u_n\left(t\right)∥}_{Q_u}^2\hfill \\ \hfill & J_N(t_{k+N},\zeta \left(t_{k+N}\right))=\left|\right|{\zeta }^{\mathrm{ref}}\left(t_{k+N}\right)-\zeta \left(t_{k+N}\right){\left)\right||}_{P_N}^2\hfill \end{array}$$

where $t_N$ denotes the time horizon for an N number of predictions and $Q_{\zeta }$ and $Q_u$ are positive definite matrices weighting control objectives of the cost function, J. The cost function, J, is associated with predetermined trajectory-tracking objectives. The matrix $Q_{\zeta }$ imposes a penalty on the trajectory-tracking errors $({\zeta }^{\mathrm{ref}}\left(t\right)-\zeta \left(t\right))$, thereby enabling prioritisation of tracking of the mobile base over the control effort on the robotic arm. Matrix $Q_u$ penalises the control input effort, minimising abrupt changes in the control actions. On the other hand, $J_N$ is the terminal cost function, which is in charge of guaranteeing global optimal responses. In this paper, $J_N$ is defined for the position and orientation coordinates of SSMM, with the objective of penalising the controlled system states at the end of the prediction horizon. The function $J_N$ addresses the system towards the reference and converge to zero error. The system states are constrained based on the physical limits of the SSMM joints. The mobile base can translate and rotate without restrictions, beyond that of lateral motion constraints. However, the joints of the robotic arm are constrained by their physical construction, requiring a range of operation. Constraints on the control inputs are set according to the maximum value that can be driven on each motor associated with the wheels and joints of the SSMM. The mechanical coupling of the SSMM generates a bidirectional distribution of disturbances, demanding an additional constraint in the optimisation problem to ensure system convergence. This constraint is closely related to the concept of passivity, which asserts that a system must not generate more energy than it receives. By incorporating this passivity constraint, the optimisation process effectively handles energy exchange within the system, thereby enhancing stability and ensuring that the controlled system can robustly handle disturbances without exhibiting unstable behaviour.

The interaction between an input control space, $\mathbb{U}$, to an output space, $\Omega$, has been defined by Equation (7). With that consideration and using the definition of Hatanaka [22], the robotic system is set to be passive if there exists a positive semi-definite storage function, V, such that the following inequality is satisfied:

$$V\left(\zeta \left(t_1\right)\right)-V\left(\zeta \left(t_0\right)\right)\le {\int }_{t_1}^{t_2}{\Omega }^T\left(t\right)u_n\left(t\right)dt$$

(10)

Equation (10) is required for all $t_0\le t_1$ when the set $(u_n\left(t\right),\zeta \left(t\right),\Omega \left(t\right))$ satisfies the system dynamics of Equation (7). If V is differentiable as a function of time, then the inequality in (10) can be written by:

$$\dot{V}\left(\zeta \left(t_1\right)\right)\le u^T\left(t\right)\Omega \left(t\right)$$

(11)

Accounting for the dynamic system of the SSMM with 5-DoF, as defined by Equation (7), the gravity vector, G, is found to have a relationship with the potential energy matrix, P, as defined through Equation (12) [22].

$$G\left(q\right)={\left({\displaystyle \frac{\partial P\left(q\right)}{\partial q}}\right)}^T$$

(12)

The sum of the kinetic energy and the potential energy, P, of the robotic system is used to define the storage function, V, as follows:

$$V\left(\zeta \right)={\displaystyle \frac{1}{2}}{\dot{q}}^{T}M\left(q\right)\dot{q}+P\left(q\right)$$

(13)

Then, the time derivative of V satisfies:

$$\dot{V}\left(\zeta \right)={\dot{q}}^T\tau -{\displaystyle \frac{1}{2}}\left(\dot{M}\left(q\right)-2C(q,\dot{q})\right)\dot{q}={\dot{q}}^T\tau$$

(14)

Equation (14) is used to determine the passivity function of the SSMM from torque input $\tau$ at joint speed $\dot{q}$. Then, the inequality constraint (11) is satisfied, and the last equality constraint in (14) may be integrated into the optimisation problem (9) to minimise the input control, $u_n$.

5. Passivity-Based Robust Control Strategy

The proposed robust controller was designed to be able to maintain suitable robust performance even in the presence of external disturbances, variations of system parameters, and mismatch model errors. Robust control strategies set up an operating range that limits the extent to which the controller can mitigate adverse effects on the robotic system. This range can also be viewed as ensuring passivity compliance, where the system’s behaviour is constrained to remain energetically stable, even when faced with external disturbances. By limiting the energy introduced into the system, passivity leads to the avoidance of excessive complexity in the control analysis and algorithms. This work examines the operating range for robustness testing in relation to the variation of model parameters. In particular, the impact of varying the inertia and mass values of the mobile manipulator components is examined. In addition, external disturbances that could impact the robot speed when the mobile base is displaced over terrain is also addressed. Such disturbances are related to the terrain type, sliding, and the presence of rocks or skidding situations, and they reflect the terra-mechanical interaction between the wheel and the surface. These interactions are modelled as passive exchanges of energy between the wheel and the terrain, ensuring the system remains dynamically stable.

The ADRC strategy is adaptable for multi-input/multi-output systems and can be suitable when certain parameters remain unknown. In this paper, the ADRC approach is used as a compensator based on the error model. The core concept of ADRC involves encapsulating the unmodelled or unknown dynamics of the robotic system as a total disturbance, which is subsequently estimated and actively compensated using an Extended State Observer (ESO)-based controller [24,63]. The disturbance compensation can also be interpreted as maintaining energy balance in the system, analogous to passivity-based control principles, where external energy inputs are regulated to avoid destabilization. The ADRC control loop, as proposed by [24], consists of three key components: (i) a transient profile generator, (ii) an ESO, and (iii) a disturbance-rejection law. The initial component of the ADRC strategy takes as input the reference r and generates a transient profile from reference $r_1$ and its derivatives, ${\dot{r}}_1,{\ddot{r}}_1,\dots ,r_1^n$ [59]. Within the proposed controller, the compensator is designed to address the discrepancies between the nominal model and the actual response of the SSMM. Ideally, when the error is zero the transient profile generator component can be omitted, allowing the ESO to directly provide estimated state information to generate the control output. This behaviour reinforces passivity principles, as the control strategy ensures that no additional energy is unnecessarily introduced into the system once disturbances are intended to be counteracted.

An ESO is designed here to estimate the robotic system output, its derivatives, and disturbances, including external disturbances and unmodelled system dynamics. In the proposed strategy, an ESO requires a nominal model, which represents the SSMM system response without disturbances or uncertainties. The nominal model is assumed to exhibit passive characteristics, ensuring that, in the absence of disturbances, the energy exchanges in the system remain balanced and stable. The uncertainty-free model, derived from Equation (7), is then described as follows:

$$\dot{\overline{\zeta }}\left(t\right)=f(\overline{\zeta }\left(t\right),\overline{u}\left(t\right))$$

(15)

where $\overline{\zeta }$ is the nominal state vector and $\overline{u}$ is the nominal input control vector of the SSMM. The error model is defined by the difference between the real system (7) and the nominal model (15). Hence, The difference between the SSMM and the nominal model response generate an error variable, $▵\zeta$, which is defined by Equation (16). The error variable captures energy imbalances caused by disturbances or uncertainties, which are addressed to restore system passivity.

$$▵\zeta \left(t\right)=\overline{\zeta }\left(t\right)-\zeta \left(t\right)$$

(16)

It is considered a state-space representation for ESO, as shown in (17):

$$\begin{array}{ccc}\hfill \dot{{\zeta }_1^{*}}& =& {\zeta }_2^{*}\hfill \\ \hfill \dot{{\zeta }_2^{*}}& =& -a_0{\zeta }_1^{*}-a_1{\zeta }_2^{*}+b^{*}{u}^{*}+w\hfill \\ \hfill {\Omega }^{*}& =& {\zeta }_1^{*}\hfill \end{array}$$

(17)

where the notation ${(}^{*})$ is included in the variables related to the observer and w represents a weight indicating the load disturbances affecting the system dynamics. Additionally, the parameters $a_0$ and $a_1$ are considered unknown, while an approximate value for the critical gain, $b^{*}$, is assumed to be available. The first two terms in the expression for $\dot{{\zeta }_2^{*}}$ are combined into a single function, denoted by the total disturbance, h. This term h encompasses both w and the discrepancy between the actual value of $b^{*}$ and its nominal value, $b_0^{*}$, formulated as:

$$h=-a_0{\zeta }_1^{*}-a_1{\zeta }_2^{*}+(b^{*}-b_0^{*})u^{*}+w$$

(18)

Then, the new state-space model for the ESO is as follows:

$$\begin{array}{ccc}\hfill \dot{{\zeta }_1^{*}}& =& {\zeta }_2^{*}\hfill \\ \hfill \dot{{\zeta }_2^{*}}& =& h+b_0^{*}{u}^{*}\hfill \\ \hfill {\Omega }^{*}& =& {\zeta }_1^{*}\hfill \end{array}$$

(19)

The total disturbance is unknown; therefore, h is assigned to an additional state, ${\zeta }_3$. This state is estimated and compensated by the inner control loop proposed by the ADRC strategy (see Figure 2). Thus, the extended state-space model (20) is obtained assuming that ${\zeta }_3\triangleq h$ and $k=\dot{h}$ is unknown:

$$\begin{array}{ccc}\hfill \dot{{\zeta }_1^{*}}& =& {\zeta }_2^{*}\hfill \\ \hfill \dot{{\zeta }_2^{*}}& =& {\zeta }_3^{*}+b_0^{*}{u}^{*}\hfill \\ \hfill \dot{{\zeta }_3^{*}}& =& k\hfill \\ \hfill {\Omega }^{*}& =& {\zeta }_1^{*}\hfill \end{array}$$

(20)

The evolution of the system states in (20) is obtained from the Extended State Observer (ESO), whose inputs correspond to the control action $u^{*}$ acting on the state-space system and measured output ${\Omega }^{*}$, as follows:

$$\begin{array}{ccc}\hfill \dot{{\xi }_1^{*}}& =& {\xi }_2^{*}-{\beta }_1{\gamma }_1\left(e\right)\hfill \\ \hfill \dot{{\xi }_2^{*}}& =& {\xi }_3^{*}+b_0^{*}{u}^{*}-{\beta }_2{\gamma }_2\left(e\right)\hfill \\ \hfill \dot{{\xi }_3^{*}}& =& -{\beta }_3{\gamma }_3\left(e\right)\hfill \\ \hfill e& =& {\xi }_1^{*}-{\Omega }^{*}\hfill \end{array}$$

(21)

where ${\xi }_i$ denotes the estimate of the i-th state, ${\zeta }_i$, ${\beta }_i$ are gain coefficients, and ${\gamma }_i\left(e\right)$ corresponds to nonlinear functions of the estimation error, e, that define the observer correction terms. In the proposed strategy, the measured output signal is represented as ${\Omega }^{*}=▵\zeta$. Active rejection of disturbances is achieved by subtracting the total disturbance contained in ${\xi }_3^{*}$ from the control law, $u_0$, according to (22) [24].

$$u_c={\displaystyle \frac{u_0-{\xi }_3^{*}}{b_0^{*}}},b_0^{*}\ne 0$$

(22)

Replacing (22) in (19) and assuming that ${\xi }_3^{*}\approx h$, the new state-space system can be written as follows:

$$\begin{array}{ccc}\hfill \dot{{\zeta }_1^{*}}& =& {\zeta }_2^{*}\hfill \\ \hfill \dot{{\zeta }_2^{*}}& =& u_0\hfill \\ \hfill {\Omega }^{*}& =& {\zeta }_1^{*}\hfill \end{array}$$

(23)

The state-space model (23) represents a disturbance-free modified model, and the control action was proposed by:

$$u_0={\overline{\beta }}_{1}fal(e_1,{\alpha }_1,\delta )+{\overline{\beta }}_{2}fal(e_2,{\alpha }_2,\delta )$$

(24)

Different combinations of arguments make it possible to evaluate the nonlinear expressions of the ESO according to ${\gamma }_i\left(e\right)=fal(e;{\overline{\alpha }}_i;\overline{\delta })$.

$$fal(e,{\alpha }_i,\delta )=\left\{\begin{array}{ccc}\hfill {\displaystyle \frac{e}{{\alpha }^{1-{\alpha }_i}}}& ,& \left|{\zeta }^{*}\right|\le \delta \hfill \\ \hfill {\left|e\right|}^{{\alpha }_i}sign\left(e\right)& ,& \left|{\zeta }^{*}\right|>\delta \hfill \end{array}\right.$$

(25)

The ADRC requires the selection of the following parameters: the observer gains, ${\beta }_i$, constants ${\overline{\alpha }}_i$ and $\overline{\delta }$ for the evaluation of the nonlinear observer functions, gains ${\overline{\beta }}_i$ together with constants ${\alpha }_i$, as well as $\delta$ for the design of the control law. The ADRC gains correspond to the bandwidth response of both the observer and the controller. However, in this work, heuristic methods are applied to determine the gains for $u_0$, which define the disturbance-rejection capability of the compensator, $u_c$, thus forming the proposed robust controller for the SSMM.

6. Experimental Tests and Results

This section presents the results of trajectory-tracking tests conducted in a 5-DoF SSMM. Three trajectories have been proposed: (i) squared-type, (ii) circle-type, and (iii) Lemniscata-type trajectories. The square trajectory enables the assessment of abrupt orientation changes of the robot when reaching corners. The other two trajectories allow one to assess the SSMM response for recurrent robot position and orientation changes over time. This test is aimed at evaluating the trajectory-tracking performance of the robotic arm with respect to both longitudinal tracking control and three-dimensional positioning. The assessment encompasses the mobile manipulator’s responses, along with the evaluation of the arm tracking 3D trajectories. This entails measurement of deviations from the target coordinates. The manipulator response is evaluated in accordance with the defined travel speed conditions, which are considered to ascertain the manipulator’s performance.

To tune the control gains of the PID controller, the process initially used the auto-tuning tool provided in MATLAB. However, considering the sensitivity of PID controllers to the system’s dynamic characteristics, additional manual tuning was necessary to further enhance performance. To achieve this, systematic simulations and field experiments were conducted using step reference inputs with both positive and negative variations. This approach allowed for a detailed observation of the system’s response to abrupt input changes, enabling an evaluation of the controller’s capability to handle rapid transitions and maintain stability effectively. Key performance metrics, such as settling time and steady-state error, were closely monitored during these simulations and experiments. Then, the PID parameters were iteratively adjusted to refine the controller’s performance. The final set of tuned PID gains used in this study were $5.2,5.7,6.1,7.03,5.01$ for the proportional term, $0.02,0.1,0.03,0.007,0.015$ for the integral term, and $1.1,20.07,10.51,30.1,20.15$ for the derivative term, respectively. These values were carefully selected to ensure the best possible performance under the tested scenarios, providing a robust basis for comparison with the proposed control strategies.

To tune the parameters of the NMPC controller, a heuristic method was used by considering the trade-off between the tracking errors and control input effort of both base and arm (see

Table 2. Weight matrices and configuration parameters used in NMPC and R-NMPC.

Weight Matrix	Values
$Q_{\zeta }$	diag(60, 100, 150, 70, 90, 0.1, 1, 1, 0.1, 0.1)
$Q_u$	diag(0.1, 1, 1, 0.1, 0.1)
$P_N$	diag(60, 100, 150, 70, 90)
${\beta }_i$	[0.9, 0.9]
${\overline{\alpha }}_i$	[1.2, 1.2]
$\overline{\delta }$	[0.2, 1.2]
${\overline{\beta }}_i$	[0.5, 0.8]
${\alpha }_i$	[1, 10]
$\delta$	[20, 20]

). The matrix of weights $Q_{\zeta }$, $Q_u$, and $P_N$ are used for the NMPC and R-NMPC configuration. The first five elements of the diagonal in $Q_{\zeta }$ and $P_N$ penalise the tracking error for position variables of the SSMM according to Equation (1). The variable ${\theta }_1$ is taken as a priority because it is the manipulator base and is highly susceptible with respect to changes in the other joints. The second priority is the translational motion of the robot base. The weights of the following variables were set according to non-oscillatory and stable responses. Using a similar insight, values for the penalisation of control inputs by means of $Q_u$ were found, prioritising the control of the second and third joint coordinates of the SSMM. In addition, the remaining model parameters required for the ADRC compensator in the mobile base are further detailed in

Table 3. Model parameters obtained for the articulated Skid-Steer Mobile Manipulador (SSMM).

Parameter	Value	Parameter	Value
$m_b$	12 kg	$J_b$	0.5 kg m2
$b_1$	1	$b_2$	0.07
$b_3$	0.12	$b_4$	0.12
$b_5$	0.12	$m_3$	2.867 kg
$m_4$	0.633 kg	$m_5$	0.79 kg
$d_1$	0.06 m	$d_2$	0.019 m
$d_3$	0.139 m	g	9.8062 m/s2

. For normalization purposes [24], the configuration parameters ${\beta }_i$ and ${\overline{\beta }}_i$ for the observer and control were set to values less than 1. The parameter ${\alpha }_i$ was found based on how fast the system compensates for disturbances. Parameter $\delta$ was chosen to prevent the system overshoot in the SSMM responses. The remaining parameters were chosen according to the best system response of the experimental tests.

Tests were conducted using three benchmark controllers: (i) a standard PID, (ii) NMPC, and (iii) R-NMPC. Two principal tests were carried out with such controllers. The first one consisted of assessing the performance of controllers for trajectory tracking in the presence of disturbances. Meanwhile, in the second test, a variation of the model parameters was introduced. In this case, the mass of the mobile base, its rotational inertia, and the length of the first two links of the robotic arm were modified. This second test consisted of testing the robustness of the proposed controller under the presence of an inaccurate prediction model. To assess the results of the proposed robust controller and the two additional controllers, three performance indices were used: (i) the cumulative tracking error, $C_{\zeta }$, (ii) the total control effort, $C_u$, and (iii) the total cost, $C_{tot}$ [30]. Each performance index was obtained while controlling the mobile base and the robotic arm, denoting the robot link as sub-indices (b) for the base and (a) for the arm.

6.1. Experimental Setup

The optimisation problem related to the proposed R-NMPC approach is solved using nonlinear programming. Such programming uses the fmincon function with the Sequential Quadratic Programming (SQP) algorithm, which requires the Optimisation Toolbox in Matlab R2023 update 1. To implement the algorithm, the Matlab software is used on a computer with an Intel^® Core(TM) i7-10750H @2.6 GHz processor and 16 GB of memory. For this paper, a 5-DoF skid-steer mobile manipulator was used, which has a 4 wheeled mobile base mechanically coupled with a robotic arm of three links. The mechanical coupling generates bidirectional disturbances from the base by the wheel–terrain interaction to the mobile arm and vice versa. The model used in the simulation workspace was debugged in [65]. Table 3 describes parameters of the SSMM model used here, where $m_b$ is the base mass without the robotic arm; $J_b$ is the mobile base inertia; $b_i$ is the friction coefficient; $m_i$ is the mass of each link; $d_i$ is the length of each link; and g is the gravity acceleration.

Control input constraints were set to $|{\tau }_i| \le 100$ (Nm) and $|F_b| \le 50$ (Nm). In addition, state constraints were set according to rotation limits for the joint axes by $|{\theta }_{b1}| \le \pi \left(\mathrm{rad}\right)$, $|{\theta }_{m1}| \le \pi$ (rad), $|{\theta }_{m2}| \le 0.7\pi$ (rad), and $|{\theta }_{m3}| \le 0.6\pi$ (rad).

6.2. Simulation Test for Disturbance Rejection

A circular trajectory was used due to its recurrent variation with respect to orientation and position reference and its constant speed (see Figure 3). The aim of this evaluation is to assess the performance of the robot in designated sections with terrain changes. To simulate slip corresponding to possible irregular sections through the terrain, linear and angular speed disturbances were introduced to the base of the SSMM. Linear speed disturbance was introduced at regular intervals of 20% throughout the simulation, varying in amplitude according to the vector ${[0,0.1,0.25,0.35,0.45]}^T$ (m/s). On the other hand, the disturbance in angular speed was introduced under the consideration of a step-type signal with amplitude 0.3 (rad/s) at 60% of the total simulation time. These conditions are designed to simulate challenging terrain scenarios that the mobile manipulator would traverse to allow the base to advance toward the target and ensure that the robotic arm reaches the predetermined height. A square-type trajectory for the robotic arm was used. The reference changes of each joint coordinate introduces disturbances, which are reflected in the moving base due to its mechanical coupling. A comparative analysis of the three proposed controllers was performed to show which controller achieved the greatest reduction in tracking error.

The first trajectory to be considered for the robot base is a circle with radius of 15 m. A square trajectory with a width of 0.35 m and a height 0.25 m is used for the robotic arm. In addition, a constant speed of 0.5 m/s is considered for the calculation of position and orientation coordinates. Figure 4 and Figure 5 show the performance metrics and trajectories achieved by the three controllers under analysis. It is important to note that a portion of the total test time is extracted to show the transient response of the control signal performance for each signal. Position changes in the robotic arm were given at time 15, 65, and 115 (s), which caused an overshoot in the tracking error of the robotic arm and in the manipulated variable. This phenomenon is also reflected in the base and is taken as the bidirectional disturbance due to the mechanical coupling. The NMPC and R-NMPC controllers minimise the effect of disturbances and prioritise trajectory tracking. Because of this, the $C{\zeta }_b$ of NMPC and R-NMPC is smaller than PID by 40% and 69%, respectively. The PID controller is the least effective at rejecting disturbances. The R-NMPC is the controller with the lowest error in the position coordinates of the mobile base ($C{\zeta }_b$), as shown in Figure 4. Similarly, disturbances in the moving base are reflected in the robotic arm, but with smaller overshoots. A positional discrepancy is observed in the robotic arm’s motion under the influence of the R-NMPC. This could be attributed to the controller prioritising the trajectory tracking of the base and partially sacrifices the performance of the joints. For this R-NMPC, $C{\zeta }_a$ is 19% higher than the NMPC. Despite the control efforts to prioritise the movement on the ground (see Figure 5), the NMPC and R-NMPC performance indexes $C{u}_b$ and $C{u}_a$ have an improvement of 72% and 40% compared to PID, respectively (see Figure 4).

The second trajectory used follows a lemniscate-type trajectory, with a height of 30 m and a width of 15 m. Linear speed disturbances were introduced at regular intervals of 20% throughout the simulation, while angular speed disturbances were applied at 60% of the total simulation time (see Figure 6). To assess abrupt changes in joint positions at corners, a square-type trajectory was maintained for the manipulator (see Figure 7), which indirectly propagates disturbances from the mobile base to the robot arm. A time range of 30 s was selected for the time axis to better visualize the transient behaviour of the control signal.

To evaluate the performance of the proposed controller against unexpected disturbances, a time range between disturbances followed by a change of joint position was also set. Then, there were more control requirements, which can be seen in Figure 7. In this instance, it can be observed that both PID and NMPC tend to generate control responses with overshoots. On the other hand, the R-NMPC controller maintained a stable response without compromising its performance in the mobile base. Figure 8 illustrates that the R-NMPC showed an improvement of 89% and 79% in terms of tracking error for the mobile base compared to the PID and NMPC, respectively. The robotic arm shows a tracking error at this time but meets prioritising the mobile base and the reference motion on the ground. Figure 8 shows that the R-NMPC obtained an improvement of 14% and 27% over the PID and NMPC in terms of tracking error for arm coordinates. With respect to the control effort, RNMPC and NMPC had 76% and 58% improvements, respectively, in comparison to PID. These values correspond to a reduction in the tracking error compensation. In both trajectory-tracking tests, the R-NMPC controller demonstrated better performance while minimising the trajectory-tracking error of the mobile base. Indeed, the proposed control strategy could mitigate the impact of recurrent disturbances in the SSMM system and facilitated the generation of a less oscillatory control effort.

6.3. Simulation Test Under Parameter Variations

To assess robust performance of the proposed controller, several simulation trials were conducted under conditions of mismatch model errors. Then, circular, Leminscata, and square trajectories were used as references for the SSMM, subject to model parameter variations. The same trajectories from Section 6.2 were used here. The prediction model used in the NMPC controller and the proposed R-NMPC controller is subject to parameter variations, including a 30% mass of mobile base and a 10% of robotic arm, a 30% inertia of mobile base, and a 35% of length of the first link. Such variations comprises the maximum values under which the mobile base and arm are able to operate under feasible constraints. The introduction of an off-line parameter variation resulted in the generation of a modelling error and an increase in model uncertainty. The proposed R-NMPC controller had to be capable of compensating for such variations, thereby minimising the tracking error and maintaining a stable control signal over time.

Tests were carried out on two trajectories for the mobile base. The first one was a circular trajectory. Figure 9 shows the tracking error for the mobile base. The PID controller still presented overshoots when the controlled system was affected by lineal and rotational speed disturbances, as well as under indirect disturbances caused by the motion of the robotic arm joints. The NMPC and R-NMPC mitigated this phenomenon by maintaining the tracking error close to zero. However, the NMPC tends to show larger errors and even loses efficiency. It was also shown that the tracking error in $x_b$ increased by up to 0.6 m, indicating that the system’s response was delayed and that it did not return to the original point. Analysing the performance indexes (Figure 10), it can be seen that the NMPC and R-NMP controller had a 55% and 76% improvement, respectively, compared to the PID. In the worst-case scenario, the R-NMPC showed a reduction in tracking error for joint coordinates, with a maximum error of 0.1 rad when prioritising base trajectory tracking. Figure 9 demonstrated that the R-NMPC controller achieved smaller overshoots compared to the NMPC and PID controllers. Overall, the proposed controller maintained the tracking error near zero during all the simulation tests. Analysing the total tracking error, Figure 10 shows that NMPC and R-NMPC achieve improvements of 52% and 37% over PID, respectively.

Figure 9 illustrates the controlled system response between 60 and 80 s. It was shown that the PID controller exhibited larger overshoots and oscillations compared to the predictive controllers, a behaviour that also raised the joint control signals of the robotic arm. When analysing these control efforts, NMPC and R-NMPC demonstrated improvements of 60% and 24%, respectively, over the PID controller for the mobile base, resulting in lower values in their control signals. Figure 10 shows that the control effort of the R-NMPC is 30% greater than that of the NMPC. This corresponds to the effort required to minimise the trajectory-tracking error. In addition, in the control effort for the joints of the robotic arm, the NMPC and R-NMPC present an improvement of 90% with respect to the PID. Figure 9 shows that the R-NMPC control signals present reduced oscillations and more stable curves over time.

The second trajectory for the mobile base follows a Lemniscate-type reference trajectory. Figure 11 illustrates the performance of the three controllers along such a path. In this scenario, the NMPC controller begins to deviate from the reference at approximately 100 s, as evidenced by tracking errors in the $x_b$ and $y_b$ coordinates. Speed disturbances, bidirectional disturbances, and modelling inaccuracies inhibit the NMPC from maintaining proper control. However, the R-NMPC compensated for these disturbances and modelling errors, then reducing tracking errors. Figure 12 indicates that both the PID and R-NMPC controllers yield improvements of 80% and 95%, respectively, over the NMPC, effectively maintaining the mobile base close to its reference. Additionally, due to mechanical coupling, disturbances in the mobile base trajectory affect the robotic arm’s response, resulting in oscillations, particularly at the third corner of the square trajectory. Figure 11 shows that the NMPC controller produced tracking errors of up to 0.3 rad, with overshoots exceeding those of the PID. The analysis of performance in Figure 12 further demonstrated that the R-NMPC achieved a 48% improvement in tracking error over the PID, while the NMPC provided only a 4% enhancement.

Figure 11 shows the response of the controllers under analysis. In the mobile base, the PID generates oscillatory control efforts in the presence of disturbances. The NMPC and R-NMPC tend to generate responses with overshoot. However, NMPC presented a more abrupt response that affects the overall system performance. Figure 11 shows that the performance index related to the control effort on the robot base for the NMPC and R-NMPC presented an improvement of 81% and 79%, respectively, with respect to the PID. On the other hand, in the robotic arm the control efforts of the NMPC and R-NMPC show a relevant improvement in comparison with the PID. The R-NMPC applied to the mobile manipulator outperformed the other test controllers with respect to trajectory tracking at the base, despite minimal addition of control effort.

6.4. Field Test for Trajectory-Tracking Control

To complement the simulated results, this section presents manoeuvrability tests conducted in field trials. Specifically, it compares the previous three motion controllers, PID, NMPC, and the proposed R-NMPC. The controllers were validated using an SSMM experimental setup, consisting of a Katana 5-DoF arm mounted on a Pioneer P3-AT robot (see Figure 13). The reader is encourage to find more details about the SSMM setup in [65,66]. In this test, the mobile base with its 2-DoF and the robot arm with only 3-DoF (shoulder, elbow, and wrist) were used to keep the experiment simple while assessing 3D point reachability in a trajectory-tracking test. It is to be noted that the SSMM was not limited to planar motion, yet assumed feasible trajectories kinematically compatible with the mobile base and arm. Specifically, the reference for the SSMM consisted of a 2D curved (S-shaped) trajectory of approximately 40 m length and 0.6 m amplitude, coupled with a stepped height reference, ranging from 0.7 to 0.4 m and then raising to 0.8 m. Such a reference was point-wise parametrized at 50 ms intervals over a total operation time of 100 s, executed sequentially with a speed profile of 0.7 m/s. The initial pose of the robot was set to ${[x_b,y_b,{\theta }_{b1}]}^T={[0,0,\pi /4]}^T$, whereas the initial reference point was given by ${[x_b,y_b,{\theta }_{b1},z_a]}^T={[4.0,0.3,\pi /4,0.7]}^T$.

Figure 14 presents the field results for the PID, NMPC, and R-NMPC strategies. The SSMM demonstrated the ability to track the reference trajectory with all three controllers. However, the trajectory achieved by the mobile base under PID control exhibited an initial drift relative to the reference position, demonstrating limitations in trajectory-tracking performance (see left plot in Figure 14). In contrast, the results highlight that the R-NMPC strategy achieved enhanced performance compared to the PID and NMPC, exhibiting improved robustness and trajectory-tracking accuracy. As expected, the NMPC and R-NMPC produced similar tracking results for the mobile base, even under disturbances caused by the internal coupling between the mobile base and the arm joints along the height reference changes. This could be explained by the fact that internal arm forces were distributed across the individual arm joints of the SSMM and prevented the full propagation to the robot base, while the control actions of the NMPC and R-NMPC for the mobile base were robust enough to similarly counteract the remaining force propagation. Unlike R-NMPC, which maintains a consistent transient arm response (see centre and right bottom of Figure 14), the instantaneous height responses of the SSMM arm were quite irregular using PID and NMPC approaches in relation to the reference changes, which can also be attributed to internal forces generated in the arm joints. On the other hand, the NMPC and R-NMPC achieved similar steady-state tracking errors in the mobile base and arm, regardless of the height variability, with errors closer to zero compared to that obtained with PID. However, the R-NMPC showed better tracking error regulation than NMPC, even with changes in the arm height position, as shown in the top and right plots of Figure 14.

As outlined in

Table 4. Performance assessment of the three motion controllers on the SSMM, tracking 3D reference trajectories at a constant speed profile.

Performance Metrics	PID	NMPC	%	R-NMPC	%
Metrics for the mobile base:
Cum. Tracking error ($C_{{\zeta }_b}$)	7.33 × 101	3.06 × 101	−58.2%	2.46 × 101	−66.4%
Cum. Control effort ($C_{u_b}$)	51.14 × 102	32.60 × 102	−36.2%	41.45 × 102	−18.9%
Total cost ($C_{to{t}_b}$)	51.87 × 102	32.91 × 102	−36.5%	41.69 × 102	−19.6%
Metrics for the robot arm:
Cum. Tracking error ($C_{{\zeta }_a}$)	8.80 × 101	7.51 × 101	−14.6%	5.07 × 101	−42.3%
Cum. Control effort ($C_{u_a}$)	31.55 × 102	33.28 × 102	5.4%	37.16 × 102	17.7%
Total cost ($C_{to{t}_a}$)	32.43 × 102	34.03 × 102	4.9%	37.66 × 102	16.12%

, the quantitative comparison of control performance provides insights into the effectiveness of the three tested controllers: PID, NMPC, and R-NMPC. First, it is shown that the cumulative tracking errors in the mobile base were reduced by 58.2% for NMPC and 66.4% for R-NMPC with respect to PID, but R-NMPC achieved the highest reduction of the cumulative tracking error regarding the other test controllers. The control effort also presented similar results for the mobile base, decreasing by 36.2% and 18.9% for NMPC and R-NMPC, respectively. When examining the robot arm, the tracking error showed a significant reduction of 14.6% with NMPC and 42.3% with R-NMPC. The tracking error reduction with R-NMPC suggests that it effectively handled the coupled dynamics between the arm and the mobile base, particularly under varying reference changes. However, R-NMPC achieved high torque effort of about 17.7%, unlike NMPC, which barely achieved a 5.4% increase. This can be attributed to the additional computational burden required by R-NMPC in the ESO and ADRC strategies to estimate and compensate for disturbances in real-time. Despite such increases in the cumulative control effort in the torque actuation, the impact on the robot’s overall resource efficiency may be minimal in the long term, as improved tracking performance and reduced trajectory errors can lead to more precise and energy-efficient operation over time.

6.5. Field Test of Robustness Under Terrain Disturbances

In this Section is provided a comprehensive evaluation of robust control performance while tracking feasible 3D reference trajectories. The given reference was composed of a straight-line shaped trajectory in the horizontal plane and constant height while terrain disturbances occurred. The reference trajectory considered a 24 m length and 0.7 m height. Similar to the previous test, the reference trajectory was parametrized with a speed profile of 0.7 m/s, and the robot sampling time was set to 50 ms. In this trial (see Figure 13), terrain unevenness was intentionally introduced through a 0.2 m high mound and a slope of approximately 35° to ensure repeatability of loss-of-traction in navigation conditions, extending the previous field experiments discussed in Section 6.4. The initial robot pose was set to ${[x_b,y_b,{\theta }_{b1}]}^T={[0,0,\pi /4]}^T$, whereas the initial reference point was given by ${[x_b,y_b,{\theta }_{b1},z_a]}^T={[1.0,0.4,\pi /4,0.9]}^T$.

The SSMM successfully maintained its trajectory across all three motion controllers (i.e., PID, NMPC, R-NMPC) despite the presence of terrain disturbances, as illustrated in Figure 15. However, as shown in the left plot of Figure 15, the robot’s trajectories revealed significant lag and deviation in the SSMM produced by the wheel–terrain interaction, with minor oscillations in the robot arm while traversing the uneven surface of the mound. Specifically, the oscillatory behaviour in the arm can be attributed to its dominant joint inertia with respect to the mobile base. Using the PID control strategy, the SSMM response was more reactive than anticipatory to terrain disturbances compared to that obtained with the NMPC and R-NMPC strategies. The SSMM response with PID exhibited larger initial errors, as the correction was reactive rather than anticipatory, occurring only after the tracking errors had emerged in a slippery situation, whereas NMPC and R-NMPC demonstrated faster correction and reduced tracking errors more effectively (see top centre and right plots of Figure 15). Specifically, NMPC and R-NMPC demonstrated faster convergence and effectively reduced the mobile base tracking errors, while the PID controller took longer to approach the tracking errors closer to zero. The PID strategy proved more sensitive to height disturbances in uneven terrain than R-NMPC, resulting in more oscillatory behaviour and larger inverse peak responses in the SSMM arm after terrain disturbances. In contrast, R-NMPC exhibited the lowest peak responses and tracking errors in both the lateral and longitudinal deviations of the mobile base, as well as in the robot arm height as the SSMM traversed terrain disturbances (see responses in shaded areas of Figure 15). This indicates that R-NMPC was more effective in counteracting the effects of terrain disturbances due to the additional ADRC compensation on the mobile base, leading to their propagation to the robot arm being mitigated.

Robustness in three control strategies was assessed with the previous performance metrics, as detailed in

Table 5. Performance assessment of the three motion controllers on the SSMM, tracking 3D reference trajectories under the presence of terrain disturbances.

Performance Metrics	PID	NMPC	%	R-NMPC	%
Metrics for the mobile base:
Cum. Tracking error ($C_{{\zeta }_b}$)	16.26 × 101	8.21 × 101	−49.5%	3.33 × 101	−79.5%
Cum. Control effort ($C_{u_b}$)	44.68 × 102	46.77 × 102	4.7%	56.07 × 102	25.5%
Total cost ($C_{to{t}_b}$)	46.30 × 102	47.59 × 102	2.8%	56.40 × 102	21.8%
Metrics for the robot arm:
Cum. Tracking error ($C_{{\zeta }_a}$)	20.04 × 101	11.34 × 101	−43.4%	10.85 × 101	−42.3%
Cum. Control effort ($C_{u_a}$)	59.77 × 102	64.99 × 102	8.7%	61.74 × 102	3.3%
Total cost ($C_{to{t}_a}$)	61.77 × 102	66.12 × 102	7.0%	62.83 × 102	1.7%

. When compared with the cumulative tracking errors, the R-NMPC controller exhibited the largest reduction for both the mobile base ($C_{{\zeta }_b}$) and robot arm ($C_{{\zeta }_a}$), with a 79.5% and 42.3% improvement, respectively. The NMPC strategy also showed enhanced performance, reducing tracking errors by 49.5% and 43.4% for the mobile base and arm, respectively, but it was outperformed by R-NMPC in both cases. In contrast, the PID technique presented the least control performance, with the lowest cumulative tracking errors for the mobile base and arm, with a lack of robustness against terrain disturbances. Despite R-NMPC presenting an increased cumulative control effort for both the mobile base (25.5% increase) and the arm (3.3% increase), only 3.3% of the control effort was enough to counteract the terrain disturbance propagation to the arm. In practice, the additional control effort is associated with the arm’s ability to retract in response to the height changes. Actually, the control effort for NMPC increased by 4.7% for the mobile base and 8.7% for the robot arm, exhibiting a less aggressive response regarding PID, but still having an efficient strategy in handling terrain disturbances. The NMPC struck a better balance between robustness and control effort while also offering significant performance improvements over PID. The PID controller exhibited the reduced effectiveness with the least control effort while handling the impact of terrain disturbances. Additionally, the R-NMPC strategy demonstrated the best robustness in terms of tracking accuracy, despite its increased control effort. This suggests that the benefits in terms of improved tracking and disturbance rejection outweigh the additional cost in control effort.

7. Conclusions

In this paper, a Robust Nonlinear Model Predictive Control (R-NMPC) has been designed and implemented for a 5-DoF Skid-Steer Mobile Manipulator (SSMM) under the presence of modelling errors and terrain disturbances. The controller was designed using a coupled dynamical model that characterized the full-body robot dynamics. The control strategy was raised within an integrated control scheme, under which the overall control combined a nominal and an ancillary control action. First, an NMPC acting as a nominal controller was used for trajectory tracking. The NMPC comprised a passivity criterion to guarantee the convergence and reachability of the control system for a given reference. Second, a compensation control action based on the Active Disturbance Reject Control (ADRC) strategy was incorporated, which uses information from the mismatch between a nominal model and the actual SSMM dynamics. The ADRC strategy encompasses disturbances and modelling errors to generate a compensation control action that minimises the effects of terrain disturbances. The ADRC leverages an Extended State Observer (ESO) to quantify the impact of external disturbances. The results indicate that the proposed R-NMPC emerges as the best control strategy when compared to non-robustified controllers, prioritising the motion compensation of the robot base over the robot arm against terrain disturbances. Simulation results exhibited a 50% improvement of the tracking error compared to the NMPC and an 80% enhancement in tracking error relative to the PID controller. Experimental results in field tests on a Pioneer 3P-AT mobile base and Katana robot arm showed that the tracking errors can be reduced with R-NMPC, approaching a reduced tracking error of 79.5% for the mobile base and 42.3% for the robot arm. Such results show the potential to strengthen robust control performance in mobile manipulators and optimise robot resources. Future work could explore the implementation of passivity-based distributed control to address time-varying uncertainties or disturbances and further enhance robust control performance separately from base–arm coupling, particularly in response to the unpredictable and heterogeneous nature of terrain conditions.