Towing an Object With a Rover

Rovers have always been the center of attention when it comes to space exploration. Moreover, with the study of possible near-future human settlements on the Moon, it is becoming more and more necessary to focus studies on how these robotic systems can be used for colony preparation, i.e., precursor missions. Among the many operations required by these efforts, the manipulation and positioning of objects on the surface may be considered of prime importance. Several examples exist of rovers capable of manipulating objects and performing sample collection [1–3]. The most recent example is the Mars 2020 Rover “Perseverance,” which has a 5 degrees-of-freedom (DOFs) robotic arm installed for use in collecting and caching samples [4,5]. The problem, although of great interest to the scientific community, presents two main challenges: (1) planning operations, i.e., how to plan the motions of the rover so as to correctly place the object in the intended position [6,7]; (2) accuracy and robustness, i.e., how to successfully position the object while avoiding the obstacles in the environment.

In the literature, a large amount of work can be found about many different kinds of object manipulation strategies achieved through mobile robots. Notably, many papers focus on cooperation strategies between multiple robots in order to complete the assigned task. For example, Yamashita et al. in 1998 studied the cooperative manipulation of objects using sticks and strings as tools for pushing or pulling respectively [8]; flexible elements tied around the object have been studied for manipulation using prehensile [9,10] and nonprehensile [11] approaches. Other transportation strategies found in the literature concern objects transported on top of a team of mobile robots [12–14]; these mostly regard control and formation control aspects. A similar strategy proposed by Hichri et al. in 2014 probes the possibility for multiple robots to cooperatively lift and transport an object [15]. Pushing is a widely explored manipulation strategy: it is either performed by single pushing mobile robots [16–18] or cooperatively [19], which can include caging [20]. Ohashi et al. in 2014 describe a hybrid strategy, which consists of the mobile robot tilting the object, in order to position handcarts under it, thus enabling transportation by pushing [21,22].

Towing is a manipulation strategy in which cables are involved; it has been largely studied and can be found in everyday contexts within the fields of marine and aerial applications. In the former, towing is used in “escort tugs” assisting large ships [23,24] or towing small submarines [25,26]. In the latter, instead, towing is mostly used to transport hanging loads [27–31], but has been studied also for the towed decoy system to protect military aircraft from radar-guided missiles [32].

While towing is widely used in the aerial and marine fields, a small number of works have been found on using this strategy applied to ground mobile robots; indeed, other types of strategies are preferred such as pushing, since cables introduce a considerable level of complexity. The Axel rover is a typical example of two rovers attached by a tether [7,33]; a similar approach is that of Cliff-bot [34]. Cheng et al. in 2008 studied the problem of towing an object via multiple cooperating robots, each tethered to the object [35]; a quasi-static approach was used to tow an L-shaped object with three point-like contact points. Kim and Shell proposed using passive tails fixed to the rear of mobile robots to hook and tow objects [36,37]. Wilson et al. in 2018 developed models for towing objects based on the behavior of ants [38].

When an object placed on a planar surface is being manipulated either by pushing or towing, it is forced to slide; this is referred to as the planar sliding problem. The motion of the object can be either translational or purely rotational around a general instantaneous center of rotation (ICR). The contact interface between two bodies in relative sliding is always frictional. When the assumption of single-point contact for the bodies contact interface is not possible and hence the contact interface is a surface, then, in addition to the usual frictional forces, a frictional moment rises up as well.

In general, the motion of an object sliding on a bi-dimensional surface is not trivial to model. Moreover, the resulting motion of the sliding object depends mainly on the pressure distribution between the object and the contact patch, which in general is undetermined and cannot be known in advance. The friction distribution between the sliding parts is another source of indeterminacy: in fact, friction behavior cannot be in general considered constant and uniform over the whole surface, but different areas may exhibit different friction properties. Additionally, friction properties may vary with time. Moreover, also external loads applied to the sliding object influence the pressure distribution.

In the modeling of the contact problem, two general approaches exist: nonsmooth and penalty methods [39]. Nonsmooth methods require contact conditions to be expressed as unilateral constraints, e.g., by solving a linear complementary problem and taking into account the Signorini noncompenetrability condition. Penalty methods (also called regularized approaches), on the other hand, allow for a small violation of Coulomb’s law and Signorini’s noncompenetrability constraint, at the obvious cost of lower accuracy. On the other hand, penalty methods tend to be numerically more efficient [40].

When dealing with space exploration rovers [41,42], the terrain is usually soft and rough rather than hard and smooth. Many works have investigated the wheel–soil interaction [43,44], and in some instances with focus on experimentation [45–47].

Many authors have focused their attention and studies on what is called the mechanics of sliding objects. A very remarkable work was proposed by Mason [48], who formulated an important theorem that determines the sense of rotation (clockwise or counterclockwise), of a pushed object regardless of the pressure distribution; Goyal et al. [49,50] introduced the concept of friction limit surface in order to establish a relationship between the net frictional load, defined as friction force and moment, and the object sliding motion; Howe and Cutkosky proposed that the friction limit surface can be approximated with an ellipsoid, thus synthesizing a relation between the net frictional load and the object sliding velocity [51]. This corpus of research has been applied for robotic pushing [52], for stable robotic pushing [53], for robotic pushing under uncertainty of anisotropic friction properties [54], and for robotic cooperative towing [35]. The same concept has then been largely used also in the field of robotic soft-fingers [55,56], which usually involves the object being pressed on the surface and moved by an external actor such as a robotic arm [57–62]. Within this field, models are simplified because usually the pressure distribution is known.

The present work focuses on the manipulability of an object of general shape via towing it with a cable. At the present stage of research, cables are considered to be massless and as a unilateral elastic constraint (traction only). Generally speaking, manipulation through towing allows to manipulate objects that cannot be lifted, e.g., because they are too heavy or bulky. Moreover, a towing manipulation strategy can be a valid alternative to the classic pick-and-place operations [63] which are already used on rovers but require the rover to be equipped with a robotic arm. This adds complexity to the robotic system, especially given the strict space mission weight constraints. This problem is exacerbated when dealing with reconfigurable or articulated mobile robots [64–66]. Moreover, adding load is a problem for the operational life of the rover: increased wear, power consumption, higher risk of embedding in loose soil, etc. In principle, the detachment of the load that comes with towing could be a valid option in critical situations as well.

The research presented in this article is related to the work presented by Cheng et al. on cooperative towing [35]. Several differences, however, separate the two works. First, in their research, the authors use a three-point contact model in order to determine the pressure distribution; in our case, we consider a continuous bi-dimensional contact surface, which is incompatible with point contact. Moreover, we consider a rover with four-steerable wheels (or four-steerable rover, for short) as the towing actor; this is subjected to more complicated kinematics constraints, which include steerable wheels and limited steering angles. This fact leads to a reduction in mobility of mobile robots. Finally, in Cheng’s work, a fleet of three robots is used to plan the pose of the towed object, while in our case, we use a single robot. Although using a multirobot system to manipulate an object is useful and in principle allows the object to assume any pose, a single robot towing the object makes the system less controllable and the planning more challenging. Using a single robot mimics the possible scenario in which the planetary rover is the only towing actor available; a space exploration mission using a multirobot system could be excessively expensive and complex to control.

Compared to the state of the art, the contributions of this paper are as follows:

• the implementation of a quasi-static approach for the path planning of an object towed by a single rover and with a continuous contact model;

• the implementation of a dynamics model of the towed object with the same contact model;

• a study of the influence of the kinematics of a four-steerable rover on the towed object trajectory;

• the implementation of a physical model for the experimental validation of the dynamics model in the context of path planning.

Based on the authors’ best knowledge, this is the first work that deals with the path planning of an object towed by a space exploration rover with four-steerable wheels, subjected to upper and lower joints limits.

The article is structured as follows: in Sec. 2, the problem statement is given together with a detailed and formally elegant model, based on a quasi-static (QS) approach for the trajectory generation. The dynamics model is reported together with the Feedforward and Feedback use cases. A kinematic model of the rover is presented, along with a path-planning method for the rover based on ICR planning and orientation look-ahead. In Sec. 3 the main results are reported for the simulated use cases; an in-depth sensitivity analysis of the main hyperparameters of the model is presented. In Sec. 4 the experimental setup and main results are presented of the Archimede rover towing an object; these are then compared with the numerical ones in order to validate the approach. Finally, in Sec. 5, the concluding remarks are reported together with possible future works along this line of research.

The main objective of this study is to understand how an object must be towed in order for it to track a given reference trajectory; above all, what should the trajectory of the towing agent be so that the object follows the reference trajectory? Within this study, we assume that the motion of the towed object is conveniently slow such that the inertial terms can be neglected; therefore the solution can be obtained by exploiting the QS approach [67].

In fact, the output of this QS approach is the generation of the towing agent trajectory. In order to validate the approach, it has to be verified that inertial effects do not cause substantial deviations from the nominal path. For this reason, in the following, we describe two dynamic models: one that is uniquely based on the forces exerted on the towed body as computed by the QS method, and one that considers the trajectory of the towing agent explicitly. The models are then used to validate the approach, by comparing the nominal path of the towed object with the actual trajectories as computed using the dynamic models.

Let us consider the problem in which a generic-shaped object is moving on a flat surface and is subjected to an external towing force. The object is assumed constrained to move along an arbitrary reference trajectory $so(t)$⁠, as shown in 1, which represents the general case of 2D motion on the ground plane. Considering the inertial reference frame (⁠$O,e^x,e^y,e^z$⁠), the triple (⁠$x,y,ϑ$⁠) describes the state of the object, referred to a generic point $P$⁠, in the inertial frame. In this context, it is assumed that $P$ is the point that has to track the trajectory $so$⁠. Let $(P,ı^o,ȷ^o,k^o)$ be a local reference frame, which is integral with the object. The mass distribution of the object, in general, is not uniform; its center of mass is denoted with $G$⁠. The object velocity $v$ is influenced by a towing force $T$ applied to the anchor point $QT$⁠; a frictional force $Fk$ applied to the center of friction $Qk$⁠, resulting from a nonuniform distribution of the friction coefficient along the contact patch; and an applied frictional moment $Mk$⁠.

In particular, 1(a) shows the general free-body diagram of an object towed and moving on a flat frictional surface, while 1(b) shows the relationship between the object and rover trajectories, i.e., an elastic unilateral cable with length at rest $l0$ and stiffness $κ$⁠.

The assumption that all forces (e.g., $T$⁠), points (e.g., $G$⁠, $QT$⁠), and motions (e.g., $v$⁠, $so$⁠) live on the ground plane was made chiefly to obtain a streamlined model for understanding the main actors that drive the kinematics of the system. This assumption, in fact, implies that the towed object is not subject to any kind of toppling moment; therefore, the distribution of its contact force on the ground becomes statically invariant with respect to the towing motion.

In this section, we describe the approach used to generate the object geometry and the modeling of the two main forces exchanged between the flat frictional surface (or ground, for short) and the object, which compose the contact interface. These forces are as follows: the normal contact force derived from the knowledge of the contact pressure distribution and the tangential force, i.e., the frictional force.

In our work, we elected to use penalty methods to model contact interactions. More specifically, regarding the Signorini condition, we allow a small degree of penetration to occur during contact, by essentially considering the ground as an equivalent spring that produces a force normal to the plane. Furthermore, for modeling friction, we implemented a relaxed Coulomb’s law in the form of a continuously differentiable function.

The geometry of the object is described as a polygonal mesh of triangles. In this work, two main geometries are chosen: an unbalanced cube of dimensions 0.4 m per side—mostly used for the sensitivity analysis—and a general geometry having contact surface as shown in Fig. 2(a).

Since the contact patch can have an arbitrary shape, rather than a closed-form solution, we implement a cell-centered finite volume method (FVM) in order to perform the integration over general domains. The FVM will be used when a surface integration is needed such as the distribution of $p(ξ,η)$⁠. As it will be seen later on, this applies to the friction components $Ff$ and $Mf$ as well.

In Sec. 2.1, it has been stated that Eqs. (1) and (2) describe, respectively, the object’s penetration and the contact pressure distribution, while Eqs. (4) and (5) describe, respectively, the friction force and the friction moments vectors. However, at this stage still, it is not clear how it is possible to determine the object penetration (thus the pressure distribution) since as already said it depends on many factors. In order to be able to compute the pressure distribution, in this section, the quasi-static model (QSM) is presented, which neglects the contribution of the inertial forces.

As shown in the previous paragraphs, the pressure distribution depends also on the external forces, which are variable. Thus, the pressure distribution needs to be recomputed continuously. In order to avoid this problem, we introduced the following assumptions, which allow to compute the pressure distribution only once, at initialization:

• The towing force is parallel to the contact patch.

• There is zero vertical offset of the towing force.

It must be pointed out that once the plane has been found, it is not necessarily the right one. This is because, as already stated, the ground reacts only to compression and not to traction. When an eccentric force is applied, it may happen that local portions of the ground are subject to traction loads, which does not happen in reality. In fact, with very eccentric vertical forces, only a small contact patch experiences a ground reaction pressure. In order to tackle this problem, an iterative process is necessary to determine the plane. Starting from the original support surface $S$⁠, the area in which the ground experiences traction is subtracted from this surface at each iteration. A new support surface $S′$ and the new mesh are therefore generated. This process keeps going until convergence of the residuals “res” is reached, where $res=‖A′x−B‖$⁠, $A′$ being the matrix $A$ computed on $S′$⁠. The process of iteratively excluding the contact patch for computation is visible in 2, where Figs. 2(a) and 2(b) show, respectively, the starting original mesh and the reduced mesh at convergence. On the other hand, Fig. 2(c) shows the convergence of the residuals and of the normalized plane coefficients. These preliminary results refer to the case of a very eccentric vertical load acting on a generally shaped object.

Once the correct plane is found, hence the contact pressure distribution, it is possible to compute the friction force and moment vectors. Again by using the QSM approach applied to the translational equations along $ı^o$ and $ȷ^o$ axis, it is possible to finally compute the required towing force $T$⁠, necessary for the point $P$ to track the given trajectory $so$⁠.

It is important to point out that the modeling of the contact surface through a mesh allows the description of a generally shaped object; furthermore, together with the modeling of the coefficient of friction and through the normalization of Eq. (3), it allows straightforward implementation of a general Coulomb friction model; this can, in turn, account for anisotropic friction, i.e., $μ(ξ,η,v)$⁠.

The task of towing the object is cast on the Archimede rover, a robotic platform developed at the University of Trieste. The rover is equipped with four independently driven and steerable wheels mounted on semi-rigid arms [71]. Full details are given by Caruso et al. in Ref. [69]. The quasi-static approach given in Sec. 2.2 provides the nominal rover trajectory $sr(t)$ starting from the object desired trajectory $so(t)$⁠. However, the constraints of the kinematics of the rover inevitably cause the real trajectory to deviate from the nominal one. In order to evaluate this discrepancy, we propose a nonholonomic model for the rover that accounts for the full kinematics, notably with the steering angle limits for each wheel. The dynamics of the rover is purposefully left out at this stage of the research.

In the following, we refer to the geometry and definitions shown in Figs. 1 and in 3. From these figures, we can immediately see the interaction between the towed object and the rover and the main geometry of the rover.

In analogy with the QSM approach defined in Sec. 2.2, the rover connection with the towed object is modeled as a unilateral elastic constraint (traction only) with stiffness $κ$ and length at rest $l0$⁠. The rover control system regulates the steering such that the point $Pr$ tracks the trajectory $sr(t)$⁠, thus maneuvering the object through the elastic constraint, for it to follow its own prescribed trajectory $so(t)$⁠.

Assuming we have a bounded span for the steering angles $ρi$ with $i=1,…,4$⁠, this leads one to observe that the location of feasible ICRs is a subset $SΩr∈R2,$ which is ultimately the intersection between the sliding constraints of each wheel obtained by sweeping its allowed $ρi$ span. A partial representation is shown in Fig. 3 in light-cyan. Indeed, the full $SΩr$ is made up of other smaller areas—e.g., under the belly of the rover—but for the purposes of this study, only the large main triangular areas at the flanks of the rover are considered. During operations, the ICR can move continuously within each of these areas, allowing for seamless nonholonomic motion of the vehicle.

These limitations concur in the emergence of steering-derived motion constraints which inhibit the overall motion of the robot.

In the following, we will separate the path planning—intended as purely positional—from the planning of the orientation of the vehicle.

However, since $SΩr$ is bounded, it can happen that $Ωr∉SΩr$⁠; in that case, the ICR is not compatible with motion and a new compatible $Ωr*$ has to be determined. This situation is shown in 3(d). As shown in Figs. 3(c) and 3(d), we can determine the following cases:

• $Ωr∈SΩr$⁠; in this case, $Ωr*=Ωr$⁠;

• $Ωr∉SΩr$ and $∃Ωr$ close to the boundary of $SΩr$⁠; in this case, $Ωr*$ is chosen as the closest point $∈SΩr$ to $Ωr$⁠;

• $Ωr∉SΩr$ and $∄Ωr$ close to the boundary of $SΩr$⁠; in this case, $Ωr*$ is chosen as the tip of the $SΩr$ area.

While case A guarantees that the vehicle reaches point $Pr(t1)$ exactly, the other cases do not. However, if $Ωr*$ is sufficiently close to $Ωr$⁠, the vehicle is able to realign after a certain number of time-steps $Δt$⁠.

Finally, when the vehicle is closer than a threshold $dt$ to $Pr(t1)$⁠, the next point $sr(t2)$ of the path is selected with time-step $t2=t1+ts$⁠. Based on these considerations, with reference to 3(c), the vehicle follows the effective trajectory, which, for sufficiently small $Δt$ deviates negligibly from the nominal one.

In order to make sure that the simulation replicates the behavior of the vehicle, we have implemented a proportional-derivative (PD) controller for the steering of the bicycle model described in Sec. 2.4.1. As such, the controller operates on the two-steerable wheels model instead of directly on the real four-wheel model. A four-wheel PD steering controller would cause steering misalignments, thus rendering the model nonsolvable using a nonholonomic formulation. The values of the gains are $Kp=2$ and $Kd=0.05$ for both wheels.

In this section, we report the results of the numerical simulations of the object towed by a rover. The results about the path planning and the QSM exclusively are reported first. Subsequently, the results of the sensitivity analysis performed on the dynamics simulations are detailed. Furthermore, an in-depth discussion is presented in order to highlight the main aspects of the models behavior.

This paragraph regards only the QSM results for the path-planning purpose, while the dynamics is considered in the next paragraph. The trajectories considered in our case study are a 1.4 m long diagonal line, a circle with a radius of 1 m, a 2 m wide lemniscate, and a square with rounded corners with sides of 4 m. The geometry of the towed object is chosen to be a general one and is shown in Fig. 2(a). Moreover, for each trajectory, three different center of mass locations have been chosen: $G1=[0.05,0.15]$ m, $G2=[0.26,0.15]$ m, $G3=$$[0.10,0.25]$ m using the object reference frame of Fig. 2(a). This choice allows us to exploit the general formulation of the model regarding continuous contact pressure distribution.

Simulations have been conducted using the QSM for the synthesis of the trajectories, such that these complete in $Tend=10$ min with a simulation time-step $ts=1×10−2$ s. These parameters have been chosen in order to account for the QSM approach.

Assuming that the motion is so slow as to be considered quasi-static, the nominal trajectories of the objects and the towing rover are reported in 4 for all the trajectory cases. In particular, the first row in Figs. 4(a)–4(d) illustrates the quasi-static solutions with the pose of the towed object, for, respectively, the line, circle, lemniscate, and rounded corner square trajectories. Figures 4(e)–4(h), instead, show the nominal trajectory of the object (black curve) and the nominal trajectories of the towing rover for each of the gravity application points $G1$⁠, $G2,$ and $G3$ for, respectively, the line, the circle, the lemniscate, and the rounded corner square trajectory. These trajectories are computed using the QSM approach detailed in Sec. 2.2. It is apparent that a nonbarycentric application point influences the path the rover takes to correctly manipulate the object, as expected. However, it seems that no clear indication of the path of the rover can be easily inferred simply from the location of $G$⁠.

In order to highlight the influence of the ICR projection method described in Sec. 2, we point to Fig. 5. In the figure, a lemniscate curve is chosen as the object path, similar to that shown in Fig. 4(e). Both the trajectories of the rover (Fig. 4(a)) and of the object (Fig. 4(b)) are shown; it should be said that this specific choice of path and planning parameters greatly stresses the accuracy of the methodology, which is in this case useful to show how the planner behaves during the process. In particular, we can see how it transitions between cases A, B, and C several times, all of which are highlighted with different colors. As expected, the planner stays in case A for the majority of the time, switching in case B or C in the most challenging segments of the path. These events are reflected in Fig. 5(b), which illustrates the real trajectory of the towed object with respect to the nominal lemniscate curve. It should be noted that there is correspondence between the planner cases between the two plots (Figs. 5(a) and 5(b)).

In this section, we report the results obtained by performing the sensitivity analysis. The aim is to characterize the influence of a subset of the parameters, especially for the FFD and FBD dynamics cases. We have elected to focus on the lemniscate and the rounded corners square. A cube with an eccentric load is used as a sample geometry for the towed object. The parameters considered in the analysis are as follows: the mean velocity of the object along the trajectory $v¯$⁠, the simulation time-step $ts$⁠, and the stiffness of the cable $κ$⁠. The parameter chosen as a metric for the sensitivity analysis is the root-mean-square (RMS) error computed between the nominal trajectory obtained through the quasi-static model and the trajectory obtained via dynamic simulation, for both the FFD and FBD cases.

The mesh used for the contact patch of the cube has a maximum element size of 0.0 m with 1322 elements. In the following, this configuration was used for all the other sensitivity analyses. However, the influence of the mesh on the computation time is noticeable, as shown in Table 1.

Taking as reference Fig. 6, we can see that Fig. 6(a) shows that the RMS decreases substantially with smaller average velocities in the case of the FFD. Instead, for the FDB case, the value remains low and roughly constant and seems not to have much influence for a large part of the considered interval of velocities. In fact, the value increases only at values of $v¯$ higher than 5 $×10−2$ m s$−1$⁠. Taking into consideration the time-step $ts$⁠, and fixing the average velocity to $v¯=5.1×10−3$ m s$−1$⁠, from Fig. 6(b), we can see that no substantial variation occurs in the RMS value. Moreover, the discrepancy of an order of magnitude between the curves can be traced back to the difference in RMS between the FFD and FBD cases, which is noticeable in Fig. 6(a). Figures 6(d) and 6(e) show the differences between the simulated object trajectories with FFD and the nominal ones for three different velocities $v¯$⁠. From both figures, it can be seen that the dynamic curves exhibit larger drift as the speed increases, with respect to the nominal one. In fact, when the velocities take reasonable values, the curves tend to the nominal one.

It is apparent from these results that the velocity $v¯$ considerably influences the accuracy of the simulation in the FFD case. This is likely due to the intrinsic limitations of the QSM model whose underlying assumption is that dynamic effects may be neglected; indeed, the method is accurate only for arbitrarily low speeds.

Furthermore, the FFD simulation tends to be unstable in the starting phase. This is likely because at $t=0,$ there exists a transient that is not captured by the QSM, which is related to the object laying still and then immediately made to accelerate. This difference in velocity cannot be modeled by the QSM because of the quasi-static nature of the method itself, where speed is supposed stable and arbitrarily close to zero. In essence, the FFD method requires that no acceleration is applied to the object at start-up. In order to carry out the simulations, we elected to set the object’s initial velocity to match the nominal tracking velocity. This removes the problem.

On the other hand, it seems that FBD trajectories are not influenced by velocity. This is possibly due to the model used to describe the system, in which the spring acts as a means to control and correct the position of the object. Indeed, in the FFD case, the forces are fed to the system blindly, which tends to exacerbate deviations. On the basis of these considerations, in Fig. 6(c), we show the influence of the spring stiffness $κ$ on both the RMS and the computation time $tc$ for the FBD dynamics simulations case. As expected, both quantities appear greatly affected by $κ$⁠, with larger values leading to smaller RMS, but higher computational time. This indicates that a certain numerical stiffness exists in the model.

A similar approach is applied to the case where the motion of the rover is not omnidirectional, but rather governed by the kinematics defined in Sec. 2.4. We have hypothesized in Secs. 2.4.3 and 2.4.4 that both the planning of the positional path and of the orientation are required for a successful path-following. In 7, we show how the towed trajectories are affected by the parameter $dt$⁠. In particular, from the second series of plots, it is clear for most cases how values between 0.1 m and 0.4 m give the best results. Notably, the case of the lemniscate curve shows a more unpredictable behavior. This is likely due to the fact that the controller is unable to adequately operate with such small curvatures. In fact, the trajectories associated with $dt=0.227$ m and $0.103$ m completely “break” the controller; the large deviations are due to the fact that the ICR that are determined are very far from optimal and cause the rover to stray by a very large margin from more sensible paths.

With reference to 8, we can see how these path-planning parameters influence the RMS for the lemniscate and the rounded square paths. The feasible area of the former appears smaller, more jagged, and in general with higher RMS values compared to the latter, as expected. In general, $ϑlh$ values of around 300 and 400 steps and values of $dt$ between 0 and 1.5 m seem to give feasible results.

In this section, the experimental setup and campaign are presented, with emphasis on the results; specifically, the Archimede rover was used to drag an object along random paths. The aim is the validation of the dynamics model described in Sec. 2.3. The friction coefficient between the object contact surface and the pavement has been estimated to be $μ=0.34∼0.36$ via direct experimentation. The experimental setup is shown in 9 where two shots are shown of the rover towing the object. The eccentric load is obtained by positioning a small weight in the rear corner of the rectangular object. The rover is controlled using the ROS (robot operating system) ecosystem. Data acquisition is performed via a SONY RX100 VII camera positioned on the ceiling. After calibration of the video feed, tracking is executed with a Kanade–Lucas–Tomasi method [72] implemented in matlab.

The tracked trajectory—after some preprocessing to decimate densely packed point clusters—is then used as input data for the FBD simulation, without further constraints. The trajectory of the object geometric center in the FBD simulation is then compared with its recreated trajectory in the experimental setup. The comparison of the two curves can be seen in 10, where a remarkable adherence can be seen between the numerical simulation and the experiments. Comparatively small deviations are visible between the curves and are likely due to discontinuities in the ground, and the elasticity of the simulated cable.

In this work, the potential of a rover to manipulate through a tether, an object that is laying on the ground, was explored. A formal and elegant methodology has been proposed to model the contact pressure distribution. A QSM approach has been used in order to synthesize both object and rover trajectories; the kinematics and joint limits of the rover as the towing actor have been taken into account. Dynamics simulations emphasize the influence of the main hyperparameters of the models, as detailed in the comprehensive sensitivity analysis. A short list of trajectories has been selected, and the analysis of the path-following performance has provided insight on the path planner flexibility. An experimental validation was presented of the full-size Archimede rover, which is equipped with steerable wheels; ad hoc image tracking software was used to analyze the motion of the rover and of the towed object, in order to validate the dynamics, friction, and contact models. Results show remarkable adherence of the models to reality.

The approach implemented in this work allowed us to model the behavior of a rover towing an object of arbitrary shape and with a nonconstant contact pressure distribution over a hard surface. We show that this kind of modeling can be exploited successfully to perform path-planning for the towed object; moreover, we illustrate how the approach is influenced by the many parameters and hyperparameters of the models, and how this influences the accuracy with which the towed object follows the planned path.

Future works foresee the replacement of the unilateral elastic model of the cable with more sophisticated models, e.g., a catenary-based model or a continuous flexible element. Moreover, the towing point will implement a nonnull vertical offset. On the other hand, the possibility of towing an object on soft and granular soil will be investigated in both simulated environments as well as experimentally.

• PRIN 2017 project “SEDUCE” (Grant No. 2017TWRCNB).

• “Microgrants 2020,” Internal Funding Program of the University of Trieste.

• Fondo Sociale Europeo (FSE) funding for “Dottorati di Ricerca 35$∘$ ciclo/P.S.89bis19—budget ricerca 10%”, “Regione Autonoma Friuli Venezia Giulia.”

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.