Automation in steep terrain agriculture: an optimal controller to prevent tipping and slipping of tethered robots on slopes

Autonomous robots have the potential to fundamentally transform conventional farming methods, e.g. by enabling economically viable farming of sloped arable land. However, navigation on slopes in harsh conditions is challenging for robots as they must be prevented from both slipping and tipping. Tethers provide one means of enabling robots to traverse steep and complex terrains with additional advantages of recoverability and connectivity for data and power transfer. Current controllers (for robot and tether) consider only the slipping condition for such robots, meaning that the tether must be mounted low on the robot. This limits the use case and can lead to entanglements and entrapment of the tether. By introducing stability criteria for tethered robots on slopes and developing a novel controller that seeks to avoid both slipping and tipping, we demonstrate how the tether mount height can be raised from 0.27 to 0.81 m on a robot, to enable, for example, the tether to be used above high crops or obstacles. This controller is demonstrated in lab and field conditions on slopes up to 0.4 radians, which translates to approximately 51% of slope. Thereby, the new controller significantly extends the practical usability of tethered robots for agricultural applications on steep slopes. GRAPHICAL ABSTRACT


Introduction
The growing global population is leading to a requirement for increasingly efficient and effective food production. However, arable land is becoming increasingly scarce and farming faces a multitude of challenges including dwindling labor forces, climate change and economic uncertainty [1]. In many regions, crops are farmed on slopes, e.g. rice fields in Vietnam, banana plantains in the Andes or vineyards in Switzerland [2,3]. For some crops such as grapes, slope grown produce is said to be of higher quality [4]. However, typically crops farmed on slopes require a lot of manual work and pose severe risk for farm workers [5]. This can render farming slopes economically unviable or reliant on pesticides. This is limiting the use of these otherwise viable and extensive natural resources. Although there are increasing examples of research and commercial robotic platforms for agricultural environments enabling weeding [6], planting or harvesting [7], they have strict limitations on slope inclinations rendering them unsuited for steep regions ( Figure 1) .
Developing robots that can robustly navigate crops in steep terrains could have significant potential for food production. It could enable an increase in the area of CONTACT Max Polzin max.polzin@epfl.ch viable agricultural land without requiring the use of pesticides, or putting workers in danger. Tether systems have been shown to be an effective means of assisting the motion of a robot on slopes in challenging environments whilst also providing a means of safe recovery, communication and power [8].
In general, tether systems have the potential to increase the workspace of robots with a wide range of geometries, e.g. vehicles, crawlers or legged platforms.
The tethers of tethered robots have to be actively managed to support and not hinder a robot's motion. Controllers have been developed for tethers that seek to minimize the slip of ground robots on complex terrains [14]. These controllers are designed for low tether mount heights that are close to the ground [9]. However, this is not feasible for agricultural scenarios, where the tether must be higher than the surrounding crops and associated infrastructure. In addition, low mounted tethers are prone to damage and entanglement [15]. Finally, the tendency of low tethers to form By developing a new active tether controller, the tether can be mounted at significant heights above the robot. This enables the tether to avoid obstacles, or, as in the case shown, move above the crop height enabling the robot to move through crop rows.
Whilst raising the tether height opens up the suitability of ground robots in agricultural scenarios, any tether which pulls on the robot induces a risk of tipping. While a robot can recover from unwanted slip, normally robots cannot recover once tipped over. Thus, tipping can be considered the more critical failure which must be prevented by a tether controller. Although existing tether controllers consider the slip of the robot [14], they do not consider this tipping effect. Thus, new approaches to tether control are required.
By analyzing the stability of tethered robots on slopes for variable mounting heights of the tether, we propose the development of a controller that optimizes the stability with respect to both slipping and tipping. A novel tether control scheme for tethered robots is proposed which in comparison to prior work not only takes the robot's stability regarding slipping into account, but also the robot's tipping stability.
The proposed tether controller enables the development of tethered robots with variable geometry and significant tether heights increasing their applicability in new domains. The applicability of the novel tether controller is demonstrated on a tethered robot in experiments in a controlled indoor environment and outdoor experiments in viticulture. In doing so, this work makes a number of contributions: • Analysis of the stability and risk of tipping of tethered robots on slopes and formulation of an optimization problem to minimize the risk of tipping • Novel tether controller to control the tether tension which takes the robot geometry and tether mount height into account • Implementation of the proposed tether controller on a robot with a variable tether mount point and demonstration and benchmarking of the applicability of the proposed tether controller in indoor and outdoor environments While the proposed system is intended to allow agriculture on sloping land, the results generalize to other domains such as extreme terrain navigation or extraterrestrial exploration. This paper is structured as follows. Section 2 analyses the stability of tethered robots on slopes, formulates a tether control strategy and introduces a tethered robot. Section 3 details experimental results from indoor and outdoor experiments. Section 4 provides concluding remarks.

Methodology
In this section the required terminology to describe the scenario of focus, tethered robots with variable tether heights moving on slopes, is first introduced. The forces acting on a robot on a slope are then analyzed to derive stability criteria for tethered robots on slopes. This analysis allows the formulation of conditions where a tether can prevent a robot from both slipping and tipping. Based on the introduced stability criteria, an optimal tether controller, which allows navigation on steep terrain, is proposed.

Scenario description
We consider a four wheel drive, skid-steered robot of width, w, length, l, and height, h, driving on a slope, where the inclination, α ∈ (0, π/2], is defined with respect to the force due to gravity, F g . The robot's center of mass can therefore be given at: A tether is attached at height, h t , in the middle of the robot, { l 2 , w 2 }. The other end of the tether is connected to a fixed point, i.e. an anchor point, further up the slope at the same height, h t . The tether is assumed to be taut and thus runs approximately parallel to the slope. The robot and acting forces are depicted in Figure 2. Note, the derived equations could generalize to other robot morphologies.
The gravitational force, F g , acting on the robot on a slope can be resolved into a parallel, inplane force, F g , and a perpendicular component, F g⊥ : The tether length can be controlled by a winch, which is either mounted to the robot or placed at the anchor point. The tether exerts a force on the robot, F t . The tether force, F t , satisfies F t ≥ 0kgf and can be divided in two components: a force pointing in the opposite direction of F g and a perpendicular one: where the arm-to-gravity angle, β ∈ [−π, π ], is defined as the angle between F t and F t . The heading angle, φ, of the robot is defined as the angle between the heading direction of the robot and F t . There are no constraints on the robot's orientation on the plane and thus φ ∈ [−π , π].

Stability analysis
In the following, two failure cases which a robot can experience are analyzed, namely slipping and tipping. Slip occurs when the robot is no longer capable of moving towards a commanded direction, e.g. its wheels do not have sufficient traction on the surface and spin in place or the robot starts sliding sideways. A robot is considered to tip when its center of mass lies outside of its support polygon. Once a robot has tipped over, it typically is impossible to further control and recover the robot.

Slip
A detailed analysis to avoid slip of tethered robots driving on slopes is provided in [14]. Generally, F t and F g are not aligned, which implies a net inplane force, Slip occurs when the wheel traction F f = μF g cos(α) > F r , where F f can be modeled as Coulomb friction with terrain dependent friction coefficient, μ. Hence, slip is avoided for tether tensions, Note, the maneuverable area, i.e. the area where the tether can prevent the robot from slipping, is constrained by tan α sin β = μ and only depends on the terrain dependent friction coefficient, μ, the arm-to-gravity angle, β, and the inclination, α, but not the robot's geometry.

Tipping
The tether and robot's gravitational forces, F t and F g , create a moment which is countered by normal force, F n . The normal force, F nψ , before the robot tips in direction, ψ ∈ [−π , π ], depends on the robots heading angle on the slope, φ, its geometry {w, l, h g , h t }, the slope inclination, α, and the orientation of the tether lead out, i.e. the arm-to-gravity angle, β. The forces acting on the robot in ψ-direction are visualized in Figure 3. The robot is stable for all heading angles, φ, when the moment induced by the tether force, F t , does not result in a negative normal force, F nψ , for tipping direction, ψ.
Depending on the robot's footprint, {w, l}, the robot is more susceptible to tipping forward/ backward (if l < w) or sideways (if w < l). In the following, the sum of moments for angles is analyzed at the shortest distance from the robot's center, The inplane forces, F g and F t , have parallel components in tipping direction, ψ, For tipping angle, ψ, the inplane gravitational force, F g ψ , and tether force, F tψ , create a moment which is countered by the normal force, F nψ , and the perpendicular gravitational force, F g⊥ , at distance, d, from the robot center, The robot tips in direction, ψ, if F nψ < 0kgf, i.e. the robot tips towards ψ for tether tensions, The robot drives stable on a slope, if it does not tip in any direction, ψ. Hence, the robot drives stable and does not tip for tether tensions, F t ≤ F t,max , where Figure 4 visualizes the maximum applicable tether force, F t,max , before a robot with tether mount height, h t , tips  A robot tips if tether force, F t , exceeds the tipping force, F t,tip , whereby F t,tip is a function of inclination, α, arm-to-gravity angle, β, tether mount height, h t , the robot's mass and geometry. The marker indicates the specific case of α = π/8 and β = π/8 for which the influence of h t on F t,tip is visualized in Figure 5. on slopes with inclinations, α, for arm-to-gravity angles, β. Figure 5 shows how F t,max varies with tether height, h t .

Tether controller
Based on the above stability criteria for tethered robots on slopes, we propose a novel tether controller with an optimal reference tension. Compared to the controller of [14] which use a reference tension to minimize the risk of slipping, the proposed controller simultaneously minimizes the risk of slipping and tipping. Figure 6 visualizes the operational constraints of a tethered robot, the proposed controller and the one of [14] for a representative scenario, i.e. a slope with inclination, α ≈ 0.4rad, and friction coefficient, μ ≈ 0.24, whereby α is a representative inclination for agricultural slopes [18], and μ is a representative friction coefficient for an agricultural soil [19].
According to (5) the robot slips downhill for tether tensions, (12) and uphill for tether tensions, F t > F + t,slip , where The robot is prone to tipping for tether tensions, F t > F t,max . For robots which tip when being pulled uphill, i.e. Figure 6. F t minimizes the net inplane force as in [14] and can cause tipping of the robot. The proposed controller, F * t , simultaneously minimizes the risk of slipping and tipping. The robot tips for F t > F t,tip and slips for F t < F − t,slip or F + t,slip < F t .
F t,max < F + t,slip , we propose the optimal reference tension, which minimizes both risks where possible, as illustrated in Figure 6, as input for a tether tension controller. Tipping is considered a worse failure than slipping. Thus, if a failure cannot be prevented, the proposed controller relaxes the constraint on not slipping and prevents the robot from tipping. Note, the reference tension from [14] can exceed the robot's tipping point and thus lead to tipping of the robot. Beyond a certain arm-to-gravity angle, β max , slip is unavoidable for the robot, regardless of the applied tether tension. If β exceeds β max , the controller minimizes the net inplane force, F r . This optimal reference tether tension, F * t , is applied as a setpoint to a tether tension controller, a PID-controller which issues control inputs according to to a motor controller, whereF t is the measured tether tension. The motor controller is controlling the reeling velocity of the winch. The robot control scheme is visualized in Figure 7. Note, that trajectory commands are solely issued to the wheeled robot, i.e. the tether controller operates independently of the robot's driving speed and direction.

Experimental setup
The proposed tether controller has been deployed on a tethered robot with variable tether mount height. The robot is a four-wheeled, skid-steered uncrewed ground vehicle with length, l = 0.4 m, width, w = 0.4 m and height, h = 0.27 m, and mass, m = 13 kg. The robot is equipped with two brushless DC motors whereby each motor directly drives the wheels on one side. Both motors are controlled by an electronic speed controller which provides up to 50A per motor. The tether can be mounted to the robot at heights between 0.27 and 1.13 m. The other end of the tether is connected to an equally high anchor point. The tether is actuated by a winch, which is located at the anchor point. The winch is powered by a DC motor whereby the motor's output is reduced to provide enough torque to support the full weight of the robot. The tether tension,F t , is measured by a load cell located at the anchor point. The arm-to-gravity angle, β, is measured with an encoder at the lead out on the robot. An Inertial-Measurement-Unit (IMU) is used to measure the inclination, α, and heading angle, φ, of the robot. Wheel odometry is obtained by integrating encoder measurements of the robot's wheels. A stereo camera, paired with the IMU, is used to obtain visual-inertial odometry, which serves as ground truth for the robot's pose in our experiments. Control, sensor and user interfaces are implemented in ROS2 [20] on an Nvidia Jetson NX Xavier development board. The optimal tether reference tension and control commands are computed and executed at 50 Hz. The used stereo camera is a Stereolabs ZED 2i with a builtin IMU. Visual odometry measurements are computed using Stereolab's ZED SDK. In experimental tests on flat ground, wheel and visual odometry coincide. If the robot slips, wheel and visual-inertial odometry diverge. If the robot tips, the IMU measures sudden changes in the robot's pitch or roll angle. Figure 8 shows the indoor experimental test bed. The indoor test bed is a plane with inclinations, α ∈ {0.35, 0.4} radians, and static friction coefficient, μ ≈ 0.24. Note, μ was determined according to [21]. Without tether, the robot slides down the test slope when driving. The robot was further tested on an outdoor slope in Lausanne, Switzerland. The outdoor slope is shown in Figure 1.

Experimental results
First, the operational constraints, i.e. the stability criteria for tipping and slipping, from Section 2.2.2 are validated on an indoor test bed. The proposed tether controller achieves comparable performance to state-ofthe-art tether controllers for tethered robots with a low tether mount point. Further, the proposed tether controller enables the tethered robot to drive on a slope with a high tether mount, in contrast to state-of-the-art tether controllers. Finally, the applicability of the proposed tether controller to robots with variable tether mount heights is demonstrated in outdoor experiments in an agricultural environment.

Slipping
To validate the operational constraints where a tethered robot does not slip, the robot was placed at the bottom of our test bed where β ≈ 0.09 rad. The tether tension is set to a fixed force, F t,fix and the robot is commanded to drive up the slope such that β gradually increases, see driven trajectories in Figure 9. Slip occurs if the sum of the fixed tether tension F t,fix and F g, exceed the robot's traction, F f . Figure 9 shows the predicted arm-to-gravity angles,  β, where slip can be avoided and the measured arm-togravity angles,β, where the robot started to slip. While in theory, the tether can prevent the robot from slipping on our indoor test bed with a static friction coefficient, μ ≈ 0.24, and inclination, α = 0.4 rad, up to β max ≈ 0.55 rad when applying a tether tension, F t = 5 kgf, in practice the robot starts slipping before, due to inaccuracy of estimated friction coefficient,μ, and limited bandwidth of the reference tension tracking controller.

Tipping
The most likely tipping direction of a tethered robot on a slope depends on its geometry, its arm-to-gravity angle, β, the inclination, α, and the tether tension. The presented wheeled robot most likely tips forward or backward, since l < w. To validate the operational constraints on tipping for different tether mount heights, h t , the robot's front wheels are fixed on a test slope. Then, the tension on the tether is increased in direction, β, until the robot started to tip. Figure 10 shows the measured and predicted tipping force, F t,tip , for the presented robot at different arm-togravity angles, β, and inclinations, α. On level ground, a lower force is required to make the robot tip forward, respectively backward, than sideways. Thus, the robot is more likely to tip forward, respectively backward, on slopes and Figure 10 only shows the tipping force, F t,tip , for the robot tipping forward, respectively backward. Further, Figure 10 shows that Equation (11) can predict the tether force at which the robot tips. Finally, the experiments confirm that the maximum tension before a tethered robot starts to tip on a slope increases inverse to the tether mount height. Figure 10. The left plot shows the required tether force to tip the robot forward or sideways for three tether heights, h t1 = 0.27 m, h t2 = 0.8 m, h t3 = 1.13 m, on level ground. The two right plots show the maximum tether force before the robot tips on slopes with α = π/9 and α = π/4 for h t1 , h t2 , h t3 , and varying arm-togravity angles, β. The lines indicate predicted and the markers measured forward tipping force.

Low tether mount
The proposed tether controller minimizes the risk of tipping without compromising the prevention of slip. For low tether mounts, the proposed tether controller is expected to perform equally well as state-of-the-art controllers which do not take the robot's geometry into account. To verify matching performance, the robot with a low tether mount, h t = 0.27 m, is placed at the bottom of the indoor slope with α = 0.4 rad, at a small armto-gravity angle, β ≈ 0.09 rad. The robot is commanded to drive up the slope in a straight line until it gets too close to the anchor point to drive further. It starts to slip when β = β max . Figure 11 shows the driven trajectories, the applied tether tension and measured trajectory errors over multiple experiments. The trajectory error at time, t, is calculated as euclidean distance between the position of the robot estimated by wheel odometry, p w := (x w , y w ), and visual odometry, p v := (x v , y v ). Note, for visualization purposes, the trajectories of the controller of [14] are mirrored in Figure 11. During the experiments the robot started driving at the same position, x = 0.4. The trajectory errors are small and comparable for both controllers until β = β max . At β = β max , the error rapidly increases, since slip can no longer be prevented and the robot slides towards the anchor point for both controllers. Note, the proposed controller applies a lower reference tether tension throughout the trial runs as expected.

High tether mount
For robots with a high tether mount, the proposed tether controller prevents the robot from tipping. The robot with a tether mounted at h t = 0.8 m, is placed at the bottom, center of the test bed with an inclination of α = 0.35 rad. It is then commanded to drive sideways. The robot stops driving if it either tips, reaches β max and starts slipping or reaches the end of the test bed. Figure 12 shows the driven trajectories for multiple runs with the proposed tether controller and for two exemplary runs with the controller of [14]. Note, only two trajectories are shown since the robot tips after a certain distance in each trial for the controller of [14]. For the proposed controller, the average recorded trajectory is shown and two standard deviations. Controller [14] applies a higher tether force than the proposed controller. It does not take into account the robot's geometry and leads to tipping of the robot. Generally, driving sideways on a slope is more challenging than driving up or down and the trajectory errors are higher than for a tethered robot driving up or down a slope.
When driving sideways, the pulling direction of the tether and robot's driving direction differ most. Hence, errors from the delayed response of the tether controller, i.e. if the tether controller cannot reach the reference tension fast enough, accumulate the fastest. This leads to the robot slipping down and deviating from its commanded path. Note, these errors are linked to limitations of the implemented hardware and not to theoretical limitations.

Outdoor experiments / general applicability
The proposed tether controller enables stable operation of novel tethered robots with high tether mounts on slopes. This opens a new domain of applications for tethered robots, e.g. farming crops on slopes. The feasibility of the proposed tether controller is demonstrated in outdoor experiments on a test field in Lausanne, Switzerland. The test field is located on a slope with inclination α ≈ 0.35 rad and estimated friction coefficient, μ = 0.24. Crops are planted in parallel rows, see Figure 1. Three robot configurations were evaluated regarding their suitability for farming crops on the test slope: an untethered configuration, a configuration with a low tether mount and a configuration with a high tether mount. In each scenario the robot starts driving at the top of the slope, Figure 12. The left plot shows the trajectories of a robot with a high tether mount, h t = 0.8 m, driving sideways with the proposed and the tether controller of [14]. The top right plot shows the predicted maximum tipping force and the applied reference tensions. The bottom right plot shows trajectory errors. Only two trajectories for the controller of [14] are shown; the robot tips in each experiment after driving 0.4 m. as indicated by the white dot in Figure 1. The robot is then commanded to drive down a crop row in line with the anchor point, i.e. β ≈ 0rad, turn 90 degrees right, traverse towards the next row, turn 90 degrees right again and drive back up. Figure 13 shows the approximately driven trajectories for each scenario. The robot with no tether cannot navigate on the test slope and tips after driving one meter down. It is impossible for the robot to recover from tipping. The robot with a low tether mount successfully drives down the slope and starts traversing. However, the low tether interferes with the crops and prevents the robot from traversing to the next row. The robot with a high tether mount successfully drives down the slope, traverses to the next row and drives up in the neighboring row. Due to the low traction of the wheels on the dry soil, slip is inevitable while traversing between rows.

Discussion
The presented tether controller optimizes the stability of tethered robots on slopes with respect to both slipping and tipping. It takes into account the robot's geometry and the tether mount height. By introducing criteria to prevent tipping into the tether controller, the height of the tether can be raised significantly, enabling use in more scenarios, and also preventing a tethered robot from undergoing catastrophic tipping failures. The proposed tether controller was implemented on a robot with a variable tether mount point and demonstrated in indoor and outdoor experiments. The results showed that the controller improved the robot's stability and performance compared to a tether controller that only minimizes the risk of slipping and an untethered scenario. Going forward, a possible extension of this work is to incorporate a feedforward component in the tether controller, as suggested by [14]. This would reduce the required bandwidth for a feedback tether controller and enhance the robustness to disturbances and uncertainties. Another direction for future work is to deploy the presented tether controller to other designs and structures of tethered robots, such as tethered quadrupeds or crawlers. This would increase the applicability of the proposed controller to further domains.

Conclusion
In this paper, we proposed an optimal tether controller to prevent the tipping and slipping of tethered robots on slopes. We analyzed the stability of tethered robots on slopes and formulated an optimization problem to minimize the risk of tipping. The proposed approach enables the development of tethered robots with variable geometries and significant tether heights that can operate and navigate on steep slopes. The proposed approach has potential applications in steep terrain agriculture and could be expanded to other domains where tethered robots have been advantageous over untethered systems. The results of this work are relevant for advancing the state-of-the-art in tethered robotics and providing novel solutions for agricultural automation. In future work, we plan to implement the proposed controllers on novel robot platforms, to verify the generality of the proposed stability criteria, and to validate their applicability in large-scale outdoor experiments on an experimental farm.

Disclosure statement
No potential conflict of interest was reported by the author(s).

Funding
This work was enabled by a CROSS -Collaborative Research on Science and Society EPFL/UNIL -Switzerland grant.

Notes on contributors
Max Polzin is a doctoral student at the Swiss Federal Institute of Technology Lausanne (EPFL) in Switzerland. He is affiliated with the Computational Robot Design & Fabrication Lab, focusing on the development of innovative mobile robots for challenging environments such as forests, mountains, and glaciers. With over eight years of experience in robotics research and development, Max has worked in both industrial and academic settings. He holds a master's degree in robotics, systems, and control from the Swiss Federal Institute of Technology Zurich (ETHZ) and has contributed to several publications in the field of field robotics. Max is driven by his passion to advance the capabilities of todays robots and making robot development more accessible to a broader audience.