Agua que da vida

Slip on deformable terrain is the consequence of the soil yielding due to excessive shear stresses, leading to a lower robot displacement than that expected from motor odometry. On wheels, a moderate level of slip usually improves the draw-bar pull on frictional soils, but excessive slip can seriously hinder motion, increase sinkage and even immobilise the vehicle. Slip not only affects mobility, but also localisation and path following. Traditional approaches to solve this use an absolute localisation technique such as GPS or, if not avail- able, range sensing or visual odometry [70]. Given the computational burden imposed by visual odometry other techniques have been developed which use proprioceptive sensors to estimate slippage from motor current sensors [71, 72], wheel encoders and Inertial Mea- surement Unit (IMU) indicators using empirical thresholds [73], Extended Kalman Filters (EKF) and state space models [74].

Slip and sinkage detection methods as those presented previously can be used for traction control, to optimise the performance of the rover without the need of complex wheel-soil interaction models. For instance, in [75] the stick-slip phenomenon in a stair climbing robot is detected through wheel angular speed sensing and used for independent traction control. Combined slip and sinkage detection could be used to switch between different traction control criteria depending on the roughness of the terrain [76].

2.3

Modelling Off-Road Vehicle-Terrain Interaction

The different approaches to model terrain behaviour and its interaction with locomotion systems need to be studied to make an effective use of the indirect in-situ soil sensing techniques described in the previous section. For this purpose, the basic principles of the field of Terramechanics and its different methodologies are introduced. The implications and advances on wheel-soil interaction applied to planetary exploration are then presented. Finally, efforts in applying Terramechanics to leg-soil interaction are reviewed and studies of rotary walking on granular media are presented.

The term Terramechanics, first used in [61] by Bekker, refers to the field of study coping with the interaction between machines or locomotion mechanisms and soil. Much research has been carried out since, with particular interest in the study of the interaction between

2.3. Modelling Off-Road Vehicle-Terrain Interaction

wheels and soil. Four main methods for terrain behaviour modelling are usually employed in Terramechanics, each with its own strengths and weaknesses.

Theoretical methods analyse the failure of the soil purely based on a theory describing its behaviour, e.g. critical state theory [77], soil mechanics and elasto-plasticity theory [78] or visco-elastic models [79]. These theories are often based on assumptions that limit their scope of use, yielding results that significantly deviate from reality when not applied cor- rectly. Empirical methods predict the mobility of a vehicle based on direct comparison of sensor measurements on similar terrains through experimental models, e.g. the Brixius Ter- rain Model [80] used to predict the mobility and traction of a wheeled vehicle by measuring sinkage and slip. While useful for on-the-fly trafficability assessment on specific vehicles these methods cannot be extrapolated to other vehicle configurations and present serious difficulties when applied to certain types of soils and tyres.

Semi-Empirical methods combine theoretical analysis and experimentation to develop wheel- soil interaction models. This approach attempts to measure and model the stress distribu- tions created by the wheel on the soil and relate them with the resistive and tractive forces generated. Bekker made the first major contributions to this approach by formulating the relation between wheel sinkage and normal stress and between soil deformation and shear stress on brittle soils [81]. Various adaptations, expansions and simplifications have been made to these formulae. Important contributions were made by Reece, Wong and Onafeko [82, 83, 84], who measured and modelled stress and shear failure zones below driven and towed wheels, and Janosi [85], who adapted the shear stress-soil deformation formula to plastic soils.

Numerical simulation methods prove that using computer technology can be very helpful to simulate wheel-soil interaction. The Finite Element Method (FEM) [86, 87] can deal with heterogeneous terrains and the Discrete Element Method (DEM) [88, 89] permits abandon- ing the terrain-continuum hypothesis to better predict the behaviour of granular soils. The biggest challenge for both methods is to find a reliable behaviour model and to accurately estimate the parameters of each terrain element. Their main drawback is the required computational power which constrains the use of these methods to off-line applications and limits the size and number of simulated elements, affecting the realism of the simulation

2.3. Modelling Off-Road Vehicle-Terrain Interaction

on fine granular soils. Hybrid approaches, e.g. Smoothed Particle Hydrodynamics (SPH) [90, 91], have the potential to mitigate these drawbacks to some extent and provide a better accuracy-efficiency trade-off .

These methods can be applied not only in terrestrial applications, but also to planetary surface exploration. Such applications can be separated into an off-line phase and an on- line phase, as schematically depicted in Fig. 2.11. The role of Terramechanics in each phase is described below.

During the Research and Development (R&D) Phase of the rover, Terramechanics can provide detailed models to develop computer assisted, high fidelity and accuracy simulation tools to optimise the design parameters of the rover according to mission requirements and constraints. Some examples of such applications are reviewed in [92], including the LocSyn framework [93], the RCAST [94] and RCET [95] sets of tools for the ExoMars rover design, the ROAMS [96] physics-based simulator for the MER rovers, the RPET [97] systematic analysis framework, and the RoSTDyn [98] Terramechanics and dynamics simulation platform.

Figure 2.11: Application of Terramechanics to planetary rovers during the R&D phase (left) and the exploration phase (right)

2.3. Modelling Off-Road Vehicle-Terrain Interaction

During the Planetary Exploration Phase the models derived from Terramechanics research can be adapted and simplified to develop high speed and efficient algorithms that use sensed interaction forces and phenomena under given operating conditions for an on-line estimation of terrain parameters. These algorithms can be used to improve closed-loop motion control and autonomous path planning.

A useful simplification is to assume the soil stress distributions to be piecewise linear and symmetric, and apply least squares fitting techniques to estimate the soil’s physical and semi-empirical parameters [99, 100]. The inputs for such estimation methods are the normal load, wheel torque, wheel sinkage, wheel rotational speeds and traversal speed. In [101] the limitations of these simplification assumptions are discussed and an alternative method is proposed combining an EKF to estimate interaction forces and slip and a Bayesian Multi- Model Estimator to assign the most likely terrain parameters from a set of a-priori model hypothesis, being able to generate new candidate models using a Genetic Algorithm as proposed in [102].

The emerging field of Terramechanics applied to wheeled planetary rovers [103] has put for- ward many new challenges and successfully applied tools from the mobile robotics field such as advanced sensing and data fusion techniques, significantly contributing to the advance- ment of Terramechanics. Differences in morphology, payload, control mode, environment and soil behaviour create a gap between classical and planetary Terramechanics. To ad- dress this gap it is common practice to combine single-wheel tests [104], using single wheel test beds as those seen in Fig. 2.12, and full rover tests on planetary soil simulants for Terramechanics model development and validation.

Some examples of this research methodology include the study of stress distributions at Massachusetts Institute of Technology (MIT) [107], modelling of steering resistance at To- hoku University [106], Bekker-based modelling of rigid and flexible wheel at the German Aerospace Centre (DLR) [108], Rankine-based theoretical analysis of diverse wheel and grouser configurations at the Harbin Institute of Technology (HIT) [109] and parameter configuration for rover design at Carnegie Mellon University (CMU) [93].

Bekker’s classical model has limited application in this field due to significant errors for small wheel dimensions, small loads and high slippage. Therefore, most of the cited research

2.3. Modelling Off-Road Vehicle-Terrain Interaction

works on improving the modified Wong-Reece models. As pointed out in [103], each of these advancements has limited applicability due to inaccuracies in modelling non-linear phenomena, e.g. slip-sinkage relation, influence of wheel width or the effect of grousers. Another limitation is that these models account only for the steady state conditions, not being applicable to transitory performance.

While the interaction of wheels with deformable terrain has been thoroughly researched, leg- soil has been given less attention. Although the importance of analysing the performance of legged robots in deformable soils was already acknowledged in [110], the higher prioritising of other critical challenges in legged locomotion, such as those mentioned in Section 2.1, has relayed leg-soil interaction to a second plane.

Unexpected leg sinkage can cause serious issues in terms of robot performance and stability. Most of the work done regarding leg-soil interaction has focused on active sensing and compliance of legged mechanisms, using force sensing feedback to compensate the effect of

(a) (b)

(c) (d)

Figure 2.12: Single-wheel test beds for Terramechanics research applied to planetary exploration: (a) DLR [105], (b) Tohoku University [106] , (c) CMU [93] and (d) MIT [68]

2.3. Modelling Off-Road Vehicle-Terrain Interaction

sinkage on body motion [111], carry out pressure sinkage tests [112] and generate a perfectly plastic model of the soil using Bernstein’s pressure-sinkage equation [113]. Other than empirical and active control approaches little work has been done on theoretical modelling of leg-soil interaction and force prediction.

In [114] a trafficability model of legs on deformable soil was formulated based on Terza- ghi’s universal earth moving equation, dependent on factors such as blade dimensions, soil weight, surcharge, cohesion, adhesion and soil bearing capacity factors. Several empirical methods have been researched to determine these factors, showing very limited validity out- side of their specific application assumptions. To predict traction forces developed by a leg the traditional Terramechanics approach performs a quasi-static force equilibrium between all generated forces in the tool-soil interface, which includes: the soil thrust force due to shear forces developed at the foot’s leg-soil interface, the draught force that supports leg transversal motion, the active force from the soil in front of the leg and the frictional force of the sides of the leg which provides further traction. The differences shown by this model between predicted and observed forces [115] indicate that the Terramechanics approach is not applicable to small tools and loose soils. A review and experimental validation of the rigidity-dependent approaches to model foot-terrain interaction is presented in [116], adapting classic Terramechanics wheel-soil to feet with different geometries. Similarly to wheel-based research, these legged locomotion studies use the simplified single leg test beds shown in Fig. 2.13.

(a) (b)

Figure 2.13: Single-leg test beds for terramechanics research applied to planetary exploration: (a) University of Surrey [117] and (b) HIT [116]

2.3. Modelling Off-Road Vehicle-Terrain Interaction

Regarding hybrid wheel-legged robots, research has been focused on improving traversabil- ity of irregular terrain and enhancing climbing capabilities while simplifying the mechanic designs as much as possible in order to better exploit the high mobility-to-complexity ratio of wheel-legs.

These primary challenges have eclipsed the research of wheel-leg interaction with deformable and granular soil. However, there have been some recent advances in this field. In [118] SandBot, a small version of the RHex robot with single-legged wheel-legs pictured in Fig. 2.14 (b), was tested on a fluidised bed of granular media, with volume fraction control. Properties of the granular media, i.e. compaction, and of the robot gait, i.e. stride frequency, showed significant influence on locomotion, transitioning between slow swimming and fast walking behaviours. This variable behaviour was explained through the fluidisation and solidification phenomena occurring in granular physics.

These observations led to the formulation of a penetration dynamics model [119] based on simple shape and even weight distribution assumptions where the penetration force was considered to be proportional to sinkage. The simple shape assumption is disregarded in [120], and a generalised model was formulated to predict horizontal and vertical penetration forces considering the leg’s attack and intrusion angles.

(a) (b)

Figure 2.14: Wheel-legged locomotion: (a) visco-elastic multi-legged rimless wheel on rigid ground [121] and (b) single-legged wheel-leg Sandbot on granular media [120]