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Motion planning in unstructured and low-speed environments is a fundamental and difficult task for all mobile robotics. If we view motion planning as an optimization problem, the non-convex collision avoidance constraints and the nonlinear vehicle dynamic constraints make motion planning challenging and time-consuming. In this paper, we propose a Successive Linearization in Feasible Set (SLiFS) algorithm to address these two difficulties. SLiFS consists of two steps. The first step is to iteratively construct convex feasible sets around the current trajectory to approximate non-convex collision avoidance constraints. The second step is to successively linearize the nonlinear dynamic constraints along the current trajectory and further penalize them into the objective function to avoid infeasible linearized constraints, so that the current trajectory can be reshaped within the obtained convex feasible sets by iteratively solving the linearized optimization problem. The main innovation of SLiFS algorithm is that we consider the L₁ norm type penalty function in the second step. We find that the sparsity of the L₁ norm might help to satisfy the robotic dynamic constraints by numerical experiments. Numerical testing results show that our proposed SLiFS algorithm has a high success rate to find feasible trajectories and costs much less time than the classical interior-point method.

As a result of earthquakes, the deformation and failure caused by anti-dip bedding rock slopes are large, and their seismic dynamic response law is complex. Using the three-dimensional discrete element numerical analysis software 3DEC, the influence of seismic wave parameters on the dynamic response of anti-dip bedding rock slopes was systematically studied, with special focus on the influence of the angle between seismic wave incidence direction and slope trend on the dynamic response of anti-dip bedding rock slopes under three-dimensional conditions. The orthogonal test was designed to conduct sensitivity analysis of five seismic parameters, including seismic wave amplitude, incidence angle of the S-wave, frequency, duration, and the time difference between the P-wave’s and the S-wave’s peak. The results revealed that the S-wave’s amplitude As and the holding time T of the seismic wave are positively correlated with the acceleration amplification factor of the slope, and the incident direction γ of the S-wave is negatively correlated with the acceleration amplification factor of the slope. The increase of seismic wave frequency f and the time difference Δt between the P-wave’s and the S-wave’s peak lead to the first increase and then decrease of the Y-directional displacement of the slope. The sensitivity of each seismic wave parameter to the Y-directional acceleration amplification factor at the shoulder of anti-dip bedding rock slopes in earthquake conditions is ordered as follows: S-wave’s amplitude As > frequency f > S-wave’s incidence angle γ > the time difference Δt > holding time T. the study results provide reference and basis for stability evaluation and engineering design of anti-dip bedding rock slopes in areas with high seismic intensity.

Underground mining operations using the sublevel caving method often result in discontinuous ground deformations involving surface cracks. Determining the limit of the discontinuous deformation zone is important for the relocation of neighbouring residential areas, mining infrastructure, and the planning of underground mining operations. The limit of the discontinuous deformation zone is characterised by the outermost surface crack, which is a tensile crack when it occurs and is correlated with the horizontal ground tensile strain. In this study, a mathematical procedure combining the horizontal displacement, as periodically monitored by total station and GPS, multiquadric radial basis function (MQ-RBF) interpolation, and continuum mechanics is provided to calculate the strain state at any point on the ground. Then the relationship between a new tensile crack and horizontal ground tensile strain is studied to determine the criterion representing the presence of a new tensile crack. This means that, when the criterion at any point on the ground is satisfied, the point will have the potential to crack. Based on this, the potential discontinuous deformation zone where surface cracks may occur can be identified, potential break angles at representative sections can then be determined. A case study is presented to illustrate the results given by the above methodology. The result shows that when major principal horizontal ground strain e1 at any point on the ground reaches the potential cracking major principal strain, PCMPS (6 mm/m in this case study), the point will have the potential to crack. Subsequently, the area within the outermost contour of the PCMPS can be defined as the potential discontinuous deformation zone. Moreover, the approximate matching of the limits of potential discontinuous deformation zones and surface cracks verifies the reliability of the mathematical procedure and the rationality of the PCMPS in identifying the discontinuous deformation zone. Additionally, in this case study, the potential break angles at two respective sections are significantly different, indicating that the break angles should be further studied and care taken in design of works in these two zones, respectively.

Block-flexure toppling failure is frequently encountered in rock slopes. In this paper, we first proposed two theoretical models to calculate the factor of safety against block-flexure toppling, namely, step by step analysis model (LEM-SSAM) and overall analysis model (LEM-OAM). Then, we developed two special MATLAB codes for a quick stability estimation based on the proposed methods, respectively. Next, the universal distinct element code (UDEC) was used to study the failure mechanisms of block-flexure toppling. In order to realistically simulate the excavation of slope and obtain the detailed damage information of joints, we developed two FISH functions in UDEC. Finally, we conducted a comparative analysis of the limit equilibrium method and the numerical simulations through twelve theoretical models. The results show that many aspects of block-flexure toppling failure have been captured using UDEC combined with the developed FISH functions. The LEM-OAM and UDEC agree well while a larger factor of safety is obtained using the LEM-SSAM. It is found that the dip angles and friction angles of the joints dipping steeply into face have a great influence on block-flexure toppling failure while the influence of the cross joint spacing is less significant.

Bi-planar failure is the commonest type of failure in steep and high footwall slopes. This paper presents an investigation of the bi-planar failure mechanism and a stability analysis model for footwall slopes based on the study of eleven diabase mines in Hunyuan County, Shanxi Province, China. First, a detailed investigation of the site and tests were carried out to obtain a geological model and calculation parameters for the slope. Then, the failure process in footwall slopes was investigated by a trigon approach performed using the Universal Distinct Element Code (UDEC) software package. A mechanism was then obtained for bi-planar compression–shear failure by combining the results obtained for the stress, displacement, and crack development. Subsequently, based on the limit equilibrium and frictional plasticity theories, a new columnar mechanical model was proposed for bi-planar footwall slope failure. The results demonstrate that the UDEC trigon approach was well suited to simulating bi-planar failure in steep and high footwall slopes. Employing this approach, a sliding failure surface that initiates, develops, and propagates in the discontinuity and intact rock could be correctly captured and the evolution of the damage in the toe of the slope could be quantitatively investigated. The results also indicate that shear failure first occurs at persistent discontinuities and then compression–shear failure occurs in the intact rocks. In addition, the failure surface of the rock emanates from the toe of the slope, is oriented upwards at an angle to the persistent discontinuity thus forming the through sliding surface (this angle can be determined using frictional plasticity theory). The critical heights and safety factors of footwall slopes could be readily obtained by employing this new columnar mechanical model giving results that are consistent with the results of numerical simulations. Furthermore, the results of a parameter-sensitivity analysis show that the dip angle of the toe breakout surface increases as the slope angle increases. However, the angle between the toe breakout and internal shear surfaces remains unchanged as the slope angle increases. Moreover, the critical height of the footwall slope decreases linearly as the slope angle increases. The method proposed provides theoretical guidance for the stability analysis of steep and high footwall slopes. The results produced should also help engineers to gain a better understanding of the mechanisms of bi-planar failure in such rock slopes.