You are provided with the abstract of a scientific paper and the abstracts of the papers it references. Your task is to write a Related Work section for the main paper. This section should concisely summarize and integrate the key points from the cited abstracts, reflecting how they relate to the main paper's contributions and context. In your summary, use numerical citation markers like [1], [2], etc., to refer to the corresponding papers.

Main Abstract:
Despite the great progress in quadrupedal robotics during the last decade, selecting good contacts (footholds) in highly uneven and cluttered environments still remains an open challenge. This paper builds upon a state-of-the-art approach, already successfully used for humanoid robots, and applies it to our robotic platform; the quadruped robot ANY-mal. The proposed algorithm decouples the problem into two subprob-lems: first a guide trajectory for the robot is generated, then contacts are created along this trajectory. Both subproblems rely on approximations and heuristics that need to be tuned. The main contribution of this work is to explain how this algorithm has been retuned to work with ANY-mal and to show the relevance of the approach with a variety of tests in realistic dynamic simulations.

Cited Abstract(s):
[1]: Traditional motion planning approaches for multilegged locomotion divide the problem into several stages, such as contact search and trajectory generation. However, reasoning about contacts and motions simultaneously is crucial for the generation of complex whole-body behaviors. Currently, coupling theses problems has required either the assumption of a fixed gait sequence and flat terrain condition, or nonconvex optimization with intractable computation time. In this letter, we propose a mixed-integer convex formulation to plan simultaneously contact locations, gait transitions, and motion, in a computationally efficient fashion. In contrast to previous works, our approach is not limited to flat terrain nor to a prespecified gait sequence. Instead, we incorporate the friction cone stability margin, approximate the robot's torque limits, and plan the gait using mixed-integer convex constraints. We experimentally validated our approach on the HyQ robot by traversing different challenging terrains, where nonconvexity and flat terrain assumptions might lead to suboptimal or unstable plans. Our method increases the motion robustness while keeping a low computation time.

[2]: The contribution of this work is to show that real-time nonlinear model predictive control (NMPC) can be implemented on position controlled humanoid robots. Following the idea of “walking without thinking,” we propose a walking pattern generator that takes into account simultaneously the position and orientation of the feet. A requirement for an application in real-world scenarios is the avoidance of obstacles. Therefore, this letter shows an extension of the pattern generator that directly considers the avoidance of convex obstacles. The algorithm uses the whole-body dynamics to correct the center of mass trajectory of the underlying simplified model. The pattern generator runs in real-time on the embedded hardware of the humanoid robot HRP-2 and experiments demonstrate the increase in performance with the correction.

[3]: We present a motion synthesis framework capable of producing a wide variety of important human behaviors that have rarely been studied, including getting up from the ground, crawling, climbing, moving heavy objects, acrobatics (hand-stands in particular), and various cooperative actions involving two characters and their manipulation of the environment. Our framework is not specific to humans, but applies to characters of arbitrary morphology and limb configuration. The approach is fully automatic and does not require domain knowledge specific to each behavior. It also does not require pre-existing examples or motion capture data. At the core of our framework is the contact-invariant optimization (CIO) method we introduce here. It enables simultaneous optimization of contact and behavior. This is done by augmenting the search space with scalar variables that indicate whether a potential contact should be active in a given phase of the movement. These auxiliary variables affect not only the cost function but also the dynamics (by enabling and disabling contact forces), and are optimized together with the movement trajectory. Additional innovations include a continuation scheme allowing helper forces at the potential contacts rather than the torso, as well as a feature-based model of physics which is particularly well-suited to the CIO framework. We expect that CIO can also be used with a full physics model, but leave that extension for future work.

[4]: The goal of this paper is to demonstrate the capacity of model predictive control (MPC) to generate stable walking motions without the use of predefined footsteps. Building up on well-known MPC schemes for walking motion generation, we show that a minimal modification of these schemes allows designing an online walking motion generator that can track a given reference speed of the robot and decide automatically the footstep placement. Simulation results are proposed on the HRP-2 humanoid robot, showing a significant improvement over previous approaches.

[5]: We present a new method for planning footstep placements for a robot walking on uneven terrain with obstacles, using a mixed-integer quadratically-constrained quadratic program (MIQCQP). Our approach is unique in that it handles obstacle avoidance, kinematic reachability, and rotation of footstep placements, which typically have required non-convex constraints, in a single mixed-integer optimization that can be efficiently solved to its global optimum. Reachability is enforced through a convex inner approximation of the reachable space for the robot's feet. Rotation of the footsteps is handled by a piecewise linear approximation of sine and cosine, designed to ensure that the approximation never overestimates the robot's reachability. Obstacle avoidance is ensured by decomposing the environment into convex regions of obstacle-free configuration space and assigning each footstep to one such safe region. We demonstrate this technique in simple 2D and 3D environments and with real environments sensed by a humanoid robot. We also discuss computational performance of the algorithm, which is currently capable of planning short sequences of a few steps in under one second or longer sequences of 10–30 footsteps in tens of seconds to minutes on common laptop computer hardware. Our implementation is available within the Drake MATLAB toolbox [1].

[6]: Most animal and human locomotion behaviors for solving complex tasks involve dynamic motions and rich contact interaction. In fact, complex maneuvers need to consider dynamic movement and contact events at the same time. We present a hierarchical trajectory optimization approach for planning dynamic movements with unscheduled contact sequences. We compute whole-body motions that achieve goals that cannot be reached in a kinematic fashion. First, we find a feasible CoM motion according to the centroidal dynamics of the robot. Then, we refine the solution by applying the robot's full-dynamics model, where the feasible CoM trajectory is used as a warm-start point. To accomplish the unscheduled contact behavior, we use complementarity constraints to describe the contact model, i.e. environment geometry and non-sliding active contacts. Both optimization phases are posed as Mathematical Program with Complementarity Constraints (MPCC). Experimental trials demonstrate the performance of our planning approach in a set of challenging tasks.

[7]: Contact constraints, such as those between a foot and the ground or a hand and an object, are inherent in many robotic tasks. These constraints define a manifold of feasible states; while well understood mathematically, they pose numerical challenges to many algorithms for planning and controlling whole-body dynamic motions. In this paper, we present an approach to the synthesis and stabilization of complex trajectories for both fully-actuated and underactuated robots subject to contact constraints. We introduce a trajectory optimization algorithm (DIRCON) that extends the direct collocation method, naturally incorporating manifold constraints to produce a nominal trajectory with third-order integration accuracy—a critical feature for achieving reliable tracking control. We adapt the classical time-varying linear quadratic regulator to produce a local cost-to-go in the manifold tangent plane. Finally, we descend the cost-to-go using a quadratic program that incorporates unilateral friction and torque constraints. This approach is demonstrated on three complex walking and climbing locomotion examples in simulation.

[8]: We present a single trajectory optimization formulation for legged locomotion that automatically determines the gait sequence, step timings, footholds, swing-leg motions, and six-dimensional body motion over nonflat terrain, without any additional modules. Our phase-based parameterization of feet motion and forces allows to optimize over the discrete gait sequence using only continuous decision variables. The system is represented using a simplified centroidal dynamics model that is influenced by the feet's location and forces. We explicitly enforce friction cone constraints, depending on the shape of the terrain. The nonlinear programming problem solver generates highly dynamic motion plans with full flight phases for a variety of legged systems with arbitrary morphologies in an efficient manner. We validate the feasibility of the generated plans in simulation and on the real quadruped robot ANYmal. Additionally, the entire solver software TOWR , which used to generate these motions is made freely available.

[9]: We present a trajectory optimization framework for legged locomotion on rough terrain. We jointly optimize the center of mass motion and the foothold locations, while considering terrain conditions. We use a terrain costmap to quantify the desirability of a foothold location. We increase the gait's adaptability to the terrain by optimizing the step phase duration and modulating the trunk attitude, resulting in motions with guaranteed stability. We show that the combination of parametric models, stochastic-based exploration and receding horizon planning allows us to handle the many local minima associated with different terrain conditions and walking patterns. This combination delivers robust motion plans without the need for warm-starting. Moreover, we use soft-constraints to allow for increased flexibility when searching in the cost landscape of our problem. We showcase the performance of our trajectory optimization framework on multiple terrain conditions and validate our method in realistic simulation scenarios and experimental trials on a hydraulic, torque controlled quadruped robot.
For simple cases like flat terrains, optimization techniques are able to correctly solve, simultaneously, the motion of the main body of the robot and its footstep placements [4] [2] . Using more complex models and solvers, the problem can be reformulated to solve motion on other terrains [9] [8] . However, this algorithm is doomed to fall into local minima, e.g. ignoring intermediate steps on stairs or trying to jump over impassable obstacles. Such behaviors can be reduced by first relaxing the complementary constraint of contacts then slowly converging back to the initial problem [6] [7] [3] . Alternatively, one can decompose the non-smooth terrain into different convex and even patches, and rely on Mixed-Integer Programming to find the best patches for each footstep [5] [1] . However the computation time of those approaches make them difficult to use on a real robot. Overall, optimization techniques are not well suited for collisions and all of the presented approaches ignore this problem.