Whole-body Control (WBC) has emerged as an important framework in locomotion control for legged robots. However, most of WBC frameworks fail to generalize beyond rigid terrains. Legged locomotion over soft terrain is difficult due to the presence of unmodeled contact dynamics that WBCs do not account for. This introduces uncertainty in locomotion and affects the stability and performance of the system. In this paper, we propose a novel soft terrain adaptation algorithm called STANCE: Soft Terrain Adaptation and Compliance Estimation. STANCE consists of a WBC that exploits the knowledge of the terrain to generate an optimal solution that is contact consistent and an online terrain compliance estimator that provides the WBC with terrain knowledge. We validated STANCE both in simulation and experiment on the Hydraulically actuated Quadruped (HyQ) robot, and we compared it against the state of the art WBC. We demonstrated the capabilities of STANCE with multiple terrains of different compliances, aggressive maneuvers, different forward velocities, and external disturbances. STANCE allowed HyQ to adapt online to terrains with different compliances (rigid and soft) without pre-tuning. HyQ was able to successfully deal with the transition between different terrains and showed the ability to differentiate between compliances under each foot.
In contrast to the aforementioned work, other approaches relax the rigid ground assumption (hard contact constraint) but not for soft terrain adaptation purposes. For instance, Kim al @cite_5 implemented an approach to handle sudden changes in the rigid contact interaction. This approach relaxed the hard contact assumption in their wbc formulation by penalizing the contact interaction in the cost function rather than incorporating it as a hard constraint. For computational purposes, Neunert al @cite_26 and Doshi al @cite_14 proposed relaxing the rigid ground assumption. Neunert al used a soft contact model in their nonlinear mpc formulation to provide smooth gradients of the contact dynamics to be more efficiently solved by their gradient based solver. The soft contact model did not have a physical meaning and the contact parameters were empirically chosen. Doshi al proposed a similar approach which incorporates a slack variable that expands the feasibility region of the hard constraint.
@cite_5 @cite_14 @cite_26
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In this paper, we present a convex optimization problem to generate Center of Mass (CoM) and momentum trajectories of a walking robot, such that the motion robustly satisfies the friction cone constraints on uneven terrain. We adopt the Contact Wrench Cone (CWC) criterion to measure a robot's dynamical stability, which generalizes the venerable Zero Moment Point (ZMP) criterion. Unlike the ZMP criterion, which is ideal for walking on flat ground with unbounded tangential friction forces, the CWC criterion incorporates non-coplanar contacts with friction cone constraints. We measure the robustness of the motion using the margin in the Contact Wrench Cone at each time instance, which quantifies the capability of the robot to instantaneously resist external force torque disturbance, without causing the foot to tip over or slide. For pre-specified footstep location and time, we formulate a convex optimization problem to search for robot linear and angular momenta that satisfy the CWC criterion. We aim to maximize the CWC margin to improve the robustness of the motion, and minimize the centroidal angular momentum (angular momentum about CoM) to make the motion natural. Instead of directly minimizing the non-convex centroidal angular momentum, we resort to minimizing a convex upper bound. We show that our CWC planner can generate motion similar to the result of the ZMP planner on flat ground with sufficient friction. Moreover, on an uneven terrain course with friction cone constraints, our CWC planner can still find feasible motion, while the outcome of the ZMP planner violates the friction limit. Making use of a perturbed Lagrangian formulation, a finite element procedure for contact problems is developed for the general case in which node-to-node contact no longer holds. The proposed procedure leads naturally to a discretization of the contact interface into contact segments. Within the context of a bilinear interpolation for the displacement field, a mixed finite element approximation is introduced by assuming discont naous contact pressure, constant on the contact segment. Because of this piece-wise constant approximation, the gap function enters into the formulation in an ‘average’ sense instead of through a point-wise definition. Numerical examples are presented that illustrate the performance of the proposed procedure. Current finite element formulations for contact problems based on either the classical Lagrange parameter procedure [l-3, 12, 203 or the penalty-function method [4-6,11], are characterized by a point-wise enforcement of the contact-constraint condition, in the sense that penetration of the bodies is established on a nodal basis. Moreover, in this methods the recovery of the contact pressure over the element from the contact nodal forces generally requires an additional procedure. Within the framework of classical Lagrange multiplier methods the contact condition is exactly satisfied by transforming the constrained problem into an unconstrained one with the introduction of additional variables (Lagrange multipliers). These extra variables add computational effort to the solution process which often requires special procedures to handle the presence of zero diagonal terms. Penalty methods, on the other hand, enable one to transform the constrained problem into an unconstrained one without introducing additional variables. The constraint condition is now satisfied only approximately for finite values of the penalty parameter. The main difhculty associated with these methods, however, lies in the poor conditioning of the problem as the penalty is increased to more accurately enforce the constraint condition. This is a well-understood phenomenon, particularly in the context of the incompressible and nearly incompressible problem in solid and fluid mechanics (e.g. see [l&22,25] for a review). Recently, augmented Lagrangian procedures have been proposed as a promising way to partially overcome these difficulties and ‘regularize’ the penalty formulation (e.g. see the survey in [7,8]). Within the framework of Zinearited e u c , it is possible to restrict the finite element formulation of contact problems by assuming that node-to-node contact occurs. This is in fact the case often considered in the literature [l, 2, 4, 6, 12, 20, 211. In the general context of fully This paper proposes a method to realize desired contact normal forces between humanoids and their compliant environment. By using contact models, desired contact forces are converted to desired deformations of compliant surfaces. To achieve desired forces, deformations are controlled by controlling the contact point positions. Parameters of contact models are assumed to be known or estimated using the approach described in this paper. The proposed methods for estimating the contact parameters and controlling the contact normal force are implemented on a LWR KUKA IV arm. To verify both methods, experiments are performed with the KUKA arm while its end-effector is in contact with two different soft objects. The present work extends the non-smooth contact class of algorithms introduced by to the case of friction. The formulation specifically addresses contact geometries, e.g. involving multiple collisions between tightly packed non-smooth bodies, for which neither normals nor gap functions can be properly defined. A key aspect of the approach is that the incremental displacements follow from a minimum principle. The objective function comprises terms which account for inertia, strain energy, contact, friction and external forcing. The Euler–Lagrange equations corresponding to this extended variational principle are shown to be consistent with the equations of motion of solids in frictional contact. In addition to its value as a basis for formulating numerical algorithms, the variational framework offers theoretical advantages as regards the selection of trajectories in cases of non-uniqueness. We present numerical and analytical examples which demonstrate the good momentum and energy conservation characteristics of the numerical algorithms, as well as the ability of the approach to account for stick and slip conditions.