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Many gait generation approaches for humanoids guarantee that balance is maintained during locomotion by enforcing the condition that the Zero Moment Point (ZMP, the point where the horizontal component of the moment of the ground reaction forces becomes zero) remains at all times within the support polygon of the robot. Due to the complexity of full humanoid dynamics, however, direct control of the ZMP is very difficult to achieve. In view of this, simplified models are generally used to relate the evolution of the ZMP to that of the Center of Mass (CoM) of the robot, which can be instead effectively controlled. A common choice is the Linear Inverted Pendulum (LIP).
There is a potential instability issue at the heart of the problem. In particular, a certain ZMP trajectory may be realized by an infinity of CoM trajectories, which, due to the nature of the CoM/ZMP dynamics, will in general be divergent with respect to the ZMP trajectory itself. In this situation, dynamic balance can be in principle achieved by properly choosing the ZMP trajectory, but internal instability indicates that such motion will not be feasible in practice for the humanoid.
The gait generation problem is often formulated in a Model Predictive Control (MPC) setting. This is convenient because it allows to generate simultaneously the ZMP and the CoM trajectories while satisfying constraints, such as the ZMP balance condition as well as kinematic constraints on the maximum step length and foot rotation. Moreover, the MPC approach guarantees a certain robustness against perturbations. A way to guarantee the stability of an MPC scheme is to enforce a terminal state constraint (i.e., a constraint on the state at the end of the control horizon), classically used for closed-loop stability in set-point control problems. However, MPC-based gait generation is neither a set-point nor a tracking problem. In fact, since the ZMP control objective is encoded via time-varying state constraints, there is no error to be regulated to (or close to) zero. The only significant stability issue in this context is internal stability, i.e., the boundedness of the CoM trajectory with respect to the ZMP trajectory.
In a preliminary work [1], we developed an Intrinsically Stable MPC (IS-MPC) for gait generation, which relies on the inclusion of an explicit stability constraint in the formulation of the problem. In particular, the idea was to enforce a condition on the future ZMP velocities (representing the control inputs) so as to guarantee that the generated CoM trajectory remains bounded with respect to the ZMP trajectory. Since the control horizon of the MPC algorithm is finite, only part of the future ZMP velocities are decision variables and can, therefore, be subject to a constraint; the remaining part, called tail, must be conjectured. In this work we exploited the periodicity of a typical regular gait and obtained the tail by infinite replication of the ZMP velocities over the control horizon.
The following clip shows simulations that highlight the benefits of the proposed method. We have first considered the case of given footsteps for a MATLAB-simulated LIP. The standard MPC scheme is unable to generate a stable gait with a shorter prediction horizon: the resulting ZMP trajectory is always feasible, but the associated CoM trajectory diverges with respect to it, because the control horizon is too short to allow sorting out the stable behavior via jerk minimization. With IS-MPC, instead, boundedness of the CoM trajectory with respect to the ZMP trajectory is preserved in spite of the shorter control horizon, thanks to the embedded stability constraint. Robot simulations are performed in V-REP on the NAO humanoid. In this case, the footsteps are automatically placed by the MPC to guarantee reactivity to disturbances, and different scenarios are tested (variable reference direction, variable reference speed, push recovery).
The work in [2] expands and develops the analysis of the IS-MPC scheme. A candidate footstep generator is included, which is in charge of choosing a set of footsteps (timing, position and orientation) realizing high-level reference velocities over a preview horizon (in general, longer than the MPC control horizon). The IS-MPC stage will be able to adapt the position of the footsteps for reacting to disturbances. The theoretical analysis of the stability constraint is extended to show that it is equivalent to a terminal constraint. Depending on the available preview information on the commanded motion, we discuss several versions of the tail (truncated, periodic, and anticipative) to be used in the constraint, and show that each of them corresponds to a specific terminal constraint. In particular, the stability constraint with the truncated tail is equivalent to the capturability constraint (from the concept of capture point), that requires the unstable dynamics of the LIP to stop at the end of the horizon. Further analysis is dedicated to the anticipative tail, which conjectures future ZMP velocities by using the available planned information over the preview horizon. In order to achieve recursive feasibility (i.e., the MPC is always able to find a solution that satisfies the constraints if properly initialized), one should always choose the anticipative tail. Once recursive feasibility is guaranteed, CoM/ZMP stability is automatically ensured in IS-MPC.
Simulations on the complete scheme are shown in the following clip. MATLAB simulation on the LIP introduce the instability problem. The stability constraint is shown to address this problem, but can introduce feasibility issues if an inaccurate tail is used, i.e., if the conjectured velocities are too different from the ones that are actually executed. The anticipative tail is able to guarantee both recursive feasibility and stability of the CoM/ZMP trajectories. Then, simulations and experimental results on NAO and HRP-4 are presented.
[1] N. Scianca, M. Cognetti, D. De Simone, L. Lanari, G. Oriolo, Intrinsically Stable MPC for Humanoid Gait Generation. 16th IEEE-RAS International Conference on Humanoid Robots, Cancún, Mexico, pp. 601-606, 2016 (pdf).
[2] N. Scianca, D. De Simone, L. Lanari, G. Oriolo, MPC for Humanoid Gait Generation: Stability and Feasibility. IEEE Transactions on Robotics, 2020 (pdf).