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CoM state-space cascading manifolds for planning dynamic walkingin very rough terrain

Luis Sentis* and Benito Fernandez**lsentis@austin.utexas.edu , benito@mail.utexas.edu

The University of Texas in Austin; 1 University Station C2200, Austin, TX 78712, USA

1 IntroductionFormulating rough terrain locomotion implies dealing withnonlinear/nonintegrable models of Center of Mass (CoM)behavior with respect to the contact state. Our hypothesis isthat center of mass phase curves can be created using per-turbation theory and then used to nd contact transitions toproduce dynamic walking in very rough terrains. This ex-tended abstract aims at validating this hypothesis.

In dynamic walking we can classify techniques in variouscategories: (1) trajectory-based techniques, (2) limit cycle-based techniques, (3) prediction of contact, and (4) hybridsof the previous three.

Trajectory-based techniques: These are techniques thattrack a time-based joint or task space trajectory accordingto some locomotion model such as the Zero Moment Point (ZMP). [6] and [8] represent some of the latest develop-ments in this area.

Prediction of contact placement: These are techniques thatuse dynamics to estimate suitable contact transitions. In[14], simple dynamic models are used to predict the place-ment of next contacts to achieve desire gait patterns.Someof the most recent work can be found in in [2, 4, 13].

Limit cycle based techniques: McGeer [11] pioneered theeld of passive dynamic walking . In [5] the authors study or-bital stability, and the effect of feedback control to achieveasymptotic stability. Optimization of open-loop stability isinvestigated in [12]. In [15], the authors analyze the ener-getic cost of bipedal walking and running. In [19], the au-thors developed a dynamic walker using articial muscles.In [20], a methodology for the analysis of state-space be-havior and feedback control are presented. Step recovery inresponse to perturbations is studied in [18]. In [16], theinterplay between robustness against perturbations and legcompliance is investigated.

Hybrid methods: In [21], the stability of passive walkers isstudied. Stochastic models of stability and its application forwalking on moderately rough unmodeled terrain are studiedin [3]. The design of non-periodic locomotion for uneventerrain is investigated in [10]. In [17], the authors explore thedesign of pasitivity-based controllers to achieve walking on

different ground slopes. Optimization-based techniques forlocomotion in rough terrains are presented in [22]. In [7], theauthors exploit optimization as a means to plan locomotion.

Figure 1: Data extraction from human walk: A humansubject walks over a rough terrain. Marker tracking is im-plemented and used to extract approximate CoM paths aswell as Sagittal and vertical CoM trajectories and velocities.

2 Summary of our approach

We tackle here rough terrain locomotion by exploring CoM state-space manifolds and transitional contact states . Theapproach is applicable to all terrains with special emphasis

to very rough environments.Our approach, can be explained algorithmically in termsof various phases, namely (1) geometric planning, (2)perturbation-based CoM phase generation, and (3) stepsolver based on adjacent phase curves The geometric plan-ning phase consists on applying kinematic planning tech-niques to obtain initial guesses of feet contact locations andCoM geometric path. Perturbation-based CoM phase gener-ation is our rst contribution and consists on: (1) formulat-ing CoM accelerations based on the contact state, (2) includethe dependency between Sagittal and vertical accelerationsdue to the desired CoM geometric path, and (3) use per-

turbation theory to obtain phase curves of the CoM in theneighborhood of the step contact. The step solver, is oursecond contribution and consists on tting polynomials tophase curves of adjacent steps and nding the roots of thedifferential polynomial between inmediate neighbors.

3 Perturbation-based CoM phase generation

We present here a solution for robot locomotion in the Sagit-tal/vertical plane. Using a human-size robot model, we con-sider a variable stepped terrain with height variations be-tween 40 [cm] and width variations between 30-40 [cm].

The goal of the planner is to maneuver the robot through thetotal length of the terrain. The speed specications are givento cruise the terrain at an average velocity of 0.6 [m/s], al-

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though this choice could be arbitrary. We also assume thatthe robot starts and nishes with zero velocities and it in-creases velocity through the steps according to a trapezoidalprole. Velocity specications are given only at each newstep, corresponding to the moment when the center of massSagittal position crosses the corresponding supporting foot,namely the apex of the step. As such, they are equivalent toboundary conditions to solve the state space behavior. Letsconsider steps to be spanned from apex to apex. Also forsimplicity, we consider only single-support phases, with in-stantaneous transition between feet. We also assume thatthe contact locations and the geometric path of the centerof mass are given by a kinematic planner, and we assume apoint mass model of the robot, with all the weight located atits waist.

For every contact state, we formulate dynamic equilibriumof moments, i.e. contact reactions vs. inertial and grav-

itational moments. This relationship yields a well-knownsolution that relates Sagittal accelerations with respect tocenter of mass Sagittal and vertical distances to contact lo-cations, and with the latter multiplied by vertical acceler-ations plus gravity. CoM moments do not appear due topoint mass conditions. Most researchers simplify the aboveequation by assuming xed vertical CoM and feet condi-tions. However, to walk in very rough uneven terrains thisassumption is no longer valid. Instead, we assume uncon-strained vertical CoM and feet variations, assuming theywill be kinematically feasible. We seek to nd a state-spacemanifold of CoM behavior due to the varying contact con-ditions and desired CoM kinematic path. We refer to pertur-bation theory [1, 9] to address the difculty of solving non-integrable equations. In particular, perturbation theory, hasbeen widely used to solve the trajectory of celestial bodiesand pendulums. Perturbation theory, is a set of methods thatenable to approximate solutions from problems that donthave exact solutions, by looking into the solution of an ex-act related problem. In our case, we have the exact solutionof accelerations given positions and pressure points and weseek to approximate the solution of the position versus thevelocity, i.e. the state-space trajectory. Using perturbationtheory, we obtain the incremental relationship between CoMpositions and velocities for each contact state, thus yield-ing state-space CoM specications. Because we operate instate-space we remove time as a variable.

The CoM manifolds, by construction, describe the CoM be-havior before and after each apex. If we combine neighbor-ing manifolds, the contact transition state can be determinednding the interseccion of the curves. This approach is thekey contribution of this work (see Figure 2).

Using the prescribed CoM kinematic path, it is now possibleto extract the corresponding CoM manifold in the verticaldirection. Moreover, given the contact transition states it is

also possible to derive feet state-space curves. This informa-tion in turn, can be utilized to create joint velocity or torquefeedback controllers to make the CoM manifold an attractor.

Figure 2: Concatenation of steps: The top graph showsthe kinematic trajectory of the human CoM (see Figure 1for the extraction of motion capture data) versus a piecewiselinear approximation that we use to generate the automaticwalking simulation. The red dots correspond to the posi-tion of the foot contacts. The bottom gure shows Matlabplots of Sagittal phase curves for the human and the auto-matic simulation. The red circles correspond to apexes of the steps. The green squares correspond to contact transi-tions of the automatic walk. The blue squares correspondto contact transitions of the human walk. Notice, that dur-ing the climbing of the rst step of the stairs results in asmooth CoM pattern for the human walk. This is due to thesmoothening effect of dual contact during the stance phase.This is not the case during the automatic walk because wehave neglected the stance phase and therefore the transitionsbetween contacts are instantaneous. Besides this difference,the rest of the walk correlates well.

4 Open questions

For this seminal work, we would like to address two of theseissues: robustness and the role of internal forces during mul-ticontact phases. Robustness issues are critical to the suc-cess of implementing dynamic locomotion. We can studythe stability robustness of a specic manifold to parameteruncertainty or to external disturbances. That is, how muchparametric uncertainty is allowed before the CoM manifoldis no longer attractive. The least-known parameters are fric-tion forces coefcients in the joints and with the ground. Bylooking at the parameter region around the estimated values,this technique can be used to determine if and where the al-gorithm would fail to obtain a suitable transitional state. Wecould also address the effect of joint compliance in the sta-

bility robustness of the system. Furthermore, the proposedapproach can be used to design the stiffness requirementsthat guarantee disturbance rejection (maintaining stabilityrobustness) while traversing rough terrains.

Locomotion under multi-contact states provides advantagesin the decision-making scenario on how and where to deter-mine the transition state. Rather than a single phase mani-fold, multi-contact conditions lead to manifolds where manysolutions are plausible. As a result, the range of motions isincreased and optimization algorithms can be carried out tomaximize robustnes, minimize energy consumption, etc. Weplan to discuss our take on this problem and propose poten-tial extensions of the algorithm to multi-contact scenarios.

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