Optimization-based walking generation forhumanoid robot

Kai Henning Koch ∗ Katja Mombaur ∗∗ Philippe Soueres ∗∗∗

∗ IWR, Heidelberg University, INF 368, 69120 Heidelberg, Germany(e-mail: Henning.Koch@iwr.uni-heidelberg.de)

∗∗ IWR, Heidelberg University, INF 368, 69120 Heidelberg, Germany(e-mail: Katja.Mombaur@iwr.uni-heidelberg.de)

∗∗∗ LAAS-CRNS, Universite de Toulouse 7 avenue du Colonel Roche,31077 Toulouse, France, (e-mail: soueres@laas.fr)

Abstract: The generation of walking motions for humanoid robots is a challenging task. Fromthe infinite number of possibilities to move the body of the robot with its redundant degreesof freedom (DOF) forward, the task is to determine those motions that are stable, feasiblewithin the robots kinematic and dynamic limitations, and also resemble the way we expect ananthropomorphic system to walk. Several approaches have been developed and implementedon humanoids in the past years, however most of them require fixing several characteristics ofthe gait, such as foot placement or step time, in advance, and none has lead to truly human-like walking performance. The purpose of this paper is to show that mathematical trajectoryoptimization or optimal control can be very helpful to generate walking motions for humanoidrobots. We propose a method that uses dynamic model information of the robot as well asefficient optimal control techniques to determine joint trajectories and actuator torques at thesame time. Foot placement and step times are also left free for optimization. The method isapplied to the humanoid robot HRP-2 with 36 DOF and 30 actuators. Different optimizationcriteria are evaluated, such as maximization of efficiency, walking speed or postural stability, anda minimization of joint torques or angular amplitudes. ZMP constraints (or alternative stabilityconstraints) can be taken into account in the optimization. The results show that differentobjective functions and constraints have a considerable influence on the resulting gait.

Keywords: Optimal Control, Humanoid robot, HRP-2, Walking motion generation

1. INTRODUCTION

The walking skills of todays humanoids are still far behindthose of their human role models. However, fast, efficient,robust and versatile walking is an essential capabilitythat humanoids have to acquire if they are supposed totackle all the challenges that humans devise for them. Theproblem for this is not only linked to the present robotichardware, but also to a large extent to the software andthe control principles used. It is a common assumptionthat movements of humans and animals are optimal dueto evolution and individual development. We try to mimicthis optimality principle of human motions, by generatingoptimal motions for robot models, using optimal control.This approach allows to directly modify important gaitcharacteristics such as stability, efficiency, or speed. Itallows to determine position and velocity trajectories aswell as actuator inputs simultaneously in an optimal way,and does not require to prescribe any of these quantities apriori.

1.1 Related work

For the generation of walking motions, Buschmann et al.(2005) give a short survey over two different approaches:lumped-mass models and optimization based approaches.

The former method generally tries to plan a COG tra-jectory between fixed points based on a stability criteria(e.g ZMP) either sequentially or in parallel and computesthe whole body motion according to various motion con-straints (please refer to (Kajita et al., 2003; Morisawaet al., 2005; Takenaka et al., 2009)). These pattern gener-ators perform remarkably well, the majority in real-time,are convenient to parametrize, but suffer from differentissues (e.g. singularities, high energy consumption). Thelatter method - optimization - is normally computationallytoo expensive to work in real-time, requires initialization,but allows to optimize the gait with respect to a specificobjective or a combination of several objectives. Early ap-proaches of gait cycle optimization for whole body planarbipeds have been published by (Roussel et al., 1998; Hardtet al., 1999) the former based on control parametrizationand the latter on collocation with respect to minimumenergy consumption. (Bessonnet et al., 2004) presents ageneration method of optimal energetic gaits with entranceand exit motion cycles based on the indirect method ofPontryagin Maximum Principle. M. et al. (2003) use col-location to optimize walking motions for different objectivefunctions. Mombaur et al. (2001, 2005) and Mombaur(2009) present open-loop stable solutions produced bystability optimization for different bipedal and anthropo-morphic configurations, and Schultz and Mombaur (2010)

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Fig. 1. Generation of optimal walking trajectories forthe humanoid robot HRP-2: (a) maximum efficiencywalking with free foot placement, respecting ZMPconstraints (red circle - position of ZMP under thefoot, blue circle - position of the center of mass of therobot), (b) stick figures for same optimized solution -full cycle (left & right step)

presents realistic optimal running solutions for 2D and 3Dhuman models. Alternative approaches for the generationof walking motions based on the stack of tasks are suc-cessfully applied by (Saab et al., 2011) and (Ramos et al.,2011).

1.2 Outline of paper

This paper is organized as follows. In Section 2, we presentthe general form of modeling humanoid gaits which issuitable for optimization. In Section 3, we show how walk-ing motions can be generated by solving optimal controlproblems. Section 4 descries results for the optimizationof different objective functions. In Section 5, we give someconclusions and perspectives for future research.

2. MATHEMATICAL MODELS OF WALKINGMOTIONS OF THE HUMANOID ROBOT HRP-2

In this section, we describe how to model humanoidwalking motions to be used in optimal control problemformulations. We start by describing the properties of thehumanoid robot HRP-2 used in this study. After that thegeneral form of the equations of motions and constraintswill be given.

2.1 The humanoid robot HRP-2

HRP-2 is a 36 degree of freedom humanoid robotic plat-form with a height of 154 cm and a weight of 58 kg andis powered by 30 DC motors. See (Kaneko et al., 2004)for the detailed informations on the robot. The robot isequipped with a stabilizer (please see (Kajita et al., 2007)for further details) that prevents the robot from fallingover during its motions, compensating small modelingdeficiencies and small environmental disturbances. Thedynamic balance of the robot is controlled by keeping theZMP (Vukobratovic and Stephanenko, 1972) in a certainstability region that is smaller than the actual supportpolygon. The robot has flat, rubber coated feet the elas-ticity effects of which are sufficiently small to be neglected.It is equipped with an elastic 3 DOF joint on the ankle toabsorb shocks during locomotion (Chardonnet, 2009). Forsimplification, the ankle elasticity is not considered duringthe following discussion. The robot can has been modeledas under-actuated tree-structure with the base referenceframe fixed to the pelvis. The modeling coordinates arethe six coordinates of this base frame as well as the 30internal joint angles, which would be minimal coordinatesfor a free-floating robot. In this model, we maintain the setof coordinates during different phases (single and doublesupport), but use additional constraints to describe thecontacts. A realistic dynamic model of a robot wouldcontain not only the dynamics of the link structure, butalso the dynamic effects of the transmission unit and theactuation systems, friction as well as elastic effects of thelinks and the transmission system - please see (Sicilianoand Khatib, 2008) for a comprehensive discussion. For thepresent study, the mathematical complexity of modelingthese aspects is too high compared to the expected gain ofquality of the resulting trajectories. We adopt the followingassumptions for our model: the robot has rigid links andtransmission units, the transmission ratios are suitably

high that dynamic coupling effects of the motor inertias tothe whole body structure are negligible and joint frictionis not considered.

2.2 General formulation of the walking motion

Walking motions of robots can be modeled as hybrid dy-namic systems consisting of continuous and discrete phaseof motion. Each step of a bipedal walking motion includestwo different continuous phases: the double support andthe single support phase. During the single support phase,the swing foot is lifted above ground, travels for a steplength into forward direction and is placed on the ground,while the other foot supports the body weight. Duringthe double support phase, the distance of the two feeton the ground is a step length into walking direction anda step width to the side. During regular gaits, discretephases with discontinuities in the velocities may occurwhen a foot hits the ground (this impact is assumed to befully inelastic). Ground contact is modeled as a unilateralconstraint (i.e. the floor can only push but not pull onthe robot foot), and slipping is avoided. For the timebeing, we are only interested in walking motions whichare periodic and symmetric. This allows us to reduce theproblem formulation to one single step, with a subsequentshift of sides in a discontinuity phase, to make the leftstep fit to the right step. For details on this formulation,see (Schultz and Mombaur, 2010).

Equations of motion With the previously chosen set ofcoordinates the equations of motions of the unconstraineddynamic multi-body system are expressed as a set of ODEsarranged in the following form:

[M (q, p) +M (p)m] q +N (q, q) q

+C (q) = F (q, q, p, τ)(1)

(q, q, q) are the joint coordinates and its first and secondorder derivatives. p are the model parameters. In analogyto the Gaussian principle of least constraint the terms ofthe inertia matrix of the robots structure and the motorsare expressed by M (q, p) and Mm(p) respectively:

M (q, p) =∑k

(Jxk )

TmkJ

xk + (Jω

k )T

ΘkJωk (2)

Mm =

R21I

m1 0 0 0 · · · 0

0. . . 0 0 · · · 0

0 0 R2NI

mN 0 · · · 0

0 0 0 0 · · · 0...

......

.... . . 0

0 0 0 0 0 0

, (3)

and the nonlinear effects by N(q, q, p):

N (q, q, p) =∑k

(Jxk )

TmkJ

xk + (Jω

k )TIkJ

ωk

−(

(Jωk )

T(IkJ

ωk q))Jωk .

(4)

Jxk , J

ωk are the Jacobians of translation and rotation of

link k of the robot. Imk is the rotor inertia of the actuationsystem and Rk the ratio of transmission respectively (incase of the free flyer joint these quantities are simply setto zero). Θ is the inertia tensor of link k in the globalsystem. The term F (q, q, p, τ) = (u, 0)T , u ∈ RN expressesthe torque that is produced by the actuation system on

the joint level. C (q) contains the effects of gravity andF (q, q, p, τ) the external forces.

In walking motions of humanoids, additional constraintsare required to describe the motions in all phases. Theequations of motion therefore take the following form of adifferential algebraic equation (DAE) of index 3, reducedto an index 1 DAE:

q = v (5)

v = a (6)

M (q, p) a+N (q, q) q + C (q)−GTλ = F (q, q, p, τ) (7)

Ga = −γ (8)

where, G = ∂∂q g (q) is the Jacobian of the contact con-

straint and γ = ∂∂q (G (q) q) q the second term of the second

order derivatives of the contact constraints. The Lagrangemultipliers λ can be shown to be identical to the (negative)contact forces, and together with the accelerations a, theyform the algebraic state variables of the DAE. Positionsq and velocities v are the differential state variables. ThisDAE has to satisfy the additional invariants coming fromindex reduction:

g (q) = 0 (9)

G (q) q = 0 (10)

Single and double contact phase are characterized bydifferent constraint matrices G(q) and different vectors γ.

The equation of motion have been formulated by meansof the dynamic model generator HuMAnS (Wieber et al.,2006) that is based on the recursive Newton-Euler Algo-rithm and the Composite Rigid Body Algorithm (Feath-erstone and Orin, 2000).

Phase order and transitions The order of phases is fixedin the model, but the individual durations are free. Phasechange times are implicitly determined by state dependentswitching functions: touch-down occurs when the heightof the swing foot, coming from above reaches zero height,and take-off occurs when the vertical ground reaction forcebecomes zero. At phase transition from single support todouble support, the swing foot may collide with the groundwhich results in discontinuities in the foot velocities andconsequently also of the joint velocities. The dynamics ofthe impact are modeled as inelastic and the velocities afterimpact can be computed as:[

M (q, p) −GT (q)G (q) 0

] [q+

Λ

]=

[M (q, p) q−

0

](11)

where, q− and q+ represents the joint velocities before andafter the impact respectively. Λ are the contact impulsionsandG (q) is the contact Jacobian at the contact. Transitionfrom double support to single support, i.e. lift off of theswing foot, is smooth.

3. GENERATION OF WALKING MOTIONS BYMEANS OF OPTIMAL CONTROL

3.1 Problem set-up

The generation of periodic symmetric walking motionscan be expressed as multiphase optimal control problemoptimizing different objective functions:

minx(·),u(·)p,ti∈M

r∑i=1

∫ ti

ti−1

Φi (x(t), u(t), p) dt+ Ψi (ti, x (ti) , p)

(12)

subject to x(t)− fi (t, x(t), u(t), p) = 0 i ∈M (13)

x(t+i)− hi

(x(t−i))

= 0 i ∈M (14)

req(x(0), x(T ), x

(t0), ..., x

(ts), T)

= 0 (15)

rineq(x(0), x(T ), x

(t0), ..., x

(ts), T)≥ 0 (16)

gi (x(t), u(t), p) ≥ 0 i ∈M (17)

The problem set-up depends on the differential system

states x = (q, q)T ∈ R2(N+6) the controls of the system u ∈

RN and the model parameters p that contain informationabout mechanical properties of the dynamic model (e.g.motor inertias, transmissions). Equation (12) expresses theobjective consisting of integral Lagrange-type terms Φi

and Mayer-type Ψi terms depending only on end values.M = {1, ..., r} contains the indices of all phases. tiexpresses the time at the end of each phase, t−i representsthe phase-end time before and t+i after the phase transitionrespectively. Without loss of generality it is assumed thatt0 = 0 holds. The simulation described in the previoussections has a single and a double support phase, henceM = {1, 2}.

Objective functions In this paper a collection of inter-esting objective functions are applied in the proposedoptimal control problem setup to generate different op-timal motions. In the following these objectives are brieflyexplained. In addition to the objective functions discussed,a small penalty term was always added to stabilize thehead in a human-like way, i.e.keep the head close to ahorizontal, forward direction.

Minimum torques squared: Minimum torque criteria areknown to produce smooth solutions with quite low energyconsumption. From a biological point of view (Schultz andMombaur, 2010) discussed that the joint torque is roughlyrelated to metabolic energy consumption of muscles. Tech-nically (M. et al., 2003) describe minimal torque criteriato minimize the armature power (heat) dissipation of theactuator, hence a fraction of energy consumption that maynot be used as mechanical work. The criterion can beformulated as

min Φtorque =

∫ T

0

N∑j=1

(ωjuj)2dt (18)

Maximum forward velocity: Humanoid robot are still muchslower than their human counterparts. The purpose ofthis objective function is to determine which maximumspeeds humanoid robots can achieve if all possibilities areexploited. This can be formulated as

max Ψforward vel =lStep

T(19)

3.1.1.1. Maximum postural stability: Stability of HRP-2and several other huamanoid robots is controlled by meansof the ZMP Vukobratovic and Stephanenko (1972), seeKajita et al. (2007). M. et al. (2003) proposes a criterionminimizing the euclidean distance from the barycentre ofcontact forces of each foot to a central reference point ofthe foot-fold, which will be investigated further:

min Φpost stab =

∫ T

0

∑e={Lf,Rf}

(pCOPe− pCentre)

2dt

(20)Maximum efficiency According to Garcia et al. (1998)efficiency for bipedal walker is defined as specific cost oftransport. The energy costs of transport are computedfollowing M. et al. (2003) as the actual mechanical poweroutput of each actuator scaled by step length:

max Φefficiency =

∫ T

0

N∑j=1

|qjuj |lstep

dt (21)

Minimum Joint Velocities Several objective functions leadto very high joint velocities. This motivates the reversecase to propose an optimization criteria that minimizesthe individual angular velocity of the joints for comparisonpurposes.

min Φjoint vel =

∫ T

0

N∑j=1

(ωj qj)2dt (22)

Constraints Equation (13) describes the dynamic be-haviour of the system at hand in the form of an ODE.For each phase the equation is to be substituted with thecorresponding DAE (5) - (8).(15) and(16) describe the pointwise coupled and decou-pled equality and inequality constraints. The coupled con-straints include e.g. periodicity constraints on the shiftedsystem, and the decoupled constraints describe take.offor touch-down conditions. (17) describes the continuousinequality constraints of the problem the simplest of whichare the bounds on all state and control variables, but alsostatic friction constraints on the foot fold, foot clearancefor obstacle avoidance, self collision constraints, unilateralconstraint and ZMP stability constraint.

3.2 Solution of optimal control problems

The optimal control problem of (12)-(17) is solved usingthe powerful optimal control software framework MUS-COD II that was written by Leineweber et al. (2003);Leineweber (1995) based on the work of Bock and Plitt(1984). This method is based on a direct approach thatdiscretizes the controls by means of suitable basis func-tions with local support on a grid of multiple intervals totransform the optimization problem of infinite dimension-ality to a finite dimension. This study uses a piecewiselinear control discretization. The multiple shooting thenparametrizes the differential states by means of new vari-ables that represent the initial conditions at each intervalto transform the boundary value problem into an initialvalue problem with continuity constraints at the intervaltransitions. For structural reasons, identical interval gridsare chosen for the control discrtization and the multipleshooting discretizatio. This results in a large nonlinear pro-gramming problem (NLP) that may efficiently be solvedby a specifically tailored sequential quadratic program-ming method. It is important to note that the MUSCODapproach also includes a treatment of he whole-body dy-namics by integrating at the user’s desired accuracy andcomputing sensitivities along the trajectory using powerfulintegrators.

All optimization methods need to be initiated with suit-able initial values for all optimization variables. The betterthe initial guess, the faster is usually convergence. Aninitial guess for walking trajectories on the complete dy-namic model of HRP-2 was derived from feasible walkingtrajectories based on the preview control pattern generatorof Kajita et al. (2003).

4. OPTIMIZATION RESULTS

The goal of this section is to show some of the walkingoptimization results generated for the HRP-2 model usingthe methods described above. The cases investigated inthis study include:

• optimization of all the objective functions discussedabove, and for all these:• considering ZMP constraints as constraints of the

optimization problem as well as ignoring these con-straints,• fixing foot placement after the step to the value of the

initial solution as well as leaving foot position free tobe determined by optimization.

The discussion of all these results is not possible in thispaper due to reasons of space, but some selected resultswill be discussed. Animations of all optimal solutions havebeen produced and will be made available at www.orb.uni-hd.de.

4.1 Optimization of different criteria - comparison of freeand fixed foot position

Several classical methods for the generation of walkingmotions require a priori fixing the position of the nextfoot contact on the ground and even the timing of thestep. It is an advantage of the optimization-based approachthat foot placement and phase times can be left free. Inthis paragraph, we compare optimal solutions with freeand fixed foot placement. In all cases, symmetry andperiodicity constraints are formulated on the solution.During optimization, the ZMP is restricted to lie withinthe area requested by the stabilizer of the real HRP-2.

The initial guess gait has a step length of 0.152 [m] and astep width of 0.144 [m]. Table 1 shows that optimizationchooses different step lengths (smaller in three cases andbigger and two cases), and smaller step widths for allobjective functions if these quantities are left free.

Table 1. Step length and step width in differentsimulations with free foot position

simulation step length step width

min. torque 0.129m 0.049mmax. forw. speed 0.171m 0.027mmin. joint speed 0.098m 0.0253mmax. postural stab. 0.158m 0.038mmax. Efficiency 0.185m 0.016m

Figure 1 (a) shows a sequence of snapshots of HRP-2performing an maximally efficient trajectory with free footplacement. Part (b) of the figure shows in a saggital planestick figure representation.

In Figure 2, we show comparisons of stick figure sequencesfor postural stability optimization and for maximum speed

(a) Maximization of postural stabilityfree foot position - fixed foot position

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Fig. 2. Comparison of free and fixed foot placement for twocriteria: maximization of postural stability (top) andmaximization of walking speed (bottom). Left stickfigures show initial solution, and right stick figuresshow optimized solutions.

optimization, in both cases with fixed and free foot posi-tioning.

From table 2, one may observe that for simulations thatneed to respect a fixed foot position, the mean forwardvelocity is lower and the duty factor is higher (onlyexception is the maximization of postural stability) thanfor the free foot position.

Table 2. Mean forward velocity and duty factorof simulations with fixed foot position (fif) and

free foot position (frf)

simulation mean forward speed duty factor

min. torque (fif) 0.102ms

70.4%

min. torque (frf) 0.138ms

59.0%

max. forw. speed (fif) 0.228ms

66.1%

max. forw. speed (frf) 0.363ms

55.1%

min. joint speed (fif) 0.100ms

61.1%

min. joint speed (frf) 0.079ms

50.9%

max. postural stab. (fif) 0.122ms

65.4%

max. postural stab. (frf) 0.174ms

62.4%

max. Efficiency (fif) 0.090ms

53.0%

max. Efficiency (frf) 0.163ms

52.3%

4.2 Collisions of feet with the ground

A common problem in robotic motion trajectory executionare collisions. In particular if the robot has a rigid struc-ture and high transmission ratios, impact collisions maymechanically overload parts of the transmission units orof the dynamic structure, causing serious deterioration orcomplete failure. Therefore particular interest was given

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Fig. 3. Knee joint angle and angular velocity of left (sup-port) and right (swing) leg - Pelvis position and orien-tation in Z distinguished between strict foot positionconstraint (fixed - solid line) and relaxed foot positionconstraint (free - dashed line). Different colors indi-cate different objective functions as explained in thetop left plot. Circles indicate plase limits, and squaresindicate the end of the simulation(i.e. the end of thelast phase).

on how the different objectives influence the magnitudeof the impulsion during collision. It was found that amongthe objectives, minimum joint speed and postural stabilityproduced the lowest magnitude of momentum. The highestmagnitude of momentum was found with the objective ofmaximal forward velocity. With a free foot position themagnitude of momentum tended to increase. Alternatively,one could also force the impacts to be zero (or very small)by means of an additional constraint, and then optimizeany of the criteria.

4.3 The influence of ZMP constraints on the trajectory

For the control of HRP-2 with its stabilizer turned onit is crucial to have walking trajectories that satisfy theZMP constraints. However, these stability criteria are verystrict and prevent any truly dynamic form of walking. It isoften noted that many humanoid robots walk in a nearlyhalf-sitting position, which is not quite alike humans. Itcan be shown that once the ZMP constraints are ignored,the humanoid walking motions are much more uprightthan with the ZMP constraint. This effect occurred for allobjective functions studied, and the average pelvis heightwas increased by 2 to 4 cm. Further investigation tried to

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Fig. 4. Influence of ZMP constraint on height of Pelvis (ori-gin of Pelvis Y coordinate is reached with stretchedknees) - circles display phase transitions and squaresthe end of the simulation - for different objectives(colored). Solid line represents results with relaxedZMP constraint and the dashed line represents resultswith strict ZMP constraint

reduce oscillations of the ZMP trajectory. It was foundthat among the objectives min. torque and especiallymaximal postural stability produced the lowest oscillationsin the ZMP trajectory particular in case of a flexible footposition.

5. CONCLUSIONS & OUTLOOK

Periodic walking motions have been successfully generatedfor the humanoid robot HRP-2 using an efficient optimalcontrol approach based on direct multiple shooting. Wehave studied the effect of different objective functions suchas torque, efficiency or angular rate minimization as wellas postural stability and walking speed maximization. Inaddition, we have studied the influence of different con-straints on the ZMP or the positioning of the foot. Theseoptimizations revealed interesting aspects about humanoidwalking motions, and show that relaxing constraints bringsseveral benefits. They show in particular that it might beworthwhile to investigate alternatives to the strict ZMPbased stability control which forces the robots to walk ina half-sitting position.

The solutions computed in this paper will be implementedon the real robot HRP-2 at LAAS in the very near future.In addition to the cyclic trajectories from this paper, thiswill also require the computation of initial and final halfcycles, that bring the robot from a static standing positioninto the cycle, and at the end from the cyclic motionagain back to its static posture. For an implementation onthe real robot, it might also be useful to compute furthercyclic solutions which are characterized by very small orzero impacts. In the future, we will also integrate a modelof the ankle elasticity, a more detailed foot model andfriction effects into the optimization model and perform

new computations for objective functions and constraintcombinations.

ACKNOWLEDGEMENTS

Financial support by the Heidelberg Graduate Schoolof Mathematical and Computational Methods for theSciences and by the European FP7 project ECHORD(GOP) is gratefully acknowledged.

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