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Human-Like Walking: Optimal Motion of a Bipedal Robot With Toe-Rotation Motion David Tlalolini, Christine Chevallereau, and Yannick Aoustin
Abstract—Fast human walking includes a phase where the stance heel rises from the ground and the stance foot rotates about the stance toe. This phase where the biped becomes underactuated is not present during the walk of current humanoid robots. The objective of this study is to determine whether the introduction of this phase for a 3-D bipedal robot is useful to reduce the energy consumed in the walking. In order to study the efficiency of this new gait, two cyclic gaits are presented. The first cyclic motion is composed of successive single-support phases with a flat stance foot on the ground, and the stance foot does not rotate. The second cyclic motion is composed of single-support phases that include a subphase of rotation of the supporting foot about the toe. The single-support phases are separated by a double-support phase. For simplicity, this double-support phase is considered as instantaneous (passive impact). For these two gaits, optimal motions are designed by minimizing the torques cost. The given performances of actuators are taken into account. It is shown that, for a fast motion, a foot-rotation subphase is useful to reduce the cost criterion. These gaits are illustrated with simulation results. Index Terms—Biped robot, cyclic walking gait, parametric optimization, robot dynamics.
I. INTRODUCTION HE knowledge coming from the physiological, biomechanical, engineering, and robotic fields about human locomotion leads to the design and development of robotics structures as human-like as possible. These anthropomorphic structures are a useful way to understand the dynamic principles of human walking. Several anthropomorphic mechanical structures have been proposed, such as WABOT-1 introduced in 1973 [1], HONDA humanoid robot in 1996 [2], [3], ASIMO in 2000 [4], the anthropomorphic autonomous biped JOHNNIE [5], and the HRP series, 1, 1S, 2L, 2P, 2, and 3P [6]–[8]. However, most of these humanoid robots have been developed with 6-DoF per leg without toe joints. This limited number of degrees of freedom (DoF) cannot capture all the features of
T
Manuscript received May 29, 2009; revised September 18, 2009 and December 27, 2009; accepted December 31, 2009. Date of publication March 8, 2010; date of current version January 19, 2011. Recommended by Technical Editor V. N. Krovi. This paper was presented in part at the International Conference on Intelligent RObots and Systems (IROS 2008), Nice, France, September 22–26, 2008. D. Tlalolini is with the Laboratory of Physiology of Perception and Action, College de France, 75005 Paris, France (e-mail:
[email protected]). C. Chevallereau and Y. Aoustin are with the Institute of Research in Communication and Cybernetics of Nantes, University of Nantes, 44321 Nantes, France (e-mail:
[email protected]; yannick.aoustin@ irccyn.ec-nantes.fr). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMECH.2010.2042458
the human gait in a humanoid robot. In physiology, biomechanical, and from studies of human walking gait, authors are convinced that the role of the foot is essential to the equilibrium of the whole body. In fact, it serves as shock absorber and has a propulsive role [9], [10]. Thus, it becomes necessary to understand and reproduce the role of the foot on the humanoid robot to achieve nearly human-like gaits. In the robotic field, only a few papers have been devoted to the behavior of the foot in double-support phases and single-support phases. Most research works that include foot rotation have been focused on controller aspects [11]–[15] and energy efficiency [16], [17]. McGeer [18] and Kuo [19] are devoted to a very simple planar biped. They show the relevance of an impulse in the stance foot before the impact. Recently, Collins et al. [20] have built minimally actuated bipeds based on passive walking principals, including a toe-like propulsion. An optimized walking gait is proposed in [21] and [22] for a seven-link biped: in singlesupport phases, the stance foot is flat on the ground and the rotation about the feet appears in double-support phases only. The recent works of Nishiwaki et al. [23], Takao et al. [24], and Yoon et al. [25] show that the presence of toe joints allow to perform longer strides, climb higher steps, and walk at a higher speed. Hobbelen and Wisse [26] show that, however, to reduce the energy consumption, a passive spring can be implemented in the ankle with a stiffness that creates premature rise of the stance foot heel. Motivated by energy efficiency and versatility of locomotion, six optimal cyclic gaits have been analyzed for a planar biped robot in [27]. It is shown that, for fast motions, a rotation of the stance foot during the single-support phase is useful to reduce the energy consumed in the walking. In this paper, we extend our analysis of generation of optimal walking motions from the planar bipedal robots to the generation of optimal walking motions for an anthropomorphic 3-D biped robot with passive toe joints. The efforts are focused on the design of reference walking trajectories for a 3-D bipedal robot, including foot rotation during the single-support phase. The cyclic gaits under study consist of successive single-support phases separated by instantaneous double-support phases. Point out that it is not possible to obtain a finite-time double-support phase with a biped composed of rigid links after a passive impact with a rigid ground [28]. To obtain a finite-time double-support phase, it is necessary to impose an impactless gait. In practice, due to the elasticity of the sole, a double-support phase is likely to happen. It is shown in [27] that the benefit of the finitetime double-support phase on the energy cost is not so clear. The real benefit is to improve the stability of the gait [28]. The single-support phase may or may not be decomposed into two subphases: a flat-foot (fully actuated) subphase and a footrotation (underactuated) subphase. This study is different from
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TLALOLINI et al.: HUMAN-LIKE WALKING: OPTIMAL MOTION OF A BIPEDAL ROBOT WITH TOE-ROTATION MOTION
previous studies in two aspects: first, the kinematic structure consists of 14 motorized joints on the level of the hip–knee– ankle–foot kinematics chain. In contrast to previous bipedal structures, this structure has one additional joint to take into account the foot twist rotation. Second, in order to solve the underactuation problem and ensure the feasibility of the biped’s motion during the foot rotation subphase, we choose the joint path of the robot [29]–[32]. Therefore, a motion compatible with the dynamic model is deduced. For the compatible motions, the center of pressure (CoP ) remains strictly on the front limit of the stance foot to allow the foot to rotate. The synthesis of the reference walking trajectories is stated under the form of a constrained parameter optimization problem. The resolution of this problem is obtained by sequential quadratic programming (SQP) methods. The performance criterion, based on the square of the torques, is optimized in order to increase the autonomy of energy of the biped robot. Furthermore, some constraints, such as actuator performances and limits on the ground reaction force in single-support phases and at impacts, are taken into account. The outline of the paper is as follows. Section II presents the description of the two gaits studied, along with the geometric description and inverse dynamic model of the bipedal robot. The formulation of the optimization problems for optimal cyclic gait with and without foot rotation are defined in Section III. In Section IV, the various constraints and the cost function taken into account during the optimization processes are defined. The simulation results are presented in Section V. The conclusions and perspectives are given in Section VI.
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Fig. 1. Gait A. (a) Traditional movement of one step of the bipedal robots. (b) Kinds of contact and gait parameters.
II. ANALYSIS OF CYCLIC MOTION AND MODELING OF BIPEDAL ROBOT A. Analysis of Cyclic Motion For a normal human being, the description of walking can be confined to a cycle pattern of movement that is repeated over and over, step after step. With the assumptions that two steps (left plus right) make one cycle and that, in a normal gait, there is a natural symmetry between the left and right sides, we study the movement of a step only (the left step). 1) Gait A (Single Supports Separated by Impact): In this gait, the rear (left) foot and the front (right) foot are flat on the ground in the initial double support. Then the rear foot leaves the ground (beginning of the single-support phase), swinging through in preparation for the foot strike, while the contralateral foot, with flat contact on the ground, assures the equilibrium. At the end of the single-support phase, the swing foot touches the ground with flat foot, producing the final double-support phase, as shown in Fig. 1(a). In this gait, the double supports are instantaneous. During the single-support phase, the whole foot–ground contact assures an area of contact, which is maximum at all instants. Stability is maximized. This gait is appropriate to achieve,walking with slow speed. 2) Gait B (Single Supports With Foot Rolling Separated by Impact): In the gait shown in Fig. 2(a), the single-support phase is composed of two subphases: the stance foot is flat on the ground and then rotates around the toe. The swing foot touches
Fig. 2. Gait B. (a) Movement of one step with foot rolling during the singlesupport phase. (b) Subdivision of single-support phase, flat-foot (first), and foot-rotation (second) subphases.
the ground with flat foot. In this gait, the instantaneous double supports, initial and final, are such that the front foot is flat on the ground and the rear foot is in contact by its toe only. With the introduction of a subphase of foot rotation, an additional DoF appears. This DoF is not actuated. This means that the robot system becomes underactuated during this subphase. B. Modeling of the Bipedal Robot 1) Bipedal Robot: The general specifications of the bipedal robot in terms of size and DoF are based on an experimental device “Hydro¨ıd” built in the Laboratoire d’Ing´enierie des Syst`emes de Versailles1 [33]. Currently, only its locomotor system is realized. The torso with arms will be developed soon. The parameters of the robot are given in Table I, and the 1 This study is done at the design stage in order to check that the actuator is adequate to produce fast walking, but the device is not completely finish; thus, the simulated trajectory cannot be experimentally tested.
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TABLE I PARAMETERS OF THE BIPED USED IN THE OPTIMAL PROCESS.
characteristics of the robot have been chosen in order to be close to human being. The bipedal robot consists of seven rigid links connected by 14 motorized joints2 and two passive toe joints. This kinematic structure is designed to retain only basic human locomotion mobilities at the level of the hip–knee–ankle– foot kinematics chain. Each ankle of the bipedal robot consists of the pitch and the yaw axes (flexion/extension and abduction/adduction), and one additional roll axis to take into account the foot twist rotation. This twist rotation about the z-axis, unusual in humanoid robot structures, increases very significantly the degree of motion anthropomorphism [34]. The knees consist of the pitch axis (flexion/extension) and the hips consist of the roll, pitch, and yaw axes (rotation, flexion/extension, and abduction/adduction) to constitute a biped walking system of two 3-DoF ankles, two 1-DoF knees, and two 3-DoF hips. 2) Geometric Description of the Kinematic Chain: To define the geometric structure of the biped walking system, we used the parameterization proposed for manipulator robots. The bipedal robot is composed of N = 15 links connected by N − 1 revolute joints into an open serial kinematic chain. The stance foot is the base and the swing foot is the final link in the kinematic chain. The joint variables are denoted by q0 , . . . , q14 . These variables describe the shape and the orientation of the bipedal robot, as shown Fig. 3. The variable q0 denotes a relative rotation of the stance foot about its toe (z0 -axis). q0 is defined as the angle between the axes xs and x0 , measured along z0 . An inertial frame Rs (Os , xs , ys , zs ) is fixed at the stance toe. The origin Os of the inertial frame is fixed to the ground at the center of the foot tip, as Fig. 3 indicates. The frames have been assigned to 15 links according to the modified Denavit–Hartenberg procedure [35] (see Fig. 3). C. Dynamic Modeling In order to determine the impact model and the dynamics for the single-support phase with or without rolling of the foot, the efficient Newton–Euler formulation is chosen. When a simple planar robot is modeled as in [27], the dynamic model can be 2 Let us assume that the actuator dynamics is neglected in the optimization process. The feet mechanism of Hydro¨ıd considers one passive joint modeled on human toes with metatarsophalangeal joints. In this study, we consider the toe joint like the tip of the foot.
Fig. 3. Bipedal robot: multilink model, link frames, and coordinate convention. In foot-rotation subphase, q0 denotes a relative angle of the rotation of R 0 with respect to R s about z0 , and in flat-foot subphase, q0 = 0.
calculated via the Lagrange formalism. For humanoid robots with many joints, the Newton–Euler formalism is more efficient for the calculation cost point of view. Associated to our choice of parameterization and by performing a so-called forward– backward recursion [36], [37], the joint torques and ground reaction forces can be calculated. For each link, a forward recursion returns linear and angular velocities, accelerations, total forces, and moments, then a backward recursion gives the solution of the inverse dynamic problem by evaluating for each link joint torques and the ground reaction forces. 1) Single-Support Phase Model (Flat-Foot Subphase): During this subphase, the stance foot is assumed to remain in flat contact
TLALOLINI et al.: HUMAN-LIKE WALKING: OPTIMAL MOTION OF A BIPEDAL ROBOT WITH TOE-ROTATION MOTION
with the horizontal ground, i.e., there is no sliding motion, no takeoff and no rotation. Therefore, the configuration of the robot is described by 14 coordinates only. Let q ∈ R14 be the generalized coordinates, where q1 , . . . , q14 denote the relative angles of ¨ ∈ R14 are the velocity vector and the joints, and q˙ ∈ R14 and q the acceleration vector, respectively. Thus, the dynamic model is written in the form FG = NE(q, q, ˙ q ¨) (1) Γ where Γ ∈ R14 is the joint torque vector, FG = (fG , mG ) = (fG x , fG y , fG z , mG x , mG y , mG z ) is the wrench of ground reaction on the stance foot, and NE represents the Newton–Euler method. Note that this subphase exists under the assumption that the zero moment point (ZMP) remains inside the convex hull of the foot support region. a) ZMP condition: The ground reaction forces are known in the reference frame Rs . The ZMP is defined as the point of the sole such that the moment exerted by the ground is zero along the axes xs and ys . Thus, we have xZM P
−mG y mG x = and yZM P = . fG z fG z
(2)
The flat-foot phase exists only if the foot does not rotate; then, for a rectangular foot, the ground force and moment must satisfy −mG y −lp mG x lp ≤ and − Lp ≤ ≤ ≤0 2 fG z 2 fG z
(3)
where lp is the width and Lp is the length of the feet. 2) Single-Support Phase Model (Foot-Rotation Subphase): In this subphase, the stance heel of the robot rises from the ground and the robot begins to roll over the stance toe. Thus, the variable q0 is added. Let qr = (q0 ; q) ∈ R15 be the generalized ¨r ∈ R15 are the velocity vector coordinates, and q˙ r ∈ R15 and q and the acceleration vector, respectively. The inverse dynamic model is written as FG ¨r ). (4) = NE(qr , q˙ r , q Γ Since only 14 torques are applied and 15 variables qr describe the biped configuration, the biped can be underactuated. Note that for obtaining the inverse dynamic model, the forward recursive equations of the Newton–Euler algorithm are initialized by ω0 = q˙0 ys
ω˙ 0 = q¨0 ys and V0 = 0
V˙ 0 = 0
(5)
where V0 (ω0 ) and V˙ 0 (ω˙ 0 ) denote the linear (angular) velocities and accelerations of the stance foot (link 0). Since the stance foot rotates about its toe and there is no actuation between the stance toe and the ground, the CoP remains strictly on the front limit of the stance foot. In order to get a desired motion of the biped, which is compatible with the contact assumptions of the stance foot with the ground, the position of the ZMP is imposed to be equivalent to the position of the CoP . Therefore, the ground
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reaction moment about axis ys must be such that mG · ys = mG y = 0.
(6)
The reaction force FG on the left-hand side of (4) must be of the form: FG = (fG x , fG y , fG z , mG x , 0, mG z ) . Not any accelerations q ¨r on the right-hand side satisfies such a constraint. In other words, not any acceleration is admissible. a) Agular momentum about the toe: The angular momentum σy s about axis ys is defined by σy s = mtot (vcm x s Pcm z − s Pcm x vcm z ) + σcm y
(7) s
where mtot is the mass of the bipedal robot. The vectors Pcm and vcm are the position (in terms of reference frame Rs ) and the velocity of the center of mass, respectively. The variable σcm is the angular momentum about the center of mass. The velocity vcm and the angular momentum σcm are linear with respect to the vector of joint velocities, and depends on the robot configuration; thus, σy s can be written as σy s = J(qr )q˙ r
(8)
where J is a (1 × 15) matrix. Now, by using the angular momentum theorem and considering the rotational dynamic equilibrium of the biped as a rigid body, the rate of change of the angular momentum of the biped about its fixed toe is σ˙ y s = mtot g s Pcm x + mG y
(9)
because the external forces are gravity and ground reaction. During the foot-rotation subphase, the CoP remains under the toe line, and mG y = 0 [see (6)]; thus, the equation becomes σ˙ y s = mtot g s Pcm x .
(10)
A motion defined by qr , q˙ r , q ¨r , satisfying (10), i.e., satisfying (6), is compatible. Torque Γ and the ground reaction forces can be calculated from (4). b) ZMP condition: When the supporting foot is rotating about the toe, in order to maintain the balance in dynamic walking, the ZMP remains on the axis ys . Its position along this axis is bounded by lp , then mG x −lp lp ≤ ≤ . 2 fG z 2
(11)
In practice, the rotation of the foot will be produced by an articulated toe, since a foot is composed of two pieces, then the contact surface will never be reduced to a line. 3) Impact Modeling: An impact occurs at the end of the single-support phase when the swing foot touches the ground. This impact is assumed to be instantaneous and inelastic, i.e., the velocity of the swing foot touching the ground is zero after its impact. We assume that at the instant of the impact, the ground reaction forces are described by a Dirac delta function with intensity IF . Before the impact, the previous stance foot is motionless and does not remain on the ground after the impact. This phase can be modeled by a dynamical equation [37] and we have + V (12) = Δ(X)V− IF
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where X = (X0 , α0 , q) ∈ R20 is the generalized coordinates vector. V− and V+ denote the velocities before and after the impact.
according to the following equation: qi = Eqf where E14×14 is
III. GAIT OPTIMIZATION FOR THE CYCLIC MOTION A. Gait Without Foot Rotation 1) Constructing the Joint Trajectories: The biped is driven by 14 torques and its configuration is given in single-support phase by 14 coordinates q. To define the joint evolution, cubic spline functions [38] are used for constructing the joint trajectories. For each joint j(j = 1, . . . , 14), a cubic spline function has the form ⎧ ϕj,1 (t), if t1 ≤ t < t2 ⎪ ⎪ ⎨ ϕj,2 (t), if t2 ≤ t < t3 (13) qj (t) = .. .. ⎪ . . ⎪ ⎩ ϕj,n −1 (t), if tn −1 ≤ t ≤ tn where n is the number of selected knots. ϕj,1 (t), . . . , ϕj,n −1 (t) are polynomials of third order3 such that ϕj,k (t) =
3
aij,k (t − tk )i , for t ∈ [tk , tk +1 ],
i=0
k = 1, . . . , n − 1
(14)
with aij,k calculated so that the position, velocity, and acceleration are always continuous in t1 , . . . , tn . The cubic spline functions are uniquely defined by specifying an initial configuration qi , an initial velocity q˙ i (both at t = 0), a final configuration qf , and a final velocity q˙ f (both at t = T f ) in double support, with n − 2 intermediate configurations in single support and T f the duration of the single support. We use n = 3 and define only an intermediate configuration qint at t = T f /2. Consequently, the configurations will be defined by a small number of optimization parameters. 2) Optimization Parameters: In order to define the initial and final configurations of the biped legs, only eight independent variables are necessary because the two feet are flat on the ground. We use the twist motion of the swing foot denoted by ψf and its position (xf , yf ) in the horizontal plane, as well as the position of the trunk (xt , yt , zt ) and θt , φt the inclination in the sagittal plane, and rotation about zt -axis of the trunk. The inclination in the frontal plane is assumed to be null. The desired trajectory has the particularity of being periodic: two following steps (left plus right) must be identical and, more precisely, the legs will swap their roles from one step to the next. The condition of periodicity is used to define the trajectory only on one step to reduce the number of optimization parameters. In this way, we avoid to use two single-support models. The position of the robot is constant during the passive impact (touch down configuration), and since the legs swap their roles from one step to the next, the generalized coordinates must be relabeled 3 In
an optimization process, the adjustment of polynomial function parameters of high order can produce oscillatory movements unintended and disruptive. To avoid this, the cubic spline functions are used.
⎡
0 ⎢ 0 ⎢ E=⎢ 0 ⎣ 0 J2
0 0 0 −J3 0
0 0 J4 0 0
(15)
0 −J3 0 0 0
J2 0 0 0 0
⎤ ⎥ ⎥ ⎥ ⎦
(16)
with Jm being an antidiagonal (m × m) identity matrix. The final configuration qf is determined from the inverse kinematics solution of each leg. To obtain the initial velocity q˙ i , from (12), the 14 last rows of V+ are used q˙ i = EV(+7 : 2 0 ) .
(17)
Using (15) and (17), the cubic spline functions q(t) can be defined as function of T f , qint , q˙ f , xf , yf , ψf , xt , yt , zt , θt , φt . The optimal trajectory is defined by 37 parameters only. 3) Torques and Ground Reaction Forces: When function q(t) is chosen, joint velocities and accelerations can be deduced by the differentiation of the polynomial function. The inverse dynamic model (1) gives the torques and the ground reaction forces required to produce the motion. B. Gait With Foot Rotation 1) Flat-Foot Subphase: When a subphase with foot rotation is added, the optimization process is modified. The two subphases are separately described and the conditions of continuity in configuration and velocity between the subphases are taken into account. For each subphase, the final state of the biped is chosen to be defined from the optimization variables and the initial state is deduced from the continuity conditions. The flat-foot subphase is described as previously. The difference is just that the final configuration for this phase is not a double-support configuration, but a single support with flat stance foot configuration4 ; thus, 14 variables are used to describe this configuration qf , and 29 optimization variables define this subphase, T f , qf , and q˙ f . 2) Foot-Rotation Subphase: During the foot-rotation subphase, the biped is driven by 14 torques, and its configuration is given in single-support phase by 15 coordinates qr . Therefore, the biped is underactuated and its motion cannot be freely chosen. Studies of control of such an underactuated robot [30], [32] have shown that a geometric evolution of the robot qr (s) can be chosen. For a given function qr (s) within some limits, function s(t) corresponding to a motion compatible with the dynamic model can be deduced using (10). In the optimization process, the joint evolution is described by function qr (s). This method solves the underactuation problem and avoids the use of equality constraints as in [39]. This point is detailed in Section III-B3. 4 Since the final configuration of this subphase is a single support, an intermediate configuration q int is not necessary. Therefore, only third-order polynomial functions, defined by q i , q f , q˙ i , and q˙ f , are needed to describe the joint motion.
TLALOLINI et al.: HUMAN-LIKE WALKING: OPTIMAL MOTION OF A BIPEDAL ROBOT WITH TOE-ROTATION MOTION
We choose to define the evolution of the joint variables as a third-order polynomial of s, where s is a monotonic function from 0 to 1. qr (0) and qr (1) are the initial and final configurations of the foot-rotation support phase, respectively. Then qr j (s) =
3
aij (s)i ,
j = 0, . . . , 14 ∀s ∈ [0, 1]
i=0
where j is the joint number. The polynomial functions qr j (s), j = 0, . . . , 14, are uniquely defined by i
conditions must be satisfied: 2maxs (V (s)) I(0)2
s(0) ˙ > I(s) = 0,
(18)
f
dq r r qir , qfr , dq ds , and ds . The indexes i and f correspond to the initial (at s = 0) and final (at s = 1) states of the robot for this subphase, respectively. The velocity of the robot is such that
dqr s(s). ˙ (19) ds Since s˙ is not given, then only the direction of the joint velocity is provided, but not its amplitude. In fact, the initial state for this subphase is the final state for the flat-foot subphase qir = ((q0f = 0); qf ); thus, it is known by the 29 optimization parameters for the flat-foot subphase. The initial velocity of the robot is known, q˙ ir = ((q˙0f = 0); q˙ f ). The i r ˙ is known, and this term initial vector dq ds can be deduced if s(0) s(0) ˙ will be an optimization variable. The final configuration is a double support one with only one foot flat on the ground; thus, nine coordinates are used to define this configuration: xf , yf , ψf , xt , yt , zt , θt , φt , q0f . The joint path qr (s) during the foot-rotation subphase can be f r calculated with 25 optimization variables: 9 for qfr , 15 for dq ds , and s(0). ˙ 3) From Joint Trajectories to Joint Motions: The joint evolution is given as qr (s), but since the robot is underactuated, function s must be such that the robot motion satisfies (10). The angular momentum is defined by (8). Since qr is a function of ˙ this angular momentum can be s and q˙ r (s) is proportional to s, rewritten as q˙ r (s) =
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for 0 ≤ s ≤ 1
in order that s˙ > 0,
for 0 ≤ s ≤ 1.
σ˙ y s = mtot g s Pcm x (qr (s))
(20)
r where I(s) = J(qr ) dq ds . These two equations can be combined to have, for 0 ≤ s ≤ 1 [30],
1 1 I(0)2 s(0) ˙ 2 = I(s)2 s(s) ˙ 2 + V (s) 2 2 s V (s) = −mg I(ξ)(xg (ξ))dξ.
(21)
0
Since function I(s) and V (s) can be calculated for any given function qr (s), it follows that the initial value s(0) ˙ allows to define the function s˙ completely by I(0)2 s(0) ˙ 2 − 2V (s) s(s) ˙ = . (22) I(s)2 The polynomial functions qr (s) are defined with the assumption that s is a well-defined increasing function; thus, the following
(24)
These constraints are taken into account in the optimization process. At the end of the foot-rotation subphase, the value of s˙ can be deduced from (22) and the velocity of the robot at the end of the foot-rotation subphase is I(0)2 s(0) ˙ 2 − 2V (1) dqfr f . (25) q˙ r = I(1)2 ds Since the impact occurs in configuration qfr with velocity the initial state of the robot for the flat-foot subphase can be deduced from equations (15) and (17). The duration of the foot-rotation phase is not a direct optimization variable, it is the result of integration of the function s(s) ˙ that defines at which time s = 1, i.e., 1 1 r ds. (26) T = 0 s˙ q˙ fr ,
4) Torques and the Ground Reaction Forces: For the footrotation subphase, when function qr (s) is chosen, s(s) ˙ can be calculated by (22). Therefore, the joint velocity is calculated by (19) and the joint acceleration can be written as d2 qr 2 dqr s¨(s). (27) s˙ + ds2 ds In order to deduce s¨, we use the linearity of the torque Γ with respect to acceleration s¨5 and the fact that the reaction moment about the toe mG y is zero. Therefore, q ¨r (s) =
¯ = u¨ Γ s+v
˙ σy s = I(s)s(s)
(23)
where, from (4), we have ¯ = FG = NE(qr (s), q˙ r (s), q Γ ¨r (s)). Γ
(28)
(29)
¯ is calculated for s¨ = 0 Using the Newton–Euler algorithm, Γ and s¨ = 1; these vectors are denoted by Γ¯0 and Γ¯1 , respectively. For any s¨, we have ¯ = (Γ ¯1 − Γ ¯ 0 )¨ ¯ 0 = u¨ Γ s+Γ s+v
(30)
¯1 − Γ ¯ 0 ). ¯ 0 and u = (Γ where v = Γ Since the ground reaction moment about the axis ys is such that mG y is zero, s¨ is easily obtained from the fifth row of (30) as v (31) s¨ = − 5 . u5 5 When the dynamic model is written under the Lagrange formalism, the linearity between the vector of acceleration and the vector of torque for given values q and q˙ is clear.
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Index 5 denotes the fifth component of a vector. Then the torques required to produce the motion are computed as v5 Γ = u(7:20) − (32) + v(7:20) u5 and the ground reaction forces as v5 fG FG = = u(1:6) − + v(1:6) . mG u5
(33)
The index (7 : 20) denotes the 14 last components and the index (1 : 6) denotes the first six components.
the criterion, and g(P ) the inequality constraints to satisfy. The optimization problem can be formally stated as Minimize JΓ (P ) . (35) subject to g(P ) ≤ 0 This nonlinear constrained optimization problem is solved numerically using the fmincon function of the MATLAB optimization package. The main parameters for this humanoid robot, used in the presented study, are given in Table I.
V. SIMULATION RESULTS
IV. OPTIMAL WALK A. Constraints and Limitations The objective of this study is to define a feasible optimal trajectory for a given robot with given actuators. In order to insure that such trajectory is possible, the constraints on the maximum torques Γm ax and velocities q˙m ax of the actuators are considered. Some others limitations have to be considered as follows. 1) Constraints on ground reaction forces: Vertical components of the ground reaction must be positive; the ground reaction must be inside the friction cone. 2) Constraints on impact model: Vertical components of the impulsive ground reaction must be positive; the impulsive ground reaction must be inside the friction cone. 3) The swing foot must not touch the ground before the prescribed end of the single-support phase. 4) Constraints on the ZMP for the flat-foot and foot-rotation subphases defined by (3) and (11). 5) Constraints on the monotony of function s defined by (23). All these constraints can be easily written as inequality conditions as they can be expressed as functions of qr (s), q˙ r (s), and q ¨r (s).
In all this section, the gait described as A is without rotation of the stance foot. Moreover, there is a rotation of the foot for the gait named B. A. Walk Without Foot Rotation The chosen motion rate for the 3-D bipedal robot is 1 m/s (3.6 km/h). For this motion, the optimal walk has the following characteristics: for one step, the duration T f is 0.39 s and the step length is 0.39 m. The value of the torque cost criterion JΓ is 7090 N2 ·m·s. The flat foot presents a twist rotation of 6.5◦ . The walk without foot rotation is denoted as gait A in Fig. 4(a). Fig. 5 shows that the constraint on nonsliding and no takeoff are satisfied. In the optimization process, the Coulomb friction coefficient μ is chosen as 0.75. The results in Fig. 5 show that the motion will be realized without sliding for friction coefficient values between 0.033 and 0.23 for both gaits. The average vertical reaction force is 400.5 N, which is coherent with the weight of the biped whose mass equals 40.75 kg. For this gait, the evolution of ZMP is illustrated in Fig. 6(a). This trajectory is the result of the optimization process. The ZMP remains within the foot area, as prescribed by (3). B. Walk With Foot Rotation
B. Cost Function In electrical motors and for a cycle of walk, most part of the energy consumption is due to the loss by Joule effect, neglecting the friction. Thus, the optimized criterion is proportional to this loss of energy. It is defined as the integral of the norm of the torque for a displacement of one meter: 1 JΓ = d
Tf
Γ(t) Γ(t)dt + 0
0
1
1 Γ(s) Γ(s) ds s˙
(34)
where T f is the duration of the flat-foot subphase of one half step and d = xf is the step length. The total duration of one step is defined by T = T f + T r , with T r obtained from (26). C. Optimization Problem The objective of this optimization procedure will be to select a feasible solution by minimizing the criterion (34) for a given motion speed of the robot by satisfying the constraints associated to a walking gait. Let P be the optimization parameters, JΓ (P )
During the optimization process, the angle of rotation of the foot at the end of the rotation subphase is constrained between 5◦ and 45◦ . The chosen motion velocity for this simulation is 1 m/s (3.6 km/h). The optimal walk has the following characteristics: for one step, the duration of the flat-foot and foot-rotation subphases are T f = 0.18 s and T r = 0.24 s, respectively. The step length is 0.42 m. The value of the torque cost criterion JΓ is 3356.6 N2 ·m·s, which is much lower than the cost of the previous motion without foot-rotation subphase. During the evolution of this motion, the foot in rotation finishes with an angle equal to 36.9◦ and a twist rotation equal to 6.9◦ . Fig. 4(b) presents the stick diagram of one step of an optimal walk with rolling of the stance foot. This optimal motion regroups the flat-foot and foot-rotation subphases. The hips have less vertical motions for gait B than for the gait A. Fig. 5(b) shows that the constraints on nonsliding and no takeoff are satisfied during both subphases. The average vertical reaction force is 398.9 N, which is coherent with the weight of the biped robot. For this gait, the evolution of the ZMP is illustrated in Fig. 6(b). The trajectory remains within the foot area during the flat-foot
TLALOLINI et al.: HUMAN-LIKE WALKING: OPTIMAL MOTION OF A BIPEDAL ROBOT WITH TOE-ROTATION MOTION
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Fig. 5. Ground reaction force during the single-support phase. The dash dotted and solid curves depict the vertical and the horizontal components of the ground reaction forces, respectively. (a) Gait A. (b) Gait B.
Fig. 4. Stick diagram of gaits A and B. (a) Walking at 1 m/s without rolling of the stance foot. (b) Walking at 1 m/s with rolling of the stance foot during the single-support phase.
Fig. 6. Location of ZMP. (a) Evolution of ZMP for the gait A. (b) Evolution of ZMP for the gait B, and during the foot-rotation subphase, the ZMP is located at the stance toe, and during the flat-foot subphase, it is within the stance foot area.
subphase and at the toe during the foot-rotation subphase. The stability of the bipedal robot is guaranteed by satisfying (3) and (11) in flat-foot and foot-rotation subphases, respectively. Fig. 7 illustrates the evolution of torques of each joint of stance (right) leg and swing (left) leg as function of time. It is clear that the torques change considerably according to the studied gaits. The torques needed to performed a motion with and without foot rotation at speed of 1 m/s are below the maximum value. The torques of the swing leg is less high than the torques of the stance leg, while the highest torques concern the hip and the knee. With the introduction of a foot-rotation subphase during the single-support phase, the torque of the stance hip and the stance knee is considerably reduced. Although the torque of stance ankle joints becomes large when the foot-rotation subphase is used, the actuator constraints = defined as maximum torques produced by the actuators, Γmax 1 max = 78 N·m, and Γ = 157 N·m, were satisfied 115 N·m, Γmax 2 3 with a large margin during all the motion. For example, the maximum torques required by the ankle joint to generate a
Fig. 7. Joint torques: evolutions of torques corresponding to gait A and gait B during the motion are blue lines and red lines, respectively.
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Fig. 8. Joint angles: evolutions of joint angles corresponding to gait A and gait B during the motion are blue lines and red lines, respectively.
Fig. 10. Comparison of gait parameters: (a) step length, (b) step duration, and (c) angle of the toe joint.
Fig. 9. Cost criteria: values corresponding to gait A and gait B are circles and squares, respectively. The values specified by stars correspond to gait B, avoiding the impulsive reaction force.
motion at 1 m/s are Γ1 = 6.79 N·m, Γ2 = 13.77 N·m, and Γ3 = 53.30 N·m. The torque cost of the stance ankle decreases with increasing speed. On the other hand, the torques corresponding to the motions out of the sagittal plane, Γ1 , Γ2 , Γ7 , are less high, but Γ6 is one of the most important torques as Γ5 . The foot rotation reduces Γ5 and Γ6 simultaneously. The trajectories of the optimized joint angles for the gaits A and B are shown in Fig. 8. According to the figure, the constraints of the maximum joint angles were satisfied. C. Walk at Various Velocities In Fig. 9, the evolution of the cost criteria is drawn as a function of the motion speed for gait A and gait B. For slow motions [the average rate is less than 0.6 m/s (1.86 km/h)], a gait composed of flat-foot single-support phases separated by instantaneous double supports is the most efficient with respect to the studied criterion. Relatively to this criterion, for walking rate
higher than 0.6 m/s, a gait with foot rotation is preferable. With the same actuators, the maximum motion rate is dramatically increased by the introduction of the foot-rotation subphase. In practice, to limit the mechanical stress in the biped, the impulsive force must be limited; thus, impactless motion can be useful. This impactless motion can also help to pose the foot flat on the ground. Thus, impactless motions with foot rotation have been defined by adding a constraint to zeroed the velocity of the swing foot at the landing. In Fig. 9, the cost criterion is drawn as a function of the walking rate for such motions using stars. The difference between the stars and squares values shows that the increase of the criterion cost, due to the avoidance of the impulsive reaction force, is very low for walking rates below 1.9 m/s. In Fig. 10, the evolution of some gait parameters is illustrated. In a motion with foot rotation, the step length and the toe joint angle increase, while the step duration decreases with increasing rate. A significant toe joint angle allows a larger step length and hence faster walking; the step duration is close to a constant for walking rates above 0.8 m/s. Let us note that the optimization algorithms used may converge to local minima. However, for all our numerical tests, we tried different initial conditions for the optimization parameter and retained the best results. VI. CONCLUSION In this paper, a solution to define an optimal walking motion with flat-foot and foot-rotation subphases has been proposed.
TLALOLINI et al.: HUMAN-LIKE WALKING: OPTIMAL MOTION OF A BIPEDAL ROBOT WITH TOE-ROTATION MOTION
Since the desired motion is based on the solution of an optimal problem, and in order to use classical algebraic optimization techniques, the optimal trajectory is defined by a reasonable number of parameters. Some inequality constraints such as the limits on torques and velocities, the condition of no takeoff and no sliding during motion and impact, and some limits on the motion of the free leg are taken into account. The desired walking gait was assumed to consist of single supports and instantaneous double supports defined by passive impacts. The single-support phase can include a foot-rotation subphase or not. It is shown that this subphase allows to reduce the cost criterion for fast motions. The torques were computed using the inverse dynamic model. This model was obtained with the recursive Newton–Euler algorithm. The main contribution of the paper is to extend the optimal trajectories generation of the planar biped robots [17] to a 3-D biped robot with rotation of the feet on the ground to achieve an optimal fast motion. To evaluate the proposed optimal motion, we used the physical parameters of Hydro¨ıd. The experimental implementation on Hydro¨ıd will be performed in future works. Moreover, a future study will focus on the introduction of a finite-time double support phase to achieve a walking motion, where the back foot rotates around its toe and the front foot rotates around its heel until the foot is flat on the ground. This phase is probably very important for the stability of the gait. REFERENCES [1] I. Kato, S. Ohteru, H. Kobayashi, K. Shirai, and A. Uchiyama, “Information-power machine with senses and limbs,” in Proc. 1st CISMIFToMM Symp. Theory Pract. Robots Manipulation, 1974, pp. 11–24. [2] K. Hirai, “Current and future perspective of HONDA humanoid robot,” in Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst., 1997, pp. 500– 508. [3] K. Hirai, M. Hirose, T. Takenaka, Y. Haikawa, and T. Takenaka, “The development of honda humanoid robot,” in Proc. IEEE ICRA, 1998, pp. 1321–1326. [4] M. Hirose, Y. Haikawa, Y. Haikawa, and T. Takenaka, “Development of humanoid robot ASIMO,” presented at the Int. Conf. Intell. Robots Syst., Workshop 2, Maui, HI, 2001. [5] M. Gienger, K. L¨offler, and F. Pfeiffer, “Towards the design of biped jogging robot,” in Proc. IEEE ICRA, 2001, pp. 4140–4145. [6] K. Yokoyama, J. Maeda, T. Isozumi, and K. Kaneko, “Application of humanoid robots for cooperative tasks in the outdoors,” presented at the Int. Conf. Intell. Robots Syst., Workshop 2, Maui, HI, 2001. [7] H. Inoue, S. Tachi, Y. Nakmura, K. Hirai, N. Ohyu, S. Hirai, K. Tanie, K. Yokoi, and H. Hiru, “Overview of humanoid robotics project of METI,” in Proc. 32nd Int. Symp. Robot., 2001, pp. 1478–1482. [8] K. Akachi, K. Kaneko, N. Kanehira, S. Ota, G. Miyamori, M. Hirata, S. Kajita, and F. Kanehiro, “Development of humanoid robot HRP-3,” in Proc. IEEE-RAS Int. Conf. Humanoid Robots, 2005, pp. 50–55. [9] S. Wearing, S. Urry, and P. Perlman, “Sagittal plane motion of the human arch during gait,” Foot Ankle, vol. 19, no. 11, pp. 738–742, 1998. [10] Y. Blanc, C. Balmer, T. Landis, and F. Vingerhoerts, “Temporal parameters and patterns of the foot roll over during walking: Normative data for healthy adults,” Gait Posture, vol. 10, no. 2, pp. 97–108, 1999. [11] C.-L. Shih, “Ascending and descending stairs for a biped robot,” IEEE Trans. Syst., Man, Cybern., vol. 29, no. 3, pp. 255–268, May 1992. [12] K. Yi, “Walking of a biped robot with compliant ankle joints: Implementation with kubca,” in Proc. 39th IEEE Conf. Decis. Control, Sydney, Australia, Dec. 2000, pp. 4809–4814. [13] T. Takahashi and A. Kawamura, “Posture control for biped robot walk with foot toe and sole,” in Proc. 27th Annu. Conf. IEEE Ind. Electron. Soc., 2001, pp. 329–334. [14] F. Silva and J. T. Machado, “Gait analysis of a human walker wearing robot feet as shoes,” in Proc. IEEE Int. Conf. Robot. Autom., Seoul, Korea, 2001, pp. 4122–4127.
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David Tlalolini received the M.S. degree in automatic control from the Center for Research and Advanced Studies, National Polytechnic Institute, Mexico City, Mexico, in 2003, and the Ph.D. degree in control and robotics from the Institute of Research in Communication and Cybernetics of Nantes (IRCCyN), University of Nantes, Nantes, France, in 2008. Currently, he is with the Laboratory of Physiology of Perception and Action, College de France, Paris, France. His research interests include modeling and control of multibody mechanical systems, particularly walking robots.
Christine Chevallereau graduated and received the Ph.D. degree in control and robotics from the Ecole Nationale Sup´erieure de M´ecanique, Nantes, France, in 1985 and 1988, respectively. Since 1989, she has been with the Centre National de la Recherche Scientifique, Institute of Research in Communication and Cybernetics of Nantes, University of Nantes, Nantes. Her research interests include modeling and control of robots, especially control of robot manipulators and legged robots.
Yannick Aoustin received the Ph.D. degree and the Habilitation to supervise research from the University of Nantes, Nantes, France, in 1989 and 2006, respectively. Currently, he is an Associate Professor at the University of Nantes, where he is a member of the Institute of Research in Communication and Cybernetics of Nantes. His research interests include flexible robots, underactuated systems, and walking robots.
