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Walking Trajectory Generation for Humanoid Robots with Compliant Joints: Experimentation with COMAN Humanoid Zhibin Li, Nikos G. Tsagarakis, and Darwin G. Caldwell Abstract— This work introduces a walking pattern generator suitable for humanoids with inherent joint compliance. The proposed walking pattern generator computes the desired center of mass (COM) references on-line based on the COM state feedback. The position and velocity of the COM are the feedback variables, and the constraint ground reaction force (GRF), which is limited by the support polygon, is the control effort to drive the COM states to track the desired ones. The zero moment point (ZMP) is obtained naturally as a result of GRF interaction with robot feet. The proposed COM tracking scheme demands a lower bandwidth from the controller compared to the ZMP tracking schemes. Experimental data of the real compliant humanoid, such as ZMP, COM motion, and GRF are presented to demonstrate the validation of the proposed gait generation method.

I. I NTRODUCTION To date, most powered humanoid robots employ stiff actuated legs [1], [2], [3], [4], [5], [6], [7]. When it comes to bipedal walking, stiff legs suffer from many deficiencies. In fact, in terms of shock absorption, energy storage and reuse and dynamic characteristics [8], compliant legs can have significant advantages over the existing stiff systems. Our work aims to exploit the inherent compliance to obtain the foot-ground impact reduction, and to combine it with a suitable walking pattern generator to permit bipedal walking. For this purpose, the compliant humanoid ‘COMAN’ (COmpliant HuMANoid) was developed in IIT [9] as an experimental research platform, Fig. 1. The conventional bipedal walking control for powered bipeds with electric motors combines the zero moment point (ZMP) [10] and the use of stiff joints. This allowed the first dynamic walking gaits to be performed by the WL-10RD robot [11]. This control philosophy and hardware design formed the basis for many successful humanoids, such as the Honda robots P2, P3 [1], Asimo [2], AIST robots HRP2 to HRP4 [3], and the Waseda robot Wabian [4]. The ZMP concept can be used in different ways to generate dynamically stable gaits. One approach is to consider the ZMP as a control target, hence it relies on the preset ZMP reference. Numerical method can be used to solve the ZMP equation to obtain the COM trajectory [12] or alternatively the ‘Preview Control’ treats the bipedal walking as a ZMP tracking problem [13]. The walking patterns of these methods can be created either off-line or on-line. To cope with disturbances, This work is supported by the FP7 European project AMARSi (ICT248311) Zhibin Li, Nikos G. Tsagarakis, and Darwin G. Caldwell are with the Department of Advanced Robotics, Istituto Italiano di Tecnologia, via Morego 30, 16163, Genova, Italy [email protected], [email protected], [email protected]

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a real-time stabilizer such as the one in [3] is needed. Another method is to consider the ZMP as a control input. Sugihara [14] and Mitobe [15] manipulated the ZMP to control the reaction forces acting on the COM for bipedal gait generation. Our method builds on this approach. Although stiff robots can perform walking with the above methods, they are more susceptible to damage and instability during foot-ground impact. The impact issue can be partly addressed by using rubber bushes/soles [16], [17], or by a special foot mechanism design [18]; nevertheless, this issue can also be resolved by using intrinsically compliant actuators such as those developed in [19], [20], [21]. However, the compliance in the joints introduces oscillation modes, limits the control bandwidth of the system and deteriorates the joint tracking performance. In light of this, the conventional methods dedicated to the stiff humanoids becomes less feasible for the compliant ones. For the ‘COMAN’ robot, the gait generation control scheme should be tolerant to the joint tracking deterioration or oscillations. In contrast to stiff robots, this compliant robot has a lower control bandwidth, as the inherent compliant components act as mechanical low-pass filter reducing the control bandwidth. Since the ZMP signal is strongly related to COM acceleration, the spectrum of ZMP signal is high. Therefore the implementation of ZMP tracking scheme requires a high control bandwidth which can be hardly achieved by the compliant joints. On the other hand, the resultant motion such as the COM velocity and position are in a lower frequency spectrum. Moreover, the discussion in [22] points out that the COM states are the better candidate for the stability margin. Therefore, our work considers the COM states and not the ZMP as the variables to be controlled. This paper is organized as follows. Section II briefly provides the specification of ‘COMAN’ and presents the

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modeling and the control principle of the lateral motion. Section III describes the synchronization control for sagittal walking. In Section IV, the results of walking experiments of the real robot are analyzed. The influence of the compliance on the gait is briefly discussed in Section V. II. M ODELING AND C ONTROL OF L ATERAL M OTION The leg of the compliant humanoid robot ‘COMAN’ has six degree of freedoms (DOF) from which the knee and ankle flexion are powered by series elastic actuators with an approximate torsional stiffness of 120N m/rad. More details on robot design and its physical parameters, such as limb dimension, mass, inertia tensor, and compliant actuator design, can be found in [9]. In the gait pattern generator, the simulated GRF is instantaneously shifted from the support foot to the new support foot when the lateral COM position changes phase. The transition of GRF is omitted for simplicity. As Fig. 2 shows, the phase plane of the lateral motion (y, y) ˙ is divided into four quadrants as follows in a counterclockwise order: the 1st quadrant Q1 (y + , y˙ + ); the 2nd quadrant Q2 (y − , y˙ + ); the 3rd quadrant Q3 (y − , y˙ − ); the 4th quadrant Q4 (y + , y˙ − ). A. Formulation of Reference Velocity The main concept of the lateral motion control is to control the COM velocity to bound its kinetic energy permitting the COM to perform a cyclic motion between the two footholds without falling. In Fig. 2, the dark blue line depicts a desired stable COM state trajectory. The COM is trapped inside the discrete potential valleys [23] due to the change of support leg, as shown by the four sway motions of the robot in Fig. 2. We use the linear inverted pendulum model (LIPM) [24] to obtain an analytical solution for computing the desired velocity of the COM to maintain the stability margin, given the current state of the COM (y,y), ˙ the nominal center of pressure (COP) λy which is also the current foothold center and the desired maximum sway distance λs . In the following formulation, we consider only four different situations in Q1 and Q4 , and the similar formulation can be obtained for Q2 and Q3 . In Q1 , when y < λs , the kinetic energy is consumed because the GRF decelerates the COM and does

negative work. Therefore, a desired velocity Vyref exists with which the COM would rest exactly at λs given the current foothold λy and the constant height of the COM zc . During this deceleration period, the desired kinetic energy Ekd is totally converted to the mechanical work W which can be analytically computed according to LIPM. Ekd + W = 0 (1) g g (λ2 − y 2 ) − λy (λs − y) (2) W = 2zc s zc 1 ref 2 g λy g (Vy ) + (λ2s − y 2 ) − (λs − y) = 0 (3) 2 2z zc r c g g Vyref = 2 λy (λs − y) + (y 2 − λ2s ) (4) zc zc All the mechanical work and energy terms are normalized by mass. Note that the real kinetic energy Ek may be larger or smaller than the ideal Ekd . Based on this approach, the kinetic energy can be regulated by simply controlling the velocity. Similarly as a reversed motion in Q4 , when (y < λs ), a symmetric negative reference is formed. r g g (5) Vyref = − 2 λy (λs − y) + (y 2 − λ2s ) zc zc In Q1 , if Ek > Ekd , λs < y < λy could occur. In this case, the velocity needs to be reduced to prevent the COM from passing over the nominal support point λy . We use the following equations to compute a desired velocity Vyref with a negative sign which can force the COM to move backward to the desired sway point λs . g g E ∗ = λy (y − λs ) + (λ2 − y 2 ), y > λs (6) zc 2zc s √ (7) Vyref = − 2E ∗ In Q4 , when λs < y < λy , a passive motion produced by a fixed COP at λy would result in an excessive kinetic energy while entering the future q phase Q3 whose velocity overshoots the desired one − zgc (2λy λs − λ2s ). Hence, a positive reference velocity needs to be formed to decrease the excessive speed. √ Vyref = 2E ∗ (8) The on-line generation of velocity references based on COM state feedback can now be formulated for all phases. For Q1 and Q4 phases,  q  |2 zgc λy (λs − y) + zgc (y 2 − λ2s )|, sign( y) ˙     y ≤ λs q Vyref =  −sign(y) ˙ |2 zgc λy (λs − y) + zgc (y 2 − λ2s )|,     y > λs (9) For Q3 and Q2 phases,  q  sign(y) ˙ | zgc (λ2s − y 2 ) − 2 zgc λy (λs + y)|,     y ≥ −λs q ref Vy =  −sign(y) ˙ | zgc (λ2s − y 2 ) − 2 zgc λy (λs + y)|,     y < −λs (10)

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As shown in Fig. 3, four velocity references of different phases form a closed basin in phase portrait, which is the desired cyclic state for the lateral COM motion. In Q1 , when y < λs (case I or II), the control effort drives the state back to the desired one; when y > λs (case III), the negative reference velocity is used to generate a control force to pull the state to the next phase. In Q4 (case IV), when y > λs , the excessive velocity should be reduced. Therefore, the positive as the Vdes (Q1 )reference velocity, y comblue line shown in Fig. 3 (case IV), g can form a controld effort which decreases y. ˙ So in Q4 , even f Vdes (Q2 ) when the COM zisc leaving the support leg, the motion is still Vdes (Q3 )controlled rather than passive falling. The above references Q1 I to converge the real state Vdes (Q4 )continuouslyQ2enable the controller of the robot towards the desired state trajectory.

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, where and and are the proportional and derivative gains respectively. yref is the target rest position, g f here set as zero. The available acceleration is not arbitrary, zc because it is limited by the available GRF determined by the current support polygon as the robot cannot obtain any arbitrary force from the ground due to the limited size of the feet. Considering these constraints and given the nominal support center λy , the support polygon boundary limit ∆− y and ∆+ acceleration udy is   desired y , and the constant zc , the f f constrained as in (13).  ycom y−(λy +∆+  s y) min  g, uy < umin  uy = y zc uy , otherwise (13) udy =   max y−(λy −∆− y ) max uy = g, uy > uy zc Given the initial COM states (y(0), y(0)) ˙ and sampling time T , the next state of the COM is computed by numerical g integration based on the acceleration term udy in (13).  f zc

Fig. 4. Finite Finite State StateMachine Machine

C. Finite State Machine In Fig. 4, Q0 denotes the gait initialization starting from standing; Qf denotes the state when the robot returns to the standing equilibrium. Q1 to Q4 were defined previously. Equation (9) and (11) are used to initiate the gait. When the state approaches near the desired one, the state trajectory changes its direction to follow the cyclic orbit. The state switches to Q1 when this turning point is detected. The switch between Q1 to Q4 simply depends on the state (y, y) ˙ according to the definition. The control is continuously applied according to (9), (10), (11), and (13). When the stop command is triggered, the state machine starts to check the upcoming state (y = 0). Once the state (y ≈ 0) is detected, the robot slows down in two stages. Initially, the robot decreases the velocity in the next single support with half of the previous step length and accordingly half of the reference velocity according to (16). Secondly, it enters double support phase and further decelerates using control law (12) and (13). III. C ONTROL OF S AGITTAL M OTION Synchronization of sagittal and lateral motion is not an issue in the conventional ZMP tracking method, because the xzmp and yzmp are predefined according to the same timeline. In our approach, the reference velocity is generated online based on the real COM state feedback, and the lateral COM position is modulated by the proposed controllers. Therefore, the sagittal walking motion needs to be controlled y com and synchronized with the lateral one. A. Formulation of Sagittal Walking Parameters The LIPM model has an analytical solution for the time interval τ between an initial state (y0 , y˙ 0 ) and a future state (y1 , y˙ 1 ). Denote (λs , 0) the state  f at the maximum sway point, and substitute the position y = y0 = 0 in (4) to get the desired initial velocity s y˙ 0 in order ycomto stop at λs . Note τ is one quarter of a complete gait cycle Tcycle . r g g y˙ 0 = 2 λy λs − λ2s (14) zc zc τ = Tc ln

y0 + Tc y˙ 0 ; Tcycle = 4τ yf + Tc y˙ f

(15)

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The positions (xf (i), yf (i), zf (i)) of the support foot at discrete time iT remain the same as (xf (i − 1), yf (i − 1), zf (i − 1)). Let F : N 9 → R1 be a 5th order polynomial function with nine inputs and one output, we have

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p , where Tc = zc /g. Given the step length λx , the average forward velocity reference can be computed. 2λx (16) Vxref = Tcycle B. Control of Sagittal COM States Fig. 5 shows the spatial trajectories of the COM and feet. Equation (17) describes the control law, where x is the COM position and xf is the current support foot center. In the position control term, the position y of the real COM (black) is normalized by λs and projected to the sagittal plane, which gives the x COM reference (red), as shown in Fig. 5. This position control synchronizes the spatial motion, otherwise x COM position may drift. The velocity control term tracks the desired forward velocity. To initiate a gait, a small GRF is obtained with resultant horizontal acceleration approximately Vxref /τ to push the COM forward.  |y| λx λx  x  Kp1 [( − + xf ) − x]    λ 2 2 s   | {z }    position control    x  + Kv1 (Vxref − x) ˙ , when x ≤ xf ;    | {z }   velocity control ux = (17) λx |y| λx  x  K [( − + x ) − x]  f p1   2 λs 2  | {z }     position control    x  + Kv1 (Vxref − x) ˙ , when x > xf    | {z }   velocity control x x , where Kp1 and Kv1 are the proportional gains for the position and velocity control respectively. In the stop phase Qf , a PD position control is used as in x x ux = Kp2 (xref − x) − Kv2 x˙

(18)

x x , where Kp2 and Kv2 are the proportional and derivative gains, and xref is the middle x position between two support feet. Similarly to the lateral case, given the foothold center − xf and the boundary limits ∆+ x and ∆x , the desired output d ux can be formulated as in (19).  x−(λx +∆+ min x)  g, ux < umin  ux = x zc d ux , otherwise ux = (19)   max x−(λx −∆− max x) ux = g, u > u x x zc

, where X 8 and Z 8 are the 8 boundary constraints of the position, velocity, acceleration and time of the interpolation.  x ¨0 = 0, x˙ 0 = 0, x0 = xsj−1 , t0 = tsj−1 ,     x ¨f = 0, x˙ f = 0, xf = xsj−1 + 2λx ,    T tf = t0 + cycle 2 , for Q1 , Q3 ; (21) X8 = x ¨0 = x ¨sj , x˙ 0 = x˙ sj , x0 = xsj , t0 = tsj ,      x ¨ = 0, x˙ f = 0, xf = xsj + λx ,   f tf = tsj + (tsj − tsj−1 ), for Q2 , Q4 .  z¨0 = 0, z˙0 = 0, z0 = zankle , t0 = tsj−1 ,    z¨ = 0, z˙ = 0, z = z  f f f ankle + λz ,    T tf = t0 + cycle , for Q , Q3 ; 8 1 2 (22) Z = s s s s z ¨ = z ¨ , z ˙ = z ˙ , z = z  0 0 0 j j j , t0 = tj ,     z¨ = 0, z˙f = 0, zf = zankle   f tf = tsj + (tsj − tsj−1 ), for Q2 , Q4 . λz is the maximum swing foot height with respect to the ankle. The pattern generator stores the previous and the current foot position of the swing leg xsj−1 and xsj respectively when the lateral phase switches at tsj−1 and tsj . In Q1 and Q3 , the updated gait cycle Tcycle is used to estimate the first half of swing duration. In Q2 and Q4 , the first half of the real swing duration (tsj − tsj−1 ) is used to estimate the second half. This is because the real gait period might be changed by uncertainties, and this estimation provides a primary way to adjust. Based on the computed spatial COM and feet trajectory, the waist and feet orientation is imposed to solve the COM based inverse kinematics numerically, and obtain the joint position trajectories. IV. E XPERIMENTS To validate the gait generator, three walking experiments were carried out: (1) walk with the constant gait frequency and step length; (2) on-line modification of the gait frequency and step length; (3) walk with load disturbance. In all experiments the gait generator used the LIPM model without modeling the foot-ground collision for simplicity reason. The COM and swing foot positions are computed with respect to the stance foot based on absolute link positions (after the compliant elements). The ZMP position is defined with respect to the local coordinate originated at the ankle joint projection on the foot plate. The foot length of the robot is −0.065m to 0.135m and the foot width is −0.05 to 0.05m. In the first experiment, the robot walked with the gait frequency 0.77Hz and stopped in one step with the right foot. In Fig.6, the measured ZMP shows a consistent physics interpretation with the motion. To initiate the gait, the x ZMP shifted backward to obtain a GRF to push the COM forward; the y ZMP shifted to the right side to push the COM to

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the left, in order to enter the left single support phase. In stop phase, the x ZMP moved forward to decelerate forward speed; the y ZMP moved to the right acquiring a larger positive lateral force to stop the COM swaying to the right support foot. The ZMP trajectories were closely related to the COM acceleration which was computed by the controllers as in Section II and III without being explicitly planned. As also shown in Fig. 7, the force measured from the right foot exhibits no large force peaks which fulfills our expectations from the use of intrinsic passive compliance. In the second experiment, both the gait frequency and step length were modulated on-line by changing λs and λx . Fig.8 shows the desired and measured lateral COM position. A larger amplitude of sway motion corresponds to a lower gait frequency. The sway distance λs firstly reached the maximum (lowest frequency 0.74Hz) during 8 − 14s, then the minimum (highest frequency 0.93Hz) during 23 − 24s, and kept a medium value (0.85Hz) after 37s. The COM and feet trajectories are shown in Fig. 9. In Fig. 10, when the robot kept a steady gait frequency 0.85Hz during 55 − 80s, the measured walking velocity had more oscillations because the gait frequency was close to the range of natural frequencies of COMAN robot (0.8−1.2Hz), which indicated the effects of the joint elasticity. The step length was from 0.02m to −0.02m to walk backward, and then increased to 0.05m to walk forward. The presence of compliance reduced the joint tracking precision, hence the step width and length varied from step to step as shown in Fig. 11 and Fig. 12. The error observed was expected due to the effect of compliance. To sum up, a range of stable walking was achieved within [0.74, 0.93]Hz

gait frequencies and the step length within [−0.02, 0.05]m. Further improvement in the performance will require the real-time feedback of COM state estimation combined with an inertia measurement unit. In the third experiment, the robot was able to continue walking after adding a 1.7kg load, which corresponded to approximately 9.4% of the robot weight (18.1kg). Fig. 13 depicts the snapshots of the experiment. V. C ONCLUSION A walking trajectory generator is proposed for the humanoids with inherent joint elasticity. Despite the benefits of the impact force reduction due to passive compliance, new challenges arise for the robot control and gait generation. These include the deterioration of the joint tracking precision, the decrease of the control bandwidth and the oscillations caused by the elastic springs especially when gait frequency is close to the resonance. All these make the traditional ZMP tracking scheme more difficult to be applied. To address this, the introduced gait generators computes the desired COM trajectories based on the COM state feedback, which requires a lower control bandwidth. Hence, the proposed method is more suitable for robots with inherent elasticity. The gait generator was experimentally validated on the compliant humanoid ‘COMAN’. It was demonstrated that the compliant humanoid was able to perform stable gaits with varying step length and the gait frequency as well as to exhibit certain robustness to load variations. R EFERENCES [1] K. Hirai, M. Hirose, Y. Haikawa, and T. Takenaka, “The development of Honda humanoid robot,” IEEE International Conference on Robotics and Automation, pp. 1321–1326, 1998.

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