Dynamic Walking in a Humanoid Robot Based on a 3D Actuated Dual ...

Dynamic Walking in a Humanoid Robot Based on a 3D Actuated Dual-SLIP Model Yiping Liu, Patrick M. Wensing, David E. Orin, and Yuan F. Zheng Abstract— This paper presents a method for the generation of dynamic walking gaits with a 3D Dual-SLIP model and its application to a simulated Hubo+ based humanoid. Previous approaches with the Dual-SLIP model have only focused on the planar case, wherein self-stable gaits can be found. When extended to 3D here, this model has not been found to exhibit self-stable gaits, requiring new methods for gait optimization and control. By taking advantage of a newly discovered symmetry condition for the Dual-SLIP model, this work proposes a quarter period (half step) optimization process to find periodic walking gaits in 3D. An LQR controller is developed to regulate the state of the model at leg midstance (MS) based on its return map dynamics. The Dual-SLIP model is extended by introducing a bio-inspired leg length actuation scheme in order to describe high-speed walking gaits (up to 2 m/s for human-compatible parameters). Finally, the CoM trajectory and footstep positions from the 3D Dual-SLIP are used as a reference in a task-space controller with a Hubo+ based humanoid model. By tracking these references, the methods successfully produce human-like dynamic walking gaits in simulation which are robust to disturbances. The wholebody control system for walking can handle uneven terrain with variation up to 10% of its leg length. This represents the first humanoid dynamic walking approach based on a 3D Dual-SLIP model.

I. I NTRODUCTION Humanoids have received more and more attention in robotics research through recent years. Control of dynamic humanoid walking is one of the most important topics since it is the first step towards many potential humanoid applications. Humanoid walking continues to be an active area of research, due to many challenges which face any whole-body humanoid control strategy for locomotion. Most notably, humanoid robots usually have a high number of degrees of freedom, experience intermittent unilateral contacts with the ground, and can become underactuated during dynamic locomotion. To handle these complexities, considerable research effort has focused on using simple models as “templates” [1] for locomotion. By focusing on the key features of a complex whole-body motion (such as the dynamic walk shown in Fig. 1), the application of Y. Liu is a PhD student in the Department of Electrical and Computer Engineering, The Ohio State University: [email protected] P. M. Wensing is a Postdoctoral Associate in the Department of Mechanical Engineering, Massachusetts Institute of Technology:

[email protected] D. E. Orin is a Professor Emeritus in the Department of Electrical and Computer Engineering, The Ohio State University: [email protected] Y. F. Zheng is a Professor in the Department of Electrical and Computer Engineering, The Ohio State University: [email protected]

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Fig. 1: Dynamic walking is developed in simulation for a Hubo+ based humanoid robot model through the use of an actuated 3D Dual-SLIP model. simple models can address many of the challenges which face locomotion control. A well-known simple model developed for walking is the Linear Inverted Pendulum Model (LIPM) [2]. The dynamic equations of motion for the LIPM are linear and decoupled, making it amenable to analytical methods. Many successful walking control strategies based on LIPM have been developed since it was first introduced. One important example is the ZMP preview control approach (with either prescribed or automatic footstep placement) [3], [4], [5], which tends to generate a rather quasi-static gait. In related work, Pratt et al. [6] proposed the concept of a “capture point” for push recovery and showed its computation with the LIPM. Initially, the capture point was meant to be a fixed point on the ground where the humanoid could step to recover from a near fall. More recently, however, a dynamically changing capture point (or Divergent-Component of Motion point, DCM) has been used to develop dynamic walking gaits [7], [8]. It has been found that the CoM always converges to DCM, thus the control of CoM dynamics can be transformed to control of the DCM dynamics. In more recent developments with 3D extensions of the DCM concept, DCM-based walking control can adapt to non-constant CoM and floor height [9]. Another less-studied template model for walking is the Dual Spring-Loaded Inverted Pendulum (Dual-SLIP, or Bipedal-SLIP). The Dual-SLIP is a variant of the wellknown SLIP model, which has been applied in humanoids as a template for running [10] and jumping [11]. While

1 Hubo+ has a leg length of 0.6 m. Many results in this paper, however, employ a leg length of 1 m in order to provide more direct comparison of the Dual-SLIP results to human walking. Also, the joints of the simulated Hubo+ like humanoid are torque controlled in this work.

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compliance is not usually considered an essential component in a walking template, Geyer et al. [12] found that the 2D Dual-SLIP model can reproduce human walking dynamics with better fidelity than other simple models such as the LIPM. More specifically, the Dual-SLIP model can produce CoM vertical oscillations and ground reaction force patterns similar to those found in humans. The most useful feature of the Dual-SLIP model for robotics, however, is that it seamlessly addresses the double support period of walking with no extra mechanism required. LIPM-based walking controllers either completely ignore the double support phase [4], [9], or manually designate a double support period as in [13]. In addition, it has been observed that the CoM lateral sway during LIPM walking ([4], [13], [14]) is usually significantly larger than the CoM lateral sway in human walking [15]. For approximately the same step width (≈ 20 cm) and leg length (≈ 0.9 m), the CoM lateral sway from LIPM-based walking control can be greater than 10 cm (for example, Fig. 15 in [14]), while it’s closer to 4 cm in humans. For energetic reasons it is generally desirable to avoid the unnecessary body movements that would lead to excess lateral CoM sway. As shown later in this paper, the CoM lateral sway during Dual-SLIP based walking is comparable to human data. With such motivations, this paper develops an alternative dynamic walking controller for humanoids based on the 3D Dual-SLIP model. The major contributions of this paper are: (1) We extended Geyer’s 2D Dual-SLIP model to 3D and propose a new way to find periodic walking gaits. We highlight a newly discovered symmetry condition for the 3D Dual-SLIP model that is used to simplify optimization. (2) Since the 3D Dual-SLIP gaits are not found to be self-stable, we develop a discrete-time infinite-horizon LQR controller to regulate the state of the model. Through this approach, the controller is robust to disturbances. (3) It has also been found during the study that the regular energy-conservative 3D Dual-SLIP model is not capable of describing forward walking at speeds faster than 1.4 m/s (assuming a leg length of 1 m). To address this limitation, we propose a variant of the regular Dual-SLIP model, the Actuated DualSLIP, to address high-speed walking (2 m/s assuming a 1 m leg length). (4) The control solution developed around the simple template has been rescaled and applied to a realistic humanoid model based on Hubo+ [16]1 (see Fig. 1) through the application of whole-body task-space control. This approach is shown to be effective to produce a humanlike dynamic walking gait and enables push recovery through automatic footstep re-planning. The humanoid model can also adapt to terrain unevenness (bumps, shallow pits) up to 1/10 of its leg length during walking. The remainder of the paper is organized as follows: Section II-A and II-B present the 3D Dual-SLIP model

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Fig. 2: 3D Dual-SLIP model. m is the point mass, with the CoM position given in inertial coordinates as pc =  T xc yc zc . The swing leg touchdown position is determined by two angles: θ, the angle between the swing leg and the vertical; and φ, the angle between the swing leg ground projection and the x axis (forward direction). The foot position is denoted as pfi , i = {A, B}. The rest leg length is l0 . The leg length kli k = l0 when leg i is in swing and kli k < l0 when leg i is in support. and the details of the aforementioned quarter-period optimization. Section II-C introduces the actuated Dual-SLIP model, which significantly extends the range of possible forward walking speeds. Section II-D describes the local LQR controller. Section III shows the results of embedding the 3D Dual-SLIP into a humanoid robot model. II. 3D D UAL -SLIP M ODEL A ND C ONTROL A. The 3D Dual-SLIP Model The 3D Dual-SLIP model and its parameters are shown in Fig. 2. In contrast to the LIPM, the 3D Dual-SLIP captures footstep locations for both feet and is able to generate CoM dynamics similar to that observed during human walking, including similar CoM vertical oscillations and lateral sway. 3D Dual-SLIP walking consists of an alternation of single and double support phases. In the single support phase, the dynamics follow: m¨ pc = k(l0 − kli k)ˆli + mg ,

(1)

where pc = [xc , yc , zc ]T ∈ R3 is the position of the pointmass model, m ∈ R its mass, g ∈ R3 is the gravity vector, l0 ∈ R is the rest length of the spring, k ∈ R its spring constant, li ∈ R3 is the leg vector from the base to the mass, and ˆli ∈ R3 is the unit vector along the leg. In the double support phase, the dynamics similarly follow: m¨ pc = k(l0 − klA k)ˆlA + k(l0 − klB k)ˆlB + mg ,

(2)

where all quantities are defined as in the single support case. A walking step is defined between two subsequent midstance (MS) events, with two steps shown in Fig. 3. Midstance happens during the single support phase when z˙c = 0. In a typical human-like gait, starting from MS, the DualSLIP CoM state will sequentially experience three other events in a single step: first the swing leg touches down

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where i ∈ {A, B} is the index of the foot in single support at MS. With the control variable denoted as un , an MS state return map f can be formed as: xn+1 = f (xn , un ) ,

which maps the MS state xn to the subsequent MS state xn+1 . For a 2-step periodic walking gait we enforce

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Fig. 3: Nominal “human-like” CoM trajectory in a full gait cycle (2 steps) of Dual-SLIP walking. The ground projection of the CoM trajectory is shown both in the single support (SS) and double support (DS) phases of walking. Inset: Sagittal plane view of a quarter walking cycle (half step). (TD), then the CoM reaches lowest height (LH), and finally the original single support leg lifts off (LO). Following LO, the model continues until it again experiences MS to finish the step (see Fig. 3). Through this progression, the resultant vertical ground reaction force (GRF) pattern for each foot will normally exhibit a double-peaked shape, as shown in Fig. 4. This GRF pattern is a key difference observed in human walking [12] in comparison to the single peak pattern in human running. Assuming that a leg labeled A begins in single support, the MS event occurs when the CoM state crosses the surface: SM S = {(pc , p˙ c ) | z˙c = 0, zc > zTH , klA k < l0 } ,

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(3)

where zTH = l0 cos θ is a threshold value for CoM height that depends on the touchdown angle θ. The subsequent event surfaces crossed in a normal step can be similarly defined: ST D = {(pc , p˙ c ) | z˙c < 0 , zc = zTH } ,

(4)

SLH = {(pc , p˙ c ) | z˙c = 0 , zc < zTH } , and SLO = {(pc , p˙ c ) | z˙c > 0 , klA k = l0 } .

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ST D and SLO more specifically are switching surfaces for the hybrid dynamics of the 3D Dual-SLIP model, as transitions to and from the double support phase happen when the CoM state crosses ST D and SLO , respectively. B. Finding Periodic Gaits Through extending the approach in [10], we would like to control the MS states during 3D Dual-SLIP walking. An MS state x for regulation is chosen as a slice of the full 3D Dual-SLIP state:  T x = xc − xfi yc − yif zc x˙ c y˙ c ∈ R5 , (7)

(9)

where A = diag(1, −1, 1, 1, −1), meaning we are interested in a 1-step motion that starts and ends with constant forward position (relative to the support foot) and velocity, constant vertical position, but sign-alternating lateral position and velocity. As one straightforward choice of the control variables, u for each step is selected to be the collection of the touchdown angles and the system spring constant (details shown in Fig. 2): u = [θ, φ, k]T . (10) For a desired forward velocity at MS, we need to determine the control u and the rest of the MS state x that together will lead to periodic dynamics. Geyer et al. [12] searched for periodic, self-stable walking gaits in the 2D Dual-SLIP model through an exhaustive scanning of the parameter space with simulation over many steps to test self stability. Seipel and Holmes [17] have shown however, that no left-right symmetric periodic running gaits in the 3DSLIP model are self-stable. Our studies have empirically found this to be the case with 3D Dual-SLIP walking gaits as well. Instead of an exhaustive parameter search, nonlinear optimization provides a viable way to search for periodic 3D Dual-SLIP gaits through the inclusion of 1step simulation within objective and constraint evaluation routines [10]. In this section, gaits are generated to satisfy the 2-step periodicity constraint (Eq. 9), while Dual-SLIP control to address the lack of self stability is described in Section II-D. It should be noted that the typical human-like gait with a double force peak, as discussed in Section II-A, is only one of many possible periodic gaits that can be captured by the 3D Dual-SLIP model. Through preliminary optimization tests over a half period (one step, from MS to MS), many of the resultant periodic Dual-SLIP gaits exhibited non-human characteristics, such as multiple (3+) GRF peaks. Instead, we propose a simpler quarter-period optimization (half step, from MS to LH, see Fig. 3 Inset) which has been found to effectively confine the behavior of the 3D Dual-SLIP model to the desired double-peak force pattern.

Given a desired forward velocity at MS x˙ 0,d , and assuming leg A is initially in support, a quarter-period optimization problem can be formulated to find a CoM trajectory from MS to LH. For a symmetric trajectory about LH, it is desired that the ground projection of the CoM at LH is directly between the two support feet. This is enforced through the optimization problem:

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2 (yA + yB (x0 , u0 )) − yc (x0 , u0 , tLH ) , (11) with  T x0 = x0,d y0,d z0 x˙ 0,d y˙ 0,d ,  T u0 = φ θ k , x0,d = 0 , y˙ 0,d = 0 , where (xAf , yAf ) is the position of the foot in support at the initial MS; and (xBf , yBf ) and (xc , yc ) are the touchdown position of the swing foot and the CoM ground projection at LH, respectively. Note (xBf , yBf ) are functions of (x0 , u0 ) and (xc , yc ) are functions of (x0 , u0 , t), and they are all evaluated through simulation. Note y˙ 0,d = 0 is needed to satisfy the periodic gait conditions (Eq. 9). For a rest leg length l0 = 1 m, y0,d is set to 0.05 m here. Although Eq. 11 only optimizes for a quarter cycle (half step), a zero residual error for (x∗0 , u∗0 ) is actually a sufficient condition to achieve CoM trajectories which are symmetric about LH over one step. This symmetry in turn ensures the 2-step periodic gait conditions (Eq. 9) for (x∗0 , u∗0 ). A proof sketch of this sufficiency is provided in the Appendix. Note that the optimization problem shown in Eq. 11 still has some redundancy to achieve a zero residual. Thus, additional objectives such as the desired step length or the desired double support ratio can be added to further shape the gait. C. Modification for High-Speed Walking For a 3D Dual-SLIP model with leg length l0 = 1 m and body mass m = 80 kg, the optimization in Eq. 11 is able to find periodic walking gaits for desired forward speeds x˙ 0,d approximately between 0.7 m/s and 1.3 m/s. Geyer et al. [12] have shown (in their Fig. 4) similar results for the 2D Dual-SLIP model using an exhaustive search approach2 . Although we might need to resort to a different simple model for walking at very slow speeds, high walking speeds up to 2.0+ m/s can be generated with small modifications to the 3D Dual-SLIP model. We propose bio-inspired modifications here which also maintain human-like ground force profiles. Through study of failed cases for the 3D Dual-SLIP model at higher walking speeds, we have found that a periodic gait cannot be found due to the use of constant spring stiffness and rest leg length within a step. During the first single support phase, the spring stiffness needs to be limited to prevent the CoM trajectory from shooting upwards. But as 2 While some may see this speed limitation as a downside of the model, in fact, the property that the Dual-SLIP captures limitations of the walking mode of locomotion is an additional feature in comparison to the LIP model.

the walking speed goes up, this limited spring stiffness and rest leg length are insufficient to reverse the CoM vertical velocity during a shortened double support period. This observation suggests that additional actuation during double support is required to overcome this deficiency. One solution (Implementation I) is to discretely change the rest leg length during the double support period, as it is more effective in altering the CoM velocity than changing the spring stiffness. More specifically, we introduce a new control variable lDS with lDS ≥ l0 . Initially at MS, both rest leg lengths l0,A = l0,B = l0 ; right after TD (touchdown with leg B), we instantaneously change l0,B to lDS ; then after LH, we instantaneously change l0,A to lDS and change l0,B back to l0 ; finally, right after LO (note, LO still happens when klA k ≥ l0 , not when klA k ≥ lDS ), we resume both rest leg lengths back to l0 . Such a process is illustrated in Fig. 5(a). The application of lDS to l0,B and l0,A successively during the double support phase is necessary for a symmetric gait. This approach is bio-inspired as it is similar to introducing an “eccentric muscle activity” upon touchdown and a “toeoff push” before liftoff to the regular, energy-conservative Dual-SLIP model. Biomechanics studies have shown these added features are critical to capture the typical dynamics of human walking [18], [19]. We call this modified model the Actuated 3D Dual-SLIP model. As simulation results show, this modified Dual-SLIP model can handle up to a 2 m/s (4.5 mph) forward walking speed. The periodic, symmetric gait for the actuated Dual-SLIP model can be found through the modified quarter-period optimization problem (comparing to Eq. 11):

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x0,d = 0 , y˙ 0,d = 0 , with additional objectives on desired step length or desired double support ratio. Note, in Eq. 12, z0 is no longer an optimization variable, but instead a fixed value, z0,d . For l0 = 1 m, z0,d is set to 0.96 m. This is because the range for possible z0 is very small (just a few centimeters) and as previous trials show, if z0 is a free variable, the optimizer tends to push it towards the upper bound as the walking speed goes up, which will then result in an undesirable gait. Implementation I only applies the actuation during the double support phase. As walking speed increases and the double support period becomes shorter, it tends to create high-rising discontinuous peaks in the ground reaction force pattern, as shown in Fig. 5(a). Nilsson and Thorstensson [20] have found that during human walking (from 1 m/s to 3 m/s), the resultant vertical ground reaction force will not exceed 1.5 b.w. (multiples of body weight). We can re-engineer the way we apply the actuation changes to achieve a more

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Fig. 5: The rest leg length and vertical ground reaction force pattern for higher-speed walking (a) Implementation I and (b) Implementation II. The corresponding periodic gait is for a forward walking speed of 1.5 m/s. 0.34 θ (rad)

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Suppose we have identified a pair (x∗0 , u∗0 ) that will initiate a periodic gait with a desired forward velocity at MS for the original 3D Dual-SLIP model. The MS state in 2-step periodic locomotion follows x∗n = An x∗0 , and by symmetry requires u∗n = B n u∗0 with B = diag(−1, 1, 1) to provide the correct outward leg angle at each step. Generally, due to disturbances, the actual MS state xn 6= x∗n , thus we would like to find a feedback law to regulate the MS state to the periodic solution. A discrete-time infinite-horizon LQR controller is used here. Letting ∆xn = (xn − x∗n ) and ∆un = (un − u∗n ), we make the change of variables ∆˜ xn = An ∆xn and ∆˜ un = B n ∆un . The return map identity

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the desired step length, which is a linear function of the forward velocity. We can see that both the double support phase duty cycle and the step width decrease as speed goes up, which is consistent to that observed in human walking. It is interesting to note that some of the quantities, for instance the optimal touchdown angle θ and the CoM vertical excursion with speed, exhibit a trend change near the preferred walking speed in humans of 1.4 m/s [21]. Although just an observation, this connection warrants further study to understand the role of compliant leg operation in the selection of preferred walking speed.

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is then helpful to obtain the error dynamics. The identity can be understood by changing the y-axis sign convention for one-step simulation, then returning the result to the original convention. With this result, a first order analysis of the return map around the periodic trajectory provides:

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Fig. 6: Periodic gait optimization results (Eq. 12 using Implementation II, with additional objective on desired step length, lstep,d = 0.5 + 0.1 · (x˙ d − 1.0), which is realized perfectly by the optimizer and shown in red. human-like ground reaction force pattern with smoother evolution and a smaller maximum vertical force. As another solution (Implementation II), we are not confined to only apply the actuation during the double support phase. A new control variable βSS is introduced to replace the previous lDS . βSS determines the rate at which the rest leg length ramps up during the single support phase, as shown in Fig. 5(b). Eq. 12 can also be used to find periodic gaits for this implementation, just substituting lDS with βSS . For both implementations, the resultant periodic solutions are similar, except the GRF pattern of the second implementation bears more resemblance to that of the regular 3D Dual-SLIP model (see Fig. 5(b) and compare it with Fig. 4). Figure 6 shows a collection of periodic gait optimization results from Eq. 12 for Implementation II. The desired forward velocity at MS ranges from 1.0 m/s to 2.0 m/s. For these results, an additional objective has been added on

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∆˜ xn+1 ≈ J x ∆˜ xn + J u ∆˜ un

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δf δf where J x = A δx and J u = A δu evaluated at (x∗0 , u∗0 ). Then consider the quadratic cost:

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(16)

where P is the unique solution of the Discrete-Time Algebraic Riccati Equation (DARE). Returning to the original coordinates provides the feedback law un = u∗n + B n KAn (xn − x∗n ) .

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For the 3D Dual-SLIP model, the MS state error is in R5 ,  T (18) ∆x = ∆x ∆y ∆z ∆x˙ ∆y˙ . However, since the 3D Dual-SLIP model is a very simple model, it is not a trivial task to identify as many effective

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Fig. 7: The ground projection of the CoM trajectory of the Hubo+ based humanoid model walking at 0.75 m/s and recovering from a lateral push that occurs near the end of the 15th step. It can be observed from the plot that the CoM ground projection never falls into either foot support polygon. 1.5 1.45

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Fig. 8: Changing the walking speed of the 3D actuated DualSLIP model with the proposed LQR controller. control variables as there are state variables. The advantage of the LQR approach is that it allows us to use fewer control variables and still have a (numerically) controllable system. But the trade-off is we won’t be able to correct the state error in one step as is done in deadbeat control. For the regular 3D Dual-SLIP model,  T ∆u = ∆φ ∆θ ∆k , (19) while for the 3D actuated model with Implementation I,  T ∆u = ∆φ ∆θ 0.0 ∆lDS . (20) Note how ∆lDS is applied in control: lDS +∆lDS is applied to l0,B between the TD and LH events, and lDS − ∆lDS is applied to l0,A between LH and LO events. For Implementation II, βSS is applied in a similar differential manner. Also note that the spring stiffness k is not adjusted by the LQR controller. Tests have shown it does not have a large effect on ∆x near the periodic trajectories. The proposed LQR controller is capable to recover from relatively large lateral disturbances while maintaining forward walking speed, as shown in Section III (and in Fig. 7). Experiments have shown that with the LQR controller, the actuated Dual-SLIP model can handle larger disturbances than the regular Dual-SLIP model. The LQR controller can also be used to transition between different periodic gaits, i.e. changing walking speeds, as shown in Fig. 8. III. A PPLICATION TO A H UMANOID M ODEL This section demonstrates the application of the methods from Section II to enable dynamic walking with push recovery in a modified Hubo+ humanoid robot model. The model has 32 degrees of freedom (DoFs) with 26 out of them actuated (6 for each limb, no hands, 1 for waist, 1 for neck). The inertial properties of each link are taken from [16].

In this work, each actuated joint is assumed to be torque controlled instead of position controlled. The humanoid model has a total weight of 38.1 kg and a leg length of approximately 0.6 m. When standing straight, the distance between the two foot centers is approximately 0.17 m. Based on this information, the mass and normal rest leg length of the actuated Dual-SLIP model are updated accordingly: m = 38.1 kg, l0 = 0.63 m. In the formulation of the optimization (Eq. 12), the desired MS height is also updated: z0,d = 0.6 m. Although the possible forward walking speeds x˙ 0,d studied in Section II ranged from 0.7 m/s to 2.0 m/s, the previous results employed a Dual-SLIP model with a 1 m leg length. Since the Hubo+ based model has a shorter leg length, this speed range can be scaled through the non-dimensional Froude number [22]3 . Using the humanoid Dual-SLIP parameters, we can rerun the optimization in Eq. 12 to find periodic gaits compatible with this specific humanoid model at various forward speeds. A prioritized task-space controller along with a walking state machine was employed for the humanoid model. This whole-body controller groups the reference CoM trajectory and footstep locations provided by the LQR-regulated 3D actuated Dual-SLIP model, along with other objectives of interest, such as a nominal whole-body pose, into tasks at different priority levels. Then at each instant, the controller performs a sequence of optimizations. Each searches for a set of optimal joint torques and ground reaction forces that minimize the task at the current priority level while not compromising the already optimized higher priority level tasks. Many details about this prioritized task-space controller can be found in [10]. The task weights, however, were reselected due to the Hubo+ model and application to a walking gait. The swing leg trajectories here were generated with a 4th order B´ezier curve relative to the CoM. Figure 7 shows CoM and footstep locations for the humanoid robot model walking at 0.75 m/s with the aforementioned approach. Note that the resultant walking motion is fully dynamic as the CoM ground projection never falls inside either foot support. Further, the lateral CoM sway is no more than 5 cm during regular walking. Near the end of the 15th step, the humanoid experiences a lateral push at the hip of 300 N for 0.025 s. Then at the next MS, the state 3 The Froude number is defined as: Fr = v 2 /gl. For instance, walking at 1 m/s with 1 m leg length, and walking at 0.75 m/s with 0.6 m leg length, the Froude number for the two cases are about the same.

Vertical GRF (N)

600 500 400 300 200 100 0

6

6.1

6.2

6.3

6.4

6.5

CoMx (m)

Fig. 9: The vertical ground reaction force (GRF) measured from the penalty-based contact model in simulation of the Hubo+ based humanoid robot. The GRFs correspond to the 12th and 13th steps as shown in Fig. 7. of the 3D actuated Dual-SLIP model is synchronized to the actual humanoid CoM state (i.e. the push is captured in the actuated Dual-SLIP model) and the LQR controller for the SLIP model corrects the state error in the next few steps. This correction includes online re-planning of a sequence of foot step locations, as shown in the right half of Fig. 7. This LQR re-planning allows the humanoid to handle much larger disturbances than can be handled with the task-space controller alone. Fig. 9 shows the vertical ground reaction forces that are measured from a penalty-based spring-damper contact model in simulation during walking at 0.75 m/s. Data is shown from the 12th and 13th step in Fig. 7. Note that the task-space controller assumes rigid contact; however, the approach is robust to these unmodeled ground contact dynamics. The video attachment demonstrates the resultant humanoid walking motion and its response to a lateral push. One might notice that the humanoid seems to spend a great effort to maintain the walking speed. This is because the original Hubo+ robot has very massive legs and a relatively light torso4 . As a result, forward leg swing during walking requires more relative effort and has a larger coupled effect on the torso dynamics in comparison to human walking. Roughly, this behavior is consistent with what a human would do if extra weights were attached to their legs and they were still asked to walk as fast5 . Also, as shown in the video, uneven terrain features such as arbitrarily-shaped bumps (elevation difference up to 10% of leg length) can be handled at the task-space control level by adjusting the target swing leg trajectories based on the perceived terrain elevation at the potential touchdown location. IV. C ONCLUSION The Dual-SLIP model has been suggested as a potential motion “template” for human-like walking. This paper has proposed a new way to find periodic walking gaits in a 3D Dual-SLIP model at various speeds through optimization over a quarter walking cycle. It has been shown that the Dual-SLIP model with actuated rest leg length can walk much faster than a regular, energy-conservative Dual-SLIP model, and the resultant key walking parameters have shown trends, versus walking speed, that are consistent with biome4 In terms of percentage of total weight, for the Hubo+ based model, the torso accounts for around 28% and each leg accounts for 27%; while for an average human, the numbers are 45% and 17%, respectively [23]. 5 Note that in other tests with models having leg/torso mass ratios close to that of a human, the torso motion is more highly damped.

chanics findings. An LQR controller for this model has been introduced to correct for midstance (MS) state tracking errors and disturbances while walking. These template gaits have then enabled human-like dynamic walking gaits and push-recovery capabilities in a realistic humanoid robot model through the application of task-space control. Through modifications to the target leg trajectories, the approach also allows the humanoid to traverse slightly uneven terrain as shown in the video. Future work will study how to adapt the Dual-SLIP model to complex terrain conditions in order to increase the applicability of this control methodology. Further, centroidal angular momentum control [24], which is currently relegated to the task-space controller [25], will be studied so that angular momentum is better coordinated with the CoM trajectories and footsteps generated by the Dual-SLIP model. As more humanoid robots begin to adopt compliantly actuated joints, we look forward to the potential for this approach to better exploit the natural dynamics of compliant leg operation in comparison to traditional LIP models. V. ACKNOWLEDGMENTS This work was supported by Grant No. CNS-0960061 from the NSF with a sub-award to The Ohio State University. One of the authors, Patrick Wensing, was also supported by an NSF Graduate Research Fellowship. A PPENDIX Proof sketch of the Sufficiency Condition for 3D Dual-SLIP Symmetry Assume that we have optimized the quarter period with zero residual through Eq. 11. That is, for an optimal statecontrol pair (x∗0 , u∗0 ), the middle of the support is right below the CoM when it reaches lowest height (LH). This case is shown in Fig. 10. Note for this proof sketch, the origin of the coordinate system is set at the middle of double support. The full state of the mass is given as x = [pTc , p˙ Tc ]T . During foot A single support, double support, and foot B single support, the system follows the continuous dynamics x˙ = fA (x), x˙ = fD (x), and x˙ = fB (x),

(21) (22) (23)

respectively which follow from Eqs. 1 and 2. Following the ideas of Carver [26] we can define a symmetry mapping h     xc −xc  yc   −yc       zc   zc      . h  =  (24)  x˙ c   x˙ c   y˙ c   y˙ c  z˙c −z˙c Intuitively, this map takes states on one side of LH, and maps them to symmetric states on the other side of double support. Moreover, this mapping is special, as it follows many useful relationships with the instantaneous dynamics of the Dual

pfB

xf = Áft1B (xLO ) = h(Áf¡tA1 (xTD )) xLO = Áft2D (xLH ) = h(Áf¡tD2 (xLH ))

xLH (0 ; 0)

Middle of double support

t2

xTD = Áf¡tD2 (xLH )

t1

pfA

x0 = Áf¡tA1 (xTD )

Fig. 10: Symmetries for the 3D Dual-SLIP in the x-y plane. t1 is the time it takes from MS to TD. t2 is the time from TD to LH. x0 and xf are the CoM states at the beginning and the end of the step. 3D-SLIP model as well as their flow over time. Through some algebra, the vector fields fD , fA , and fB can be shown to satisfy h ◦ fD = −fD ◦ h ,

(25)

h ◦ fA = −fB ◦ h ,

(26)

which in turn imply vector field flow [27] symmetries (as shown in Fig. 10) D h ◦ φf−t ◦ h−1 = φft D , and

h◦

A φf−t

−1

◦h

=

φft B

.

(27) (28)

The conditions assumed on the state of the system at LH (z˙c = 0, xc = 0, and yc = 0) imply that h(xLH ) = xLH ,

(29)

which, when combined with Eq. 27 provides D h ◦ φf−t (xLH ) = φft D (xLH ) .

(30)

From this result, it follows that if the LH state occurs directly over the middle of the feet, then the CoM trajectory (flow) in double support is symmetric about LH. Although space limits full development, we can also show a corresponding symmetry between the foot A single support phase and foot B single support phase in a similar way, which is illustrated in Fig. 10. Thus, in a step where LH occurs directly over the middle of support, trajectories which pass through any state x prior to LH will also pass through h(x) after LH. Most importantly, however, this implies that for an initial state x0 at MS, Eq. 12 enforces a sufficient condition for the CoM to reach the symmetric state h(x0 ) at the subsequent MS. Thus, the quarter-period optimization (from MS to LH) ensures the desired periodicity properties (Eq. 9) for the full MS-to-MS state evolution. R EFERENCES [1] R. J. Full and D. E. Koditschek, “Templates and anchors: neuromechanical hypotheses of legged locomotion on land,” J. Exp. Biol., vol. 202, no. 23, pp. 3325–3332, 1999. [2] S. Kajita, O. Matsumoto, and M. Saigo, “Real-time 3D walking pattern generation for a biped robot with telescopic legs,” in IEEE Int. Conf. on Robotics and Automation, vol. 3, pp. 2299–2306, 2001. [3] S. Kajita, F. Kanehiro, K. Kaneko, K. Fujiwara, K. Harada, K. Yokoi, and H. Hirukawa, “Biped walking pattern generation by using preview control of Zero-Moment Point,” in IEEE Int. Conf. on Robotics and Automation, vol. 2, pp. 1620–1626, Sept 2003.

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