July 10, 2015
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1
Foot Placement Control for Bipedal Walking on Uneven Terrain: An Online Linear Regression Analysis Approach Yangwei You∗ , Zhibin Li, Nikos G. Tsagarakis and Darwin G. Caldwell Department of Advanced Robotics, Istituto Italiano di Tecnologia, Genoa, Liguria 16163, Italy ∗ E-mail:
[email protected] This paper presents a novel foot placement control algorithm for adaptive bipedal walking. In this method, the torso attitude and height are stabilized by synergic patterns so that the forward velocity and its change have a stable and nearly linear relation with the foot placement. Hence, our proposed online linear regression analysis well represents the local linear models by estimating continuously from measured data. Based on this estimation, an appropriate foot placement can be determined to control the forward velocity. Our simulation study successfully demonstrates the natural gait with accurate tracking of walking velocity, and the robustness of walking over uneven terrain. Keywords: Foot placement control; Dynamic walking; Legged locomotion.
1. Introduction To date, many humanoid robots have been built with impressive progresses in hardware and control.1–5 This demands the software and algorithms to improve the feasibility and autonomy of those robots.6 One important aspect for humanoid locomotion is to place the foot correctly,7 which remarkably affects the stability. The work in7,8 studied the relation between the foot placement and the system stability, and proposed foot placement indicator and estimator as the measure of balance. Many methods for humanoid walking were developed based on the zeromoment point,3,4,9,10 and the linear inverted pendulum model (LIPM) was often used for the foot placement which provides analytic solutions. However, the robots governed by LIPM move with unnatural bent-knee gaits. Biological research reveals that the movement of legged animals can be represented better by a spring loaded inverted pendulum (SLIP) model.11 Raibert developed legged robots from one-legged hoppers to quadrupeds12 and used the foot placement to regulate the forward velocity by a simple
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WSPC - Proceedings Trim Size: 9in x 6in
Yangwei˙CLAWAR˙2015˙Online
Author’s preprint version for academic circulation only.
2 WALKING PATTERN GENERATION equation. However, to apply his controller, the physical model needs to be carefully designed to satisfy the assumption that the leg is sufficiently light compared to the body. Guocai raised a similar controller but he used only the ankle actuators to control the walking velocity.13,14 To address this issue, we propose a novel foot placement control based on the online linear regression analysis to identify the approximated linear relation between the foot placement, forward velocity and its change. Though the real relation is non-linear due to the multi-body dynamics, the foot placement can still be represented well by the local linear estimation which is being corrected continuously using the real measurements. The local linear model can be corrected continuously to predict an appropriate foot placement for the forward velocity control. This permits more freedom for the design of stance leg motion (straight knee) and can easily realize the natural walking. Our proposed algorithm is applied to a simulated planar robot, and produces natural and stable walking on flat ground, steps and slopes. This paper is organized as follows. Section 2 presents the robot model and the pattern synergies. Section 3 elaborates the principles of the linear regression analysis for the foot placement control. Section 4 demonstrates the feasibility from our simulation study. We conclude in the last section.
0
(1.5T ) (T ) (1.5T ) 0
1.5T
t
thigh
(rad )
p
hip
0
knee +
shin
- -
sole ankle
(1.5T ) 1.2 0.3vd (1.5T ) 0
(T ) 0 (T ) 0 T fp
1.5T
t
0
T
fp
fp
1.5T
(1.5T ) 0.3 0.2vd (1.5T ) 0 t
(T ) t (T ) 0
fp
fp
(0.5T ) 0.3 0.2vd (T ) 0.1 (T ) 0 (0.5T ) 0 0.5T t fp T
(b)
(a)
Fig. 1.
t
(0.5T ) 1.2 0.3vd (T ) 0 (0.5T ) 0 (T ) 0 0.5T T fp
0
0
T
0.5T f p
0
fp
(T ) 0.1 0.28vd (T ) 0
swing leg trajectories
(rad )
torso
stance leg trajectories (T ) 0.05vd (T ) 0 T f
(rad )
-
(rad )
double stance single stance
(rad )
2. Walking Pattern Generation
(rad )
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t
(c)
Planar humanoid robot and its walking pattern.
To enable natural walking, the basic patterns of stance and swing legs are designed first, then the foot placement control can be implemented to adjust the specific trajectories for each step and keep the walking stability. The simulated planar robot is comprised of 7 parts, one torso, two 2
July 10, 2015
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Author’s preprint version for academic circulation only.
2
WALKING PATTERN GENERATION
thighs, two shins and two soles, as shown in Fig. 1(a). Rotary actuators are mounted in the hip, knee and ankle joints. γ is the pitch angle of torso. θ is the swing angle, which is the angle between the vertical line and the line across the hip and ankle. θt is the touch-down angle, which is the swing angle at the moment of touch-down. α and β are the angles of knee and ankle joints respectively. The direction convention of these angles are defined in Fig. 1(a) with ’+’ and ’-’ signs. A 3rd-order polynomial is used for trajectory planning of a period from the time Ts to Tf to ensure the continuity of angle and angular velocity: ˙ s ), φ(Tf ), φ(T ˙ f ), t) = φ(Ts ) + φ(T ˙ s )t φ(t) = fp (φ(Ts ), φ(T ˙ s ) + (Tf − Ts )φ(T ˙ f) 3φ(Ts ) − 3φ(Tf ) + 2(Tf − Ts )φ(T t2 − 2 (Tf − Ts ) ˙ s ) + (Tf − Ts )φ(T ˙ f) 2φ(Ts ) − 2φ(Tf ) + (Tf − Ts )φ(T t3 . + (Tf − Ts )3
(1)
˙ s ), φ(Tf ) and φ(T ˙ f ) are respectively the initial angle and where, φ(Ts ), φ(T angular velocity and the final angle and angular velocity. For simplification, all the 3rd-order polynomials used below are referred as fp . The touch-down event of walking is detected when the swing leg really touches on the ground and the stance leg has supported for more than half of the expected stance time. At this moment, the trajectories of legs between this touch-down and the next one are planned based on the current state and expected forward velocity. Then, the stance leg is activated to swing while the swing leg starts to support. Although there is a short period of double support, this transition is not specially planned, but taken into account while planning the trajectories of single support phase. So the trajectory patterns of a leg can be classified according to the walking phases. During stance, the hip actuator applies torque at the torso to track a planned trajectory of body pitch, which is related to forward velocity in order to imitate the leaning-forward behavior when people are walking. Meanwhile, actuators on knee and ankle are used to extend the leg to propel. Note that when the actual stance time is longer than the desired one and the swing leg still doesn’t touch down, the stance leg will actively retract. The expected trajectories of these angles are piecewise three-order polynomials and detailed in Fig. 1(b), where T is the expected stance time. During the swing phase, the swing angle is controlled by the actuator on hip joint to track the desired touch-down angle. Meanwhile, the leg is retracted first to have foot-ground clearance, and then extended by the knee and ankle actuators. When the actual swing time is bigger than expected 3
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3
FOOT PLACEMENT CONTROL
and the touch down still does not occur, then knee and ankle joints will remain at the same position until touch-down happens. The swing angle is of particular interest, because it affects the touch-down angle θt and should be updated according to the current forward velocity. More details will be discussed in Section 3. The desired swing leg trajectories are also in Fig. 1(c).
3. Foot Placement Control This section explains the foot placement control algorithm for adjusting the forward walking velocity. Previously, by observing the symmetry of running and the influence of foot placement, Raibert proposed a simple algorithm to control forward velocity as:12
xf =
xT ˙ + kx˙ (x˙ − x˙ d ), 2
(2)
where xf is the foot placement, T is the duration of the stance phase, x˙ ˙ is the current velocity, and x˙ d is the desired velocity. The item xT 2 is the neutral point where the movement is symmetric and the average traveling speed remains unchanged after one step. The item kx˙ (x− ˙ x˙ d ) is the feedback to displace the foot from the neutral point for stabilizing the forward speed against the errors and external disturbances.12 To some extent, Raibert’s method works fine due to a good linear relationship among the foot placement, forward velocity and its change. To ensure this linearity, the mass ratio between the torso and leg of the robot has to be large, which limits the application on real humanoid robots with considerable mass of the legs. Our idea is that though the mathematical function among the foot placement, forward velocity, and its change may appear as a nonlinear hyper surface, it can still be presented well by a set of local linear models. Therefore, to overcome the limitation of Raibert’s method, we use the linear regression analysis on the real measurements to improve the prediction of foot placement by online updating the coefficients during walking. Take the touch-down angle θt as the representative of foot placement. Regarding θt as the dependent variable, and forward velocity x˙ and its change (x˙ − x˙ d ) as the independent variables, their relationship can be 4
July 10, 2015
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WSPC - Proceedings Trim Size: 9in x 6in
Yangwei˙CLAWAR˙2015˙Online
Author’s preprint version for academic circulation only.
3
FOOT PLACEMENT CONTROL
estimated by the least square method as follows:
θt = k0 + k1 x˙ + k2 (x˙ − x˙ d ) −1 I · I T x˙ · I T ∆x˙ · I T I · θ Tt T = I · x˙ T x˙ · x˙ T ∆x˙ · x˙ T · x˙ · θ Tt k0 k1 k2 I · ∆x˙ T x˙ · ∆x˙ T ∆x˙ · ∆x˙ T ∆x˙ · θ Tt
(3)
I = [ 1 1 · · · 1 ]1×n ; x˙ = [ x˙ 1 x˙ 2 · · · x˙ n ]; θ t = [ θt1 θt2 · · · θtn ]; ∆x˙ = [ ∆x˙ 1 ∆x˙ 2 · · · ∆x˙ n ]; ∆x˙ i = x˙ i − x˙ di ; i = 1, 2 · · · n.
where three coefficients (k0 , k1 , k2 ) are used to approximate the linear relation, and compute the touch-down angle θt given the current velocity x˙ and the desired velocity x˙ d . Compared with (2), one more coefficient k0 is added into the linear model to resolve a possible offset. All three coefficients (k0 , k1 , k2 ) are updated based on the online linear regression analysis. For the regression analysis, n sets of data are collected online. Each data set consists of three elements: forward velocity just before touch-down x˙ i , touch-down angle θt , and the forward velocity change after one step ∆x˙ i . To ensure a good linear representation of the current local correlation, the latest data are used for the analysis. A queue that stores n sets of data is used, where the latest data is loaded while the oldest one is deleted continuously. To form an accurate local model, we avoid the singularity of data collection, by applying a data filter algorithm with thresholds (x˙ th , θtth , ∆x˙ th ) to determine whether the latest data is too similar with the existing ones. The thresholds affect the steady-state accuracy and the robustness of forward velocity tracking. If they are too big, the sampled data are too far away from each other to form a precise local linear approximation. Whereas if too small, the data concentrate on a singular point thus cannot form a linear plane well. So the thresholds are the trade-off between steady-state accuracy and robustness. The number of data sets, n, also affects the control performance. If n is too big, the estimated linear model will refresh slowly and cannot response well to the system change. On the other hand, if too small, the estimated model changes too drastically and results in instability. Given the predicted touch-down angle θt , the swing leg trajectory can be computed. The expected touch-down angle is calculated continuously and used to adjust the swing trajectory in response to the change of forward velocity. The swing angle is obtained as in (4), where θt is the expected touch-down angle calculated at the touch-down, and θtnew is the updated 5
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4
SIMULATION
1
1.5
k0 k1 k2
Coefficients
1 Velocity [m/s]
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0.5 0 −0.5
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Forward velocity tracking and online coefficient adaption of walking.
one during the stance phase, and T is the expected stance time. ˙ fp (θ(0), θ(0), θt , 0, t) + θtnew − θt , t ≤ T ; θ(t) = new θt , t > T.
(4)
4. Simulation We validated the performance of our proposed algorithm in the simulations with a humanoid as in Fig. 1(a). The mass of torso, thigh, shin and sole are 19.8, 2.8, 2.5 and 0.3 kg respectively, while their moments of inertia are 0.3, 0.02, 0.02, 3.7e-4 kgm2 , and the limb dimension of 0.4, 0.25, 0.25, 0.075 m. The motor PD controllers have the stiffness and damping of 287 Nm/rad and 5.7 Nms/rad for both knee and ankle. For the hip joint, the PD gains are 250 Nm/rad and 10 Nms/rad for controlling the torso attitude during stance, and are 100 Nm/rad and 2 Nms/rad for controlling the swing leg. We used 6 sets of measured data for the online linear regression analysis. The thresholds x˙ th , θtth , ∆x˙ th of the filter are 0.05 m/s, 0.02 rad, and 0.05 m/s. The initial data for the linear regression are listed in Table 1. They are chosen arbitrarily and cannot ensure stable walking without online update. Table 1.
Initial data for linear regression analysis.
Num.
x˙
∆x˙
θt
Num.
x˙
∆x˙
θt
1 2 3
1.000 0.000 0.000
0.000 1.000 0.000
0.400 0.100 0.000
4 5 6
1.000 0.000 0.000
0.000 1.000 0.000
0.400 0.100 0.000
The robot walked in place for 5 s to adjust the coefficients in (3), and then walked at 1 m/s within a transition of 5 s. At the beginning, the velocity fluctuated because the initial data for regression analysis was arbitrarily given. When the coefficients were adjusted well, then the walking velocity was tracked accurately. The online update of the coefficients and controlled 6
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Step off
Step on
5 cm
Walk up
Author’s preprint version for academic circulation only.
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SIMULATION 0.15 rad
Walk down
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Target Velocity
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5
(c)
15
1
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k0 k1 k2
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0
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10 Time [s]
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10 Time [s]
(e) Fig. 3.
20
(d)
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0
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5
10 Time [s]
15
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(f) Simulation of walking across uneven terrain.
walking velocity are shown in the Fig. 2. Then, the robot was controlled to walk at 1 m/s to cross an unknown step of 5 cm height. Fig. 3(a) shows the snapshots of successful walking over the step. The coefficients refreshing and walking velocity are shown in the Fig. 3(e) and 3(c). Further, the robot was controlled to walk up and down a slope of 0.15 rad at 0.8 m/s. The slope was known for the robot, and the ankle angle β(T ) was revised according to the slope to ensure a good contact between the sole and slope. In practice, the slope can also be calculated according to the orientation of sole when it was fully in touch with the ground. Figure 3(b) is the snapshots of successful walking over the slope. The coefficients refreshing and walking velocity are shown in the Fig. 3(f) and 3(d). 7
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WSPC - Proceedings Trim Size: 9in x 6in
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5
CONCLUSION
5. Conclusion We proposed a novel foot placement algorithm for online adaptive control of the bipedal walking. By controlling the torso attitude and height of the robot, the foot placement has a nearly linear relation with the forward velocity and its change. Therefore, our online regression analysis can estimate a good foot placement to regulate the walking velocity accurately, compared to the previous work.12 The simulation study demonstrates the robustness of the algorithms by three successful validations: accurate tracking of forward velocity on the flat ground, blinding walking over a stair, ascending and descending known slopes. In the future, this method is expected to be implemented on a real 3D humanoid robot.
Acknowledgments This work is supported by the FP7 EU project WALK-MAN (ICT 2013-10).
References 1. S. Kajita, K. Kaneko, F. Kaneiro, K. Harada, M. Morisawa, S. Nakaoka, K. Miura, K. Fujiwara, E. S. Neo, I. Hara et al., Cybernetic human hrp4c: A humanoid robot with human-like proportions, in Robotics Research, (Springer, 2011) pp. 301–314. 2. Z. Li, N. Tsagarakis and D. Caldwell, Walking trajectory generation for humanoid robots with compliant joints: Experimentation with coman humanoid, in IEEE International Conference on Robotics and Automation, 2012. 3. Z. Li, N. G. Tsagarakis and D. G. Caldwell, Walking pattern generation for a humanoid robot with compliant joints, Autonomous Robots 35, 1 (2013). 4. C. G. A. Salman Faraji, Soha Pouya and A. J. Ijspeert, Versatile and Robust 3D Walking with a Simulated Humanoid Robot (Atlas) a Model Predictive Control Approach, IEEE International Conference on Robotics and Automation (2014). 5. N. Paine, J. S. Mehling, J. Holley, N. A. Radford, G. Johnson, C.-L. Fok and L. Sentis, Actuator Control for the NASA-JSC Valkyrie Humanoid Robot: A Decoupled Dynamics Approach for Torque Control of Series Elastic Robots, Journal of Field Robotics (2015). 6. S. Dalibard, A. El Khoury, F. Lamiraux, A. Nakhaei, M. Ta¨ıx and J.P.Laumond, Dynamic walking and whole-body motion planning for humanoid robots: an integrated approach, The International Journal of Robotics Research (2013). 7. P. Van Zutven, D. Kostic and H. Nijmeijer, Foot placement for planar bipeds with point feet, in IEEE International Conference on Robotics and Automation, 2012.
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5
CONCLUSION
8. D. L. Wight, E. G. Kubica and D. W. Wang, Introduction of the foot placement estimator: A dynamic measure of balance for bipedal robotics, Journal of computational and nonlinear dynamics 3 (2008). 9. 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 International Conference on Robotics and Automation, 2003. 10. C. Chevallereau, D. Djoudi and J. W. Grizzle, Stable bipedal walking with foot rotation through direct regulation of the zero moment point, IEEE Transactions on Robotics 24, 390 (2008). 11. H. Geyer, A. Seyfarth and R. Blickhan, Compliant leg behaviour explains basic dynamics of walking and running, Proceedings of the Royal Society B: Biological Sciences 273, 2861 (2006). 12. M. H. Raibert, Legged robots that balance (MIT press Cambridge, MA, 1986). 13. G. Liu, M. Li, W. Guo and H. Cai, Control of a biped walking with dynamic balance, in International Conference on Mechatronics and Automation, International Conference on Mechatronics and Automation 2012. 14. G. Liu, F. Zha, M. Li, W. Guo, P. Wang and H. Cai, Control of a humanoid robot walking with dynamic balance, in IEEE International Conference on Robotics and Biomimetics (ROBIO), 2013.
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