Human-like Walking with Compliant Legs
Human-like Walking with Compliant Legs Ludo C. Visser, Wouter de Geus, Stefano Stramigioli and Raffaella Carloni
Abstract— This work presents a novel approach to robotic bipedal walking. Based on the bipedal spring-mass model, which is known to closely describe human-like walking behavior, a robot has been designed that approaches the ideal model as closely as possible. The compliance of the springs is controllable by means of variable stiffness actuators. The controllable stiffness allows the gait to be stabilized against external disturbances.
q(t)
m
where Bi collect the Fsi ,u terms and we t the system is in the single support phas z = (q, q) ˙ the system dynamics (3) can compact standard form z˙ = f (z) +
2 &
gi (z)ui
q2
i=1
where (z1 , z2 , z3 , z4 ) := (q1 , q2 , q˙1 , q˙2 ).
re n le st
I. I NTRODUCTION
h gt
B. Stabilizing Controller k0 + u2 Humans are excellent walkers, able of energy efficient k0 + u1 Given the system dynamics (4), a sta locomotion and quick adaptation to different terrains. This is has been designed. The reference for thi of the stable autonomous gaits that th achieved by a complicated musculoskeletal system, comprisα0 in the absence of control input. In part ing many muscles, tendons and joints. However, it was shown q1 an autonomous gait, it can be parameter in [1] that the human gait can be accurately modeled by aFig. 1. Ideal(a)model variable , which allows a complete Theof bipedal (b) CADq1rendering the bipedal spring-mass walker—The legs model are compliant, and the relatively simple bipedal spring-mass model. In particular,stiffness can be invidually controlled by a control input ui . The trajectory of desired gait by ∗the two reference function functionrealization. q2 (q1 ) describes the desir the hip is Fig. denoted it determines the transitionsmodel between single 1.by q(t), Theandbipedal spring-mass and theThe prototype the model explicitly allows for a double support phaseand double support phase by intersections of the surface S. A single step q˙1∗ (q1 ) its desired forward velocity. begins and ends at the VLO, at which a relabeling of the legs takes place, The reference functions can be redefine and reproduces the ground reaction force profiles that aremaking q1 a cyclic variable. observed in human walking gaits. Moreover, it was shown z2∗ (z1renders ) := q2∗ (q1 ),a z3∗ (z1 ) := In [3], we proposed a stabilizing controller that ends when the system is in Vertical Leg Orientation (VLO), that the model encodes a variety of passive gait patterns, passive desired gait stable against external disturbances by in which the hip is exactly above the supporting leg in the Then, two error functions can be defined including running. In a more detailed study, presented insinglecontrolling stiffness ki . The control law significantly support phase. the The leg transition to the double support h1 = z2 − z2∗ is determinedthe by the parameters α0of andthe L0 , i.e. the In particular, it was [2], it was shown that a large subset of the possible gaits arephaseincreases robustness gait. h2 = z3 − z3∗ angle at which the swing leg touches the ground and the rest asymptotically stable, and that these gaits have a relativelylengthshown that the controller is successful in dealing with disof the telescopic spring respectively. After each step, The system is fully actuated during the dou underactuated during the single suppor the legs are relabeled, making variable. large basin of attraction. 1 a cyclic turbances such as qswing leg dynamics andbutimpact forces that that the leg stiffness can be written as k0 + a switching controller is designed that ach In [3], we proposed to extend the bipedal spring-massui,Considering occur when legs with non-negligible mass are considered. i = 1, 2, it can be easily verified that the force generated to a neighborhood of the desired gait. model with variable compliance in the legs. By making theby the spring during the single support phase is given by: following control inputs were proposed in III. R EALIZATION OF THE ROBOT • During the single support phase, Fs = Fs ,0 + Fs ,u , leg stiffness controllable, a control input is made available ! "# $ Currently, a robot is built that approaches the ideal bipedal L ' 2 q 0 1 1 that can be used to stabilize a desired gait. Because the Fs ,0 = k0 −1 u1 = −Lf h1 − κd Lf q2 (1) L1 as closely L L h g f 1 spring-mass model as possible. The bulk of the "# $ ! passive gaits are stable and locally attractive, control input L0 q1 and u2the ≡ 0;legs are −1 Fs ,u = closely tou the hip joint, while is only required to converge to a neighborhood of the gait. mass is located q2 1 L1 • During the double support phase, % made of lightweight carbon fiber tubes. The stiffness of the Then, once convergence is achieved, no additional control iswhere L1 = q12 + q22 is the length of the leg. During the # 2 # $ −Lf h1 − κd Lf u1 stiffness legs is controlled by the novel vsaUT-II variable double support phase, both legs exert a force on the hip mass. = A−1 required to sustain the gait. u2 −Lf h2 − The additional force is given by: the prototype. In this work, we describe the realization process of a actuator [4]. Fig. 1b shows a CAD rendering of with # Fs = Fs ,0experiments + Fs ,u , Currently, " # with $ the prototype are under devel! L L h Lg L bipedal walking robot, based on the principles of the bipedal A= g f 1 L0 q1 − a − 1 of validating the simulation results. LIn g L f h2 L g L opment,Fswith aim ,0 = kthe 0 spring-mass model with variable compliant legs. q2 (2) L2 $ "# 2 the variable particular, the! Ltests are designed to show that where L h , L h and L i f i g Lf hi are th q − a 0 f 1 −1 u2 F ,u = derivatives of the error functions hi , defi II. T HE B IPEDAL S PRING -M ASS M ODEL q2 L2 stiffness s actuation increases the robustness sufficiently to the vector fields defined in (4), and κd , κp wherestabilize a denotes the touchdown point of the former swing gait of the walker. Fig. 1a schematically depicts the bipedal spring-massleg and L = %(qthe This controller achieves: 2 2 2 1 − a) + q2 is the length of this leg. model. It consists of a point mass m, and two massless tele- In addition to the forces exerted byRthe springs, the hip lim h1 (t) = 0 EFERENCES t→∞ mass is also subject to the gravitational acceleration g0 . By scopic springs with rest length L0 and controllable stiffnessrearranging (1) and (2), the complete dynamics of the system and, for some 0, [1] H. Geyer, A. Seyfarth, and R. Blickhan, “Compliant legε > behaviour ki = k0 + ui , i = 1, 2. During the transition from singlecan be written as: basic dynamics of walking and running,” Proceedings of the explains $ # $# $ # lim |h2 (t)| < ε 2 support to double support, the former swing leg is assumed m 0Royal no. 1603, pp. 2861–2867, 2006. t→∞ q¨1 Society 0 B, vol. 273, & Bi (q)ui (3) + − Fs (q) = J. Rummel, Y.0 Blum, H. M. Maus, C. Rode, and Seyfarth, 0 m q¨2 mg TheseA.properties are“Stable proven in REF . to touch down at an angle α0 . For appropriately chosen [2] i=1 and robust walking with compliant legs,” in Proceedings of the IEEE parameters m, L0 , k0 , α0 and initial conditions q(0), q(0), ˙ the International Conference on Robotics and Automation, 2010, pp. 5250– model will show a stable passive (ui ≡ 0) gait trajectory q(t). 5255. L0
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This work has been funded by the European Commission’s Seventh Framework Programme as part of the project VIACTORS under grant no. 231554. {l.c.visser,s.stramigioli,r.carloni}@utwente.nl, MIRA Institute, Department of Electrical Engineering, University of Twente, 7500 AE Enschede, The Netherlands.
[3] L. C. Visser, S. Stramigioli, and R. Carloni, “Robust bipedal walking with variable stiffness actuation,” submitted to the IEEE International Conference on Robotics and Automation 2012. [4] S. S. Groothuis, G. Rusticelli, A. Zucchelli, S. Stramigioli, and R. Carloni, “The vsaUT-II: a novel rotational variable stiffness actuator,” submitted to the IEEE International Conference on Robotics and Automation 2012.
Abstract— This work presents a novel approach to robotic bipedal walking. Based on the bipedal spring-mass model, which is known to closely describe human-like walking behavior, a robot has been designed that approaches the ideal model as closely as possible. The compliance of the springs is controllable by means of variable stiffness actuators. The controllable stiffness allows the gait to be stabilized against external disturbances.
q(t)
m
where Bi collect the Fsi ,u terms and we t the system is in the single support phas z = (q, q) ˙ the system dynamics (3) can compact standard form z˙ = f (z) +
2 &
gi (z)ui
q2
i=1
where (z1 , z2 , z3 , z4 ) := (q1 , q2 , q˙1 , q˙2 ).
re n le st
I. I NTRODUCTION
h gt
B. Stabilizing Controller k0 + u2 Humans are excellent walkers, able of energy efficient k0 + u1 Given the system dynamics (4), a sta locomotion and quick adaptation to different terrains. This is has been designed. The reference for thi of the stable autonomous gaits that th achieved by a complicated musculoskeletal system, comprisα0 in the absence of control input. In part ing many muscles, tendons and joints. However, it was shown q1 an autonomous gait, it can be parameter in [1] that the human gait can be accurately modeled by aFig. 1. Ideal(a)model variable , which allows a complete Theof bipedal (b) CADq1rendering the bipedal spring-mass walker—The legs model are compliant, and the relatively simple bipedal spring-mass model. In particular,stiffness can be invidually controlled by a control input ui . The trajectory of desired gait by ∗the two reference function functionrealization. q2 (q1 ) describes the desir the hip is Fig. denoted it determines the transitionsmodel between single 1.by q(t), Theandbipedal spring-mass and theThe prototype the model explicitly allows for a double support phaseand double support phase by intersections of the surface S. A single step q˙1∗ (q1 ) its desired forward velocity. begins and ends at the VLO, at which a relabeling of the legs takes place, The reference functions can be redefine and reproduces the ground reaction force profiles that aremaking q1 a cyclic variable. observed in human walking gaits. Moreover, it was shown z2∗ (z1renders ) := q2∗ (q1 ),a z3∗ (z1 ) := In [3], we proposed a stabilizing controller that ends when the system is in Vertical Leg Orientation (VLO), that the model encodes a variety of passive gait patterns, passive desired gait stable against external disturbances by in which the hip is exactly above the supporting leg in the Then, two error functions can be defined including running. In a more detailed study, presented insinglecontrolling stiffness ki . The control law significantly support phase. the The leg transition to the double support h1 = z2 − z2∗ is determinedthe by the parameters α0of andthe L0 , i.e. the In particular, it was [2], it was shown that a large subset of the possible gaits arephaseincreases robustness gait. h2 = z3 − z3∗ angle at which the swing leg touches the ground and the rest asymptotically stable, and that these gaits have a relativelylengthshown that the controller is successful in dealing with disof the telescopic spring respectively. After each step, The system is fully actuated during the dou underactuated during the single suppor the legs are relabeled, making variable. large basin of attraction. 1 a cyclic turbances such as qswing leg dynamics andbutimpact forces that that the leg stiffness can be written as k0 + a switching controller is designed that ach In [3], we proposed to extend the bipedal spring-massui,Considering occur when legs with non-negligible mass are considered. i = 1, 2, it can be easily verified that the force generated to a neighborhood of the desired gait. model with variable compliance in the legs. By making theby the spring during the single support phase is given by: following control inputs were proposed in III. R EALIZATION OF THE ROBOT • During the single support phase, Fs = Fs ,0 + Fs ,u , leg stiffness controllable, a control input is made available ! "# $ Currently, a robot is built that approaches the ideal bipedal L ' 2 q 0 1 1 that can be used to stabilize a desired gait. Because the Fs ,0 = k0 −1 u1 = −Lf h1 − κd Lf q2 (1) L1 as closely L L h g f 1 spring-mass model as possible. The bulk of the "# $ ! passive gaits are stable and locally attractive, control input L0 q1 and u2the ≡ 0;legs are −1 Fs ,u = closely tou the hip joint, while is only required to converge to a neighborhood of the gait. mass is located q2 1 L1 • During the double support phase, % made of lightweight carbon fiber tubes. The stiffness of the Then, once convergence is achieved, no additional control iswhere L1 = q12 + q22 is the length of the leg. During the # 2 # $ −Lf h1 − κd Lf u1 stiffness legs is controlled by the novel vsaUT-II variable double support phase, both legs exert a force on the hip mass. = A−1 required to sustain the gait. u2 −Lf h2 − The additional force is given by: the prototype. In this work, we describe the realization process of a actuator [4]. Fig. 1b shows a CAD rendering of with # Fs = Fs ,0experiments + Fs ,u , Currently, " # with $ the prototype are under devel! L L h Lg L bipedal walking robot, based on the principles of the bipedal A= g f 1 L0 q1 − a − 1 of validating the simulation results. LIn g L f h2 L g L opment,Fswith aim ,0 = kthe 0 spring-mass model with variable compliant legs. q2 (2) L2 $ "# 2 the variable particular, the! Ltests are designed to show that where L h , L h and L i f i g Lf hi are th q − a 0 f 1 −1 u2 F ,u = derivatives of the error functions hi , defi II. T HE B IPEDAL S PRING -M ASS M ODEL q2 L2 stiffness s actuation increases the robustness sufficiently to the vector fields defined in (4), and κd , κp wherestabilize a denotes the touchdown point of the former swing gait of the walker. Fig. 1a schematically depicts the bipedal spring-massleg and L = %(qthe This controller achieves: 2 2 2 1 − a) + q2 is the length of this leg. model. It consists of a point mass m, and two massless tele- In addition to the forces exerted byRthe springs, the hip lim h1 (t) = 0 EFERENCES t→∞ mass is also subject to the gravitational acceleration g0 . By scopic springs with rest length L0 and controllable stiffnessrearranging (1) and (2), the complete dynamics of the system and, for some 0, [1] H. Geyer, A. Seyfarth, and R. Blickhan, “Compliant legε > behaviour ki = k0 + ui , i = 1, 2. During the transition from singlecan be written as: basic dynamics of walking and running,” Proceedings of the explains $ # $# $ # lim |h2 (t)| < ε 2 support to double support, the former swing leg is assumed m 0Royal no. 1603, pp. 2861–2867, 2006. t→∞ q¨1 Society 0 B, vol. 273, & Bi (q)ui (3) + − Fs (q) = J. Rummel, Y.0 Blum, H. M. Maus, C. Rode, and Seyfarth, 0 m q¨2 mg TheseA.properties are“Stable proven in REF . to touch down at an angle α0 . For appropriately chosen [2] i=1 and robust walking with compliant legs,” in Proceedings of the IEEE parameters m, L0 , k0 , α0 and initial conditions q(0), q(0), ˙ the International Conference on Robotics and Automation, 2010, pp. 5250– model will show a stable passive (ui ≡ 0) gait trajectory q(t). 5255. L0
1
1
1
1
1
1
2
2
2
2
1
2
1
2
j
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0
This work has been funded by the European Commission’s Seventh Framework Programme as part of the project VIACTORS under grant no. 231554. {l.c.visser,s.stramigioli,r.carloni}@utwente.nl, MIRA Institute, Department of Electrical Engineering, University of Twente, 7500 AE Enschede, The Netherlands.
[3] L. C. Visser, S. Stramigioli, and R. Carloni, “Robust bipedal walking with variable stiffness actuation,” submitted to the IEEE International Conference on Robotics and Automation 2012. [4] S. S. Groothuis, G. Rusticelli, A. Zucchelli, S. Stramigioli, and R. Carloni, “The vsaUT-II: a novel rotational variable stiffness actuator,” submitted to the IEEE International Conference on Robotics and Automation 2012.
