An essential model for generating walking motions for humanoid robots
Introduction
In the design of walking gaits for humanoid robots a simple model is often desired. However, the simpler the model, the more inaccurate the solution that describes the behavior of the physical system. In spite of this weakness, the 3D Linear inverted pendulum (LIP) model is still the most used model to develop 3D walking gaits due to its simplicity and the fact that decoupled closed-form solutions can be found. For instance, in [1] and [2] a decomposition of the 3D LIP model is carried out to introduce a concept called Capture point which is useful for starting and stopping phases and also for designing periodic walking gaits. In [3] an optimal control problem using an approximate value function derived from the 3D LIP model while respecting the dynamic, input, and contact constraints of the full robot dynamics is solved. Recently, in [4] the 3D LIP model was used to design a biped walking pattern based on a new way of discretization named spatially quantized dynamics (SQD). In [5] the 3D LIP model is studied along with their energy-optimal gait planning based on geodesics in order to achieve a stable walking gait. As shown, the 3D LIP model is still largely used in the literature, however, as it is an approximate model, the resulting walking gaits cannot be directly implemented, since they do not have the same performance when they are realized by the complete model, therefore, as shown in these works, complementary control techniques or adjustments must be taken into account.
One of the main difficulties of walking studies is the equilibrium of the robot, i.e. to satisfy the contact hypothesis and in particular to avoid the rotation of the stance foot. Thus, the constraint on the Zero Moment Point (ZMP) is crucial (see [6]). One way to ensure this constraint is satisfied is by using high-level control to impose a desired evolution for the ZMP [7].
On the other hand, a walking gait can be an “automatic” task if the floor is flat and empty, or a task precisely defined by the environment. In the first case, the assumption that the motion of the robot depends only on its internal states is made [8], [9], [10], [11], [12]. In the second case, the modeling must be based on a reference linked to the environment to impose a precise pose of the landing foot even in presence of perturbations. In this case, a temporal evolution of the swing foot is often prescribed [7], [13], [14], [15].
The objective of this paper is to propose a new model of the same dimension as the 3D LIP model, i.e. of dimension 4, that considers the whole dynamics of the humanoid robots. An application of this model is to develop walking gaits that deal with the issues described above, i.e. fulfilling the equilibrium condition by ensuring the stance foot is in contact with the ground, and defining the body motion as a function of internal and external variables. The new model is called essential model in this paper. This model has been developed by taking into account the notion of zero dynamics, which is a very useful tool to analyze the internal dynamics of a system [16]. Among many applications, this tool has also been used to develop walking motions of underactuated systems such as in [10], [12], [17], [18], among others. However, unlike previous works, this paper is dedicated to build a reduced model for gait design and not for stability analysis. The original idea in this paper is to define a 3D dynamic relation between the two internal states (usually the horizontal position of the CoM) and the ZMP without the assumptions of the well-known 3D LIP model. Therefore, the main contribution of this work is the development of a new model for humanoid robots that can be used instead of the 3D LIP with several advantages with respect to it, as the following:
- 1.
Unlike the 3D LIP model, the essential model is not based on a mechanical approximation (concentrated mass) of the robot. Moreover, the motion and dynamics of all the robot’s body are taken into account.
- 2.
It is possible to impose a desired location for the ZMP during the whole step (as shown in Cases I and II in Section 6) or make the ZMP follow a desired path while the robot performs its motion (as shown in Case III in Section 6).
- 3.
Impacts of the swing foot with the ground can be considered in the computation of periodic walking gaits (as shown in Cases II and III in Section 6).
- 4.
By using internal and/or external information, the desired motion of the robot can be defined. Thereby, different essential models can be developed for different purposes. For instance, the motion of the robot can be defined as a function of the position of the center of mass (CoM) in horizontal plane, thus obtaining an essential model that can be used to develop autonomous walking gaits (as shown in Case II in Section 6). On the other part, by including the time as an external variable to develop the essential model, a precise pose of the landing foot can be imposed (as shown in Cases I and III in Section 6).
Furthermore, the proposed model can conceivably be used to develop motions in double support phases, However, in this paper, only a continuous single support (SS) phase and an instantaneous double support (DS) phase are taken into account.
On the other hand, unlike the 3D LIP model, the essential model is nonlinear, coupled and has a non-closed form formula. Despite this, many numerical studies can be conducted with this model, including gaits for climbing stairs, walking gaits on uneven ground, walking gaits that minimize energy consumption, rolling of the stance foot during walking, starting or stopping phases, reaction to disturbances by means of an extension of the capture point method, among others. However, since the aim of the paper is to introduce the essential model in a more general way, these tasks are not analyzed in here and they will be realized in the future.
The paper is structured as follows. In order to put this proposal in context, three dynamic models used in the literature are presented in Section 2. The new model is introduced in Section 3. Then, the connection between the essential model and the complete model in order to deduce the desired joint motions and the corresponding torques is detailed in Section 4. A brief discussion on generation of periodic walking patterns is presented in Section 5. The application of this model to the humanoid robot ROMEO as a case of study is presented in Section 6, where three cases are studied in order to show the effectiveness of the essential model. In there, a comparison with the 3D LIP model, and different walking gaits where impacts are considered and the ZMP follows a desired trajectory are found. Finally, the paper ends with several concluding remarks and perspectives in Section 7.
Section snippets
The dynamic model
The considered models of walking locomotion are inherently hybrid models. They are composed of continuous differential equations which describe the motion of the robot during SS and DS phases, and discrete components where the change of support takes place [19]. In this paper, an instantaneous DS phase is assumed. This assumption leads to a discontinuity in the velocities if the velocity of the swing foot is not zero at the transition.
The proposed essential model
A new model for generation of walking gaits for humanoid robots called essential model is proposed. A fully actuated -DOF robot is considered. Note that in SS phase, due to the unilaterally of the ground reaction force on the stance foot, a limited control of the CoM can be achieved due the fact that the ZMP must be inside of the support polygon. Therefore, in order to produce a model which is close to the 3D LIP model, the development of the essential model considers to control the position
From the essential model to joint motion
As known, it is difficult to propose some desired trajectories for the joints that produce a desired evolution of the ZMP, i.e. fulfill the conditions (21), (22). Therefore, points of interest of the robot can be chosen as controlled variables in order to define the desired trajectories for them. These points of interest can be the robot’s extremities such as the hands or the swing foot, or directly the joints.
Let us define the vector of generalized variables of the same dimension as ,
Generation of periodic walking patterns
In this section, the generation of periodic walking patterns are studied. The studied gaits are composed of a SS phase and an instantaneous DS phase. The starting and ending phases can also be studied but they are out of the scope of this paper.
Case study: The humanoid robot ROMEO
In this section, the essential model is exploited by taking into account the characteristics on the humanoid robot ROMEO developed by Softbank robotics [25]. This robot has a weight of 40.8 kg, a height of 1.46 m and it has 31 DoF (6 DoF for each leg, 1 DOF for the torso, 7 DOF for each arm and 4 DOF for the neck and head) as shown in Fig. 4.
In this paper, as explained in Section 4 the swing foot, the torso and the upper-body joints are chosen as points of interest. Thus, the configuration
Conclusion
The use of simple models, as the 3D LIP model, for building walking trajectories are useful. However, the vertical motion of the CoM is constrained and the ZMP is not keep to its desired location when the complete dynamics of the robot is considered. Therefore, further techniques for keeping the ZMP inside the convex hull of support are usually carried out. In this paper, a new model of the same dimension as the 3D LIP model, called essential model, has been proposed in order to deal with these
Acknowledgments
The authors are grateful with the ANR Robotex project in France, CONACyT in México and the China Scholarship Council for the support received.
Victor de-León-Gómez Victor de-León-Gómez received his M.S. and Ph.D. degrees both in electrical engineering from the Instituto Tecnológico de La Laguna, Mexico, in 2010 and 2014 respectively. He is a postdoctoral researcher at the Laboratoire des Sciences du Numérique de Nantes (LS2N) in France, working on dynamic modeling and walking patterns generation for humanoid robots. His research interests include bipedal locomotion, trajectory planning, modeling and control of mobile robots.
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Victor de-León-Gómez Victor de-León-Gómez received his M.S. and Ph.D. degrees both in electrical engineering from the Instituto Tecnológico de La Laguna, Mexico, in 2010 and 2014 respectively. He is a postdoctoral researcher at the Laboratoire des Sciences du Numérique de Nantes (LS2N) in France, working on dynamic modeling and walking patterns generation for humanoid robots. His research interests include bipedal locomotion, trajectory planning, modeling and control of mobile robots.
Qiuyue Luo Qiuyue LUO received the B.Eng. and M.Eng. degrees in mechanical engineering from Northwestern Polytechnical University (China) in 2013 and 2016. She is currently pursuing her Ph.D. in Robotics at the Laboratoire des Sciences du Numérique de Nantes (LS2N) in Centrale Nantes (France). Her research field is walking algorithm and control of humanoid robots.
Anne Kalouguine Born in Montpellier, France, in 1993. Studied in the CPGE Blaise Pascal (Orsay, France), ENSTA ParisTech (Palaiseau, France) and TUM (Munich, Germany). She is currently Ph.D. student in LS2N laboratory, in collaboration with Softbank Robotics, working on bipedal walking.
J. Alfonso Pámanes He was born in Torreon, Mexico, in 1953. He received the B.S. degree in 1978 from the La Laguna Institute of Technology (ITLag) in Torreon; the M.S. degree in mechanics in 1984 from the National Polytechnic Institute of Mexico (Mexico City); and the Ph.D. degree in mechanics from the University of Poitiers (France), in 1992. He is a professor in mechanics of robots at the ITLag. His research interests are in modeling and motion planning of serial, parallel and humanoid robots. He is author or co-author of more than 100 scientific papers published in national or international conferences and journals.
Yannick Aoustin Yannick Aoustin is Professor at University of Nantes, France. He earned his Ph.D. degree in Automatic in 1987 and his research degree for leading research and Ph.D. students in 2006. He has been engaged in research for more than 25 years and his research interests include mechanical systems under actuated legged robots, bipedal, nonlinear observers, and biomechanics. He has publications on journals circulated internationally and premier conferences. He is also a topic editor of “International Journal of Advanced Robotic Systems” and member of the editorial board of the journal “Multibody system dynamics”.
Christine Chevallereau Christine Chevallereau graduated from Ecole Nationale Supérieure de Mécanique, Nantes, France in 1985, and received the Ph.D. degree in Control and Robotics from Ecole Nationale Supérieure de Mécanique, Nantes in 1988. Since 1989, she has been with the CNRS in the Institut de Recherche en Communications et Cybernétique de Nantes and then Laboratoire des Sciences du Numérique de Nantes. She is deputy director of the LS2N. Her research interests include modeling and control of manipulators and locomotor robots, in particular biped, and bio-inspired robots.