3LP: a linear 3D-walking model including torso and swing dynamics

In this paper, we present a new model of biped locomotion which is composed of three linear pendulums (one per leg and one for the whole upper body) to describe stance, swing and torso dynamics. In addition to double support, this model has different actuation possibilities in the swing hip and stance ankle which could be widely used to produce different walking gaits. Without the need for numerical time-integration, closed-form solutions help finding periodic gaits which could be simply scaled in certain dimensions to modulate the motion online. Thanks to linearity properties, the proposed model can provide a computationally fast platform for model predictive controllers to predict the future and consider meaningful inequality constraints to ensure feasibility of the motion. Such property is coming from describing dynamics with joint torques directly and therefore, reflecting hardware limitations more precisely, even in the very abstract high level template space. The proposed model produces human-like torque and ground reaction force profiles and thus, compared to point-mass models, it is more promising for precise control of humanoid robots. Despite being linear and lacking many other features of human walking like CoM excursion, knee flexion and ground clearance, we show that the proposed model can predict one of the main optimality trends in human walking, i.e. nonlinear speed-frequency relationship. In this paper, we mainly focus on describing the model and its capabilities, comparing it with human data and calculating optimal human gait variables. Setting up control problems and advanced biomechanical analysis still remain for future works.Comment: Journal paper under revie

Push recovery with stepping strategy based on time-projection control

In this paper, we present a simple control framework for on-line push recovery with dynamic stepping properties. Due to relatively heavy legs in our robot, we need to take swing dynamics into account and thus use a linear model called 3LP which is composed of three pendulums to simulate swing and torso dynamics. Based on 3LP equations, we formulate discrete LQR controllers and use a particular time-projection method to adjust the next footstep location on-line during the motion continuously. This adjustment, which is found based on both pelvis and swing foot tracking errors, naturally takes the swing dynamics into account. Suggested adjustments are added to the Cartesian 3LP gaits and converted to joint-space trajectories through inverse kinematics. Fixed and adaptive foot lift strategies also ensure enough ground clearance in perturbed walking conditions. The proposed structure is robust, yet uses very simple state estimation and basic position tracking. We rely on the physical series elastic actuators to absorb impacts while introducing simple laws to compensate their tracking bias. Extensive experiments demonstrate the functionality of different control blocks and prove the effectiveness of time-projection in extreme push recovery scenarios. We also show self-produced and emergent walking gaits when the robot is subject to continuous dragging forces. These gaits feature dynamic walking robustness due to relatively soft springs in the ankles and avoiding any Zero Moment Point (ZMP) control in our proposed architecture.Comment: 20 pages journal pape

Imprecise dynamic walking with time-projection control

We present a new walking foot-placement controller based on 3LP, a 3D model of bipedal walking that is composed of three pendulums to simulate falling, swing and torso dynamics. Taking advantage of linear equations and closed-form solutions of the 3LP model, our proposed controller projects intermediate states of the biped back to the beginning of the phase for which a discrete LQR controller is designed. After the projection, a proper control policy is generated by this LQR controller and used at the intermediate time. This control paradigm reacts to disturbances immediately and includes rules to account for swing dynamics and leg-retraction. We apply it to a simulated Atlas robot in position-control, always commanded to perform in-place walking. The stance hip joint in our robot keeps the torso upright to let the robot naturally fall, and the swing hip joint tracks the desired footstep location. Combined with simple Center of Pressure (CoP) damping rules in the low-level controller, our foot-placement enables the robot to recover from strong pushes and produce periodic walking gaits when subject to persistent sources of disturbance, externally or internally. These gaits are imprecise, i.e., emergent from asymmetry sources rather than precisely imposing a desired velocity to the robot. Also in extreme conditions, restricting linearity assumptions of the 3LP model are often violated, but the system remains robust in our simulations. An extensive analysis of closed-loop eigenvalues, viable regions and sensitivity to push timings further demonstrate the strengths of our simple controller

Towards Robust Bipedal Locomotion:From Simple Models To Full-Body Compliance

Thanks to better actuator technologies and control algorithms, humanoid robots to date can perform a wide range of locomotion activities outside lab environments. These robots face various control challenges like high dimensionality, contact switches during locomotion and a floating-base nature which makes them fall all the time. A rich set of sensory inputs and a high-bandwidth actuation are often needed to ensure fast and effective reactions to unforeseen conditions, e.g., terrain variations, external pushes, slippages, unknown payloads, etc. State of the art technologies today seem to provide such valuable hardware components. However, regarding software, there is plenty of room for improvement. Locomotion planning and control problems are often treated separately in conventional humanoid control algorithms. The control challenges mentioned above are probably the main reason for such separation. Here, planning refers to the process of finding consistent open-loop trajectories, which may take arbitrarily long computations off-line. Control, on the other hand, should be done very fast online to ensure stability. In this thesis, we want to link planning and control problems again and enable for online trajectory modification in a meaningful way. First, we propose a new way of describing robot geometries like molecules which breaks the complexity of conventional models. We use this technique and derive a planning algorithm that is fast enough to be used online for multi-contact motion planning. Similarly, we derive 3LP, a simplified linear three-mass model for bipedal walking, which offers orders of magnitude faster computations than full mechanical models. Next, we focus more on walking and use the 3LP model to formulate online control algorithms based on the foot-stepping strategy. The method is based on model predictive control, however, we also propose a faster controller with time-projection that demonstrates a close performance without numerical optimizations. We also deploy an efficient implementation of inverse dynamics together with advanced sensor fusion and actuator control algorithms to ensure a precise and compliant tracking of the simplified 3LP trajectories. Extensive simulations and hardware experiments on COMAN robot demonstrate effectiveness and strengths of our method. This thesis goes beyond humanoid walking applications. We further use the developed modeling tools to analyze and understand principles of human locomotion. Our 3LP model can describe the exchange of energy between human limbs in walking to some extent. We use this property to propose a metabolic-cost model of human walking which successfully describes trends in various conditions. The intrinsic power of the 3LP model to generate walking gaits in all these conditions makes it a handy solution for walking control and gait analysis, despite being yet a simplified model. To fill the reality gap, finally, we propose a kinematic conversion method that takes 3LP trajectories as input and generates more human-like postures. Using this method, the 3LP model, and the time-projecting controller, we introduce a graphical user interface in the end to simulate periodic and transient human-like walking conditions. We hope to use this combination in future to produce faster and more human-like walking gaits, possibly with more capable humanoid robots

3LP: A linear model of locomotion including falling, swing and torso dynamics

In this article, we present a new linear model (3LP) for bipedal locomotion which can describe swing and torso dynamics as well as falling. Compared to traditional inverted-pendulum based models, 3LP produces more human-like center of mass trajectory and swing motion which is missing in point-mass models. Different periodic gaits can be found with 3LP and modulated online without any numerical integration. Using linearity properties, we derive closed-form solutions and a transition matrix which describe state evolution over footsteps. This matrix as well as linear inequalities on inputs and states can be used in online model predictive controllers (MPC) to guaranty feasibility, robustness and optimality of the motion. Besides, 3LP can also be a powerful tool in bio-mechanics as it provides human-like ground reaction and torque profiles

Designing a virtual whole body tactile sensor suit for a simulated humanoid robot using inverse dynamics

In this paper, we propose a novel architecture to estimate external forces applied to a compliantly controlled balancing robot in simulations. We use similar dynamics equations used in the controller to find mismatches in the available sensory data and associate them to an unknown external force. Then by decomposing Jacobians, we search over the surface of all body links in the robot to find the force application point. By approximating link geometries with ellipsoids, we can derive analytic solutions to solve the search problem very fast in real time. The proposed approach is tested on a complex humanoid robot in simulations where it outperforms static estimators over fast dynamic motions. We foresee a lot of applications for this method especially in human-robot interactions where it can serve as a whole body virtual suit of tactile sensors. It can also be very useful in identifying the inertial properties of objects being manipulated or mounted on the robot like a backpack

A virtual tactile sensing suit for humanoids based on dynamic equations and internal sensors

In this article, we propose a multi-staged algorithm to detect the magnitude, direction and location of a single external force applied to a humanoid robot while performing dynamic tasks. We use contact force and joint torque sensors as well as IMU to estimate accelerations and build the equation of motion. Then using a search process and based on Jacobian decomposition, we analytically find the point on the whole body which can best account for the error observed in the equation of motion. The method can provide reasonably stable and consistent estimations, even during fast dynamic motions

Time-Projection control on 3LP, a simple idea to deal with intermittent pushes online

In this article, we propose a new controller for recovering intermittent pushes during bipedal locomotion. We use 3LP as a template model which can provide closed-form solutions for state evolution. The idea behind our controller is to project the perturbed state of current time-step back to the beginning of the hybrid phase, use the expertise of a discrete controller and then apply the resulting optimal policy to the system at the current time-step. Linear properties of 3LP makes such calculation very fast and effective. By optimizing a certain cost function, we find the most robust and generic projecting configuration which outperforms the discrete controller itself

Modeling robot geometries like molecules, application to fast multi-contact posture planning for humanoids

Traditional joint-space models used to describe equations of motion for humanoid robots offer nice properties linked directly to the way these robots are built. However, from a computational point of view and convergence properties, these models are not the fastest when used in planning optimizations. In this paper, inspired by Cartesian coordinates used to model molecular structures, we propose a new modeling technique for humanoid robots. We represent robot segments by vectors and derive equations of motion for the full body. Using this methodology in a complex task of multi-contact posture planning with minimal joint torques, we set up optimization problems and analyze the performance. We demonstrate that compared to joint-space models that get trapped in local minima, the proposed vector-based model offers much faster computational speed and a suboptimal but unique final solution. The underlying principle lies in reducing the nonlinearity and exploiting the sparsity in the problem structure. Apart from the specific case study of posture optimization, these principles can make the proposed technique a promising candidate for many other optimization-based complex tasks in robotics