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

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

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

Superconducting Coil Compression by Scissor Laminations

A new system of coil compression has been designed which uses iron laminations to transfer the pressure from an outer shrink ring to the coil. The laminations are simple circular discs around the coil with the peculiarity that the rim is slightly eccentric as compared to the coil. Successive laminations are mounted with different angular orientations to oppose their eccentricities. The shrink ring pushed these discs inwards against the coil creating compression by a scissor movement. Tests on mechanical models of single as well as multiple aperture magnets have shown it to work as expected. The system has already successfully been applied to several corrector magnets for LHC. The advantages are low cost (suppression of the usual collars), increased coil compression in particular from cooling down, and field enhancement from having the iron close to the coil

Superconducting Sextupole Corrector Magnet for the LHC Main Dipoles

Each LHC main dipole will be equipped with small sextupole corrector ma g nets with a field strength of 1970 x2 T/m2 and a magnetic length of 100 mm designed to correct the sextupole field errors. The paper presents a cosine-q type of design where much emphasis has been put on the cost reduction because these magnets have to be made in large series of some 2500 pieces. We describe the design of a two-layer coil which can be wound automatically. The winding starts in the middle of the wire with the only joggle, the layer jump, which is housed in a corresponding groove in the end of the central island. The two layers are wound simultaneously turning in opposite directions to find their position without the need of local tooling. The coil ends are closely packed and need no end spacers. The 18 pole perturbation introduced by the ends is corrected by the position of the coil block in the straight part. The yoke is made of iron laminations of the "Scissors type" which transmit the pre-stress from the outer aluminium shrink ring to the coil. This allows the iron to be close to the coil for field enhancement and also boosts the pre-stress in the coil due to the cool down contractions. The paper describes the experience with the magnet construction and gives the first test results

Simulation of the Effect of a Series of Superconducting Magnets on a Quenching Magnet using a Controlled Current Pulse

In the LHC, the superconducting corrector magnets will be powered in series of up to 154 magnets. For protection in case of a quench, each magnet has been equipped with a parallel resistor as a bypass for the current. To validate and optimize the parallel resistor value, a test arrangement has been set up which allows quenching a single magnet as if it were connected in a large series of magnets. This simulation is obtained by maintaining the current for a certain time interval after the quench occurred. Calculations have shown that, depending on the magnet type, a current duration (after quench) of 0.2 s to 1 s simulates correctly the effect of the series of magnets. The paper gives calculation results comparing the real situation with the simulated one and reports on the test set-up that will be used to optimize the parallel resistors

Definition and composition of motor primitives using latent force models and hidden Markov models

In this work a different probabilistic motor primitive parameterization is proposed using latent force models (LFMs). The sequential composition of different motor primitives is also addressed using hidden Markov models (HMMs) which allows to capture the redundancy over dynamics by using a limited set of hidden primitives. The capability of the proposed model to learn and identify motor primitive occurrences over unseen movement realizations is validated using synthetic and motion capture data