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The HeiCub (iCub of Heidelberg University) humanoid robot. In red, the series elastic actuators, which are not considered in the context of this work.
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![Table 6 . [Hardware experiment result.] Key Performance Indicators...](publication/323130197/figure/tbl3/AS:631589671297058@1527594120312/Hardware-experiment-result-Key-Performance-Indicators-KPIs-for-the-ending-step_Q320.jpg)
Bipedal locomotion remains one of the major open challenges of humanoid robotics. The common approaches are based on simple reduced model dynamics to generate walking trajectories, often neglecting the whole-body dynamics of the robots. As motions in nature are often considered as optimal with respect to certain criteria, in this work, we present a...
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... robot has executed all four obtained motions, proving the feasibility of the motions and of the periodicity constraints. We can see the resulting CoM trajectories in red in Figure 10. We can observe from the results that the CoM trajectories could be followed closely by the robot; however, when the torso orientation minimization is not included, a bigger deviation can be observed in the height of the CoM trajectories in correspondence with the spikes; this is due to the impact forces as mentioned in the previous section. It seems however that the introduction of torso orientation minimization has a damping effect on the error when executed on the ...
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... the context of the European Project KoroiBot, IIT has delivered to Heidelberg University a reduced version of the standard iCub, consisting of 15 internal DOF: three in the torso and six in each leg, as shown in Figure 1. The robot is furthermore equipped with four Force Torque sensors (F/T), two in the upper legs and two in the feet. The robot has an on-board PC104 with a dual core, but has no battery; therefore, it needs to be connected to an external power supply by means of cables, which serve also as network communication cables, allowing one to use external computers to carry out bulky computations. The robot has also four custom Series Elastic Actuators (SEA) [33] with springs that can be unmounted to obtain rigid actuators. In the context of this paper, the springs are unmounted, and therefore, the joints are considered perfectly rigid. The robot has a weight of 26.4 kg, and it is 0.97 m tall. The leg length from the hip axis is 0.51 m, and the feet are 0.2 m long and 0.1 m wide. The weight distribution of the robot is about 6 kg for the torso, 5 kg for the pelvis and 7.5 kg each leg. All the kinematic and dynamic parameters of the robot are described in a URDF (Unified Robot Description Format) file, which was extracted directly from the original CAD model. The hardware limitations, including joint limits, joint velocity limits and torque limits, are as in Table 1. Please note that the velocity and torque limits have been obtained via experiments; therefore, they are not perfectly precise, i.e., these limits might be conservative, as they are set to guarantee the safety of the robot. ...
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Bipedal locomotion remains one of the major open challenges of humanoid robotics. The common approaches are based on simple reduced model dynamics to generate walking trajectories, often neglecting the whole-body dynamics of the robots. As motions in nature are often considered as optimal with respect to certain criteria, in this work we present an...
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... For decades, humanoid robots have been created and manufactured with inspiration from nature, particularly human anatomy and behavior [1]. However, due to the limits of bipedal robot control methods and modeling methodologies, the bipedal robot still needs some improvement as far as the walking algorithm is concerned. ...
... For more natural and biologically inspired motion generation that also exploits the full dynamics of the robotic system, the use of optimization, for example, an optimal control approach based on the robot's whole-body model, is favorable. Optimal control is a promising tool in motion generation as the motions are more feasible with respect to all constraints within the model and the motion task Hu and Mombaur (2018). In the past, optimization has already been successfully used to realize demanding motions on robots such as HRP-2 Koch et al. (2012a), Koch et al. (2012b), Lengagne et al. (2013), Koch et al. (2014). ...
To enable the application of humanoid robots outside of laboratory environments, the biped must meet certain requirements. These include, in particular, coping with dynamic motions such as climbing stairs or ramps or walking over irregular terrain. Sit-to-stand transitions also belong to this category. In addition to their actual application such as getting out of vehicles or standing up after sitting, for example, at a table, these motions also provide benefits in terms of performance assessment. Therefore, they have long been used as a sports medical and geriatric assessment for humans. Here, we develop optimized sit-to-stand trajectories using optimal control, which are characterized by their dynamic and humanlike nature. We implement these motions on the humanoid robot REEM-C. Based on the obtained sensor data, we present a unified benchmarking procedure based on two different experimental protocols. These protocols are characterized by their increasing level of difficulty for quantifying different aspects of lower limb performance. We report performance results obtained by REEM-C using two categories of indicators: primary, scenario-specific indicators that assess overall performance (chair height and ankle-to-chair distance) and subsidiary, general indicators that further describe performance. The latter provide a more detailed analysis of the applied motion and are based on metrics such as the angular momentum, zero moment point, capture point, or foot placement estimator. In the process, we identify performance deficiencies of the robot based on the collected data. Thus, this work is an important step toward a unified quantification of bipedal performance in the execution of humanlike and dynamically demanding motions.
... A number of studies were made for the robot HRP-2 using whole-body dynamics (18) (19) (20) , generating optimized gaits that were validated by experiments, but all studies relied on direct methods that do not estimate costates to enforce Pontryagin's Principle. More recently, Hu and Mombaur (21) also generated optimal solutions by a direct method for a whole-body model and validated it in the prototype iCub. The only study found so far that satisfied Pontryagin's Principle was made by Rostami and Bessonet (22) . ...
Optimization of robotic bipedal walking widely uses simplified dynamical models to minimize the computational cost and generate fast solutions, but this poses the problem of modeling errors and increased inaccuracy. Moreover, direct methods of optimization are almost always used; however, most are not implemented to satisfy Pontryagin's Principle, which can lead to suboptimal solutions. This study presents the optimization of planar robotic bipedal walking using exact dynamics. We apply pseudospectral optimal control to optimize gait generation using two accuracy requirements: nonlinear multi-degree-of-freedom whole-body dynamics and satisfied conditions of optimality according to Pontryagin's Principle. In order to consistently achieve solvability of optimization in this type of problem, we present a novel methodology consisting of synergic techniques that address well-known numerical difficulties of optimization.
... The previous studies focused on the solvability of optimization. Nowadays, with the improvement of computational power and the development of robust optimal solving methods and commercial softwares [6] [7], more and more studies optimizing the whole-body dynamics of biped walkers are being successfully realized [8] [9] [10] [11]. As a consequence, the solvability problem is already being addressed. ...
The two most common formulations of equation of motion for mechanical systems are derived by 1) vector analysis from Newton's Laws, and 2) energy analysis from Euler-Lagrange equations. A third alternative, derived from Hamilton's equations, is only well known in quantum mechanics and niche fields, specially if conservation of momentum is a central phenomenon. In Optimal Control, a core theorem is Pon-tryagin's Principle, whose main term-the control Hamiltonian-is inspired in the original Hamiltonian present in Hamilton's Equations. Even though there is a clear parallel between the two hamiltonians, dynamic formulations used in optimization are still largely based on Lagrangian mechanics and Classical mechanics. In this paper we present the competitiveness of Hamiltonian dynamics in the optimization of robotic bipedal walking-a complex nonlinear system-using its exact whole-body dynamics. With the use of Hamiltonian dynamics, we observed a significant increase in precision of optimization, with the trade-off of bigger computational time. Our results demonstrate the advantage of use of Hamiltonian dynamics for precise offline optimizations, and suggest the maintenance of use of Lagrangian dynamics for fast online optimizations.
... Many biped robot algorithms have been developed, such as balance control and inverse kinematics [1][2][3]. The zero moment point (ZMP) and walking dynamics analysis can be adopted to enable a robot to walk steadily [4][5][6][7]. However, these methods require copious calculations; researchers must develop an approximate or simple dynamic model. ...
In this paper, a real-time balance control method is designed and implemented on a field-programmable gate array (FPGA) chip for a small-sized humanoid robot. In the proposed balance control structure, there are four modules: (1) external force detection, (2) push recovery balance control, (3) trajectory planning, and (4) inverse kinematics. The proposed method is implemented on the FPGA chip so that it can quickly respond to keep the small-sized humanoid robot balanced when it is pushed by external forces. A gyroscope and an accelerometer are used to detect the inclination angle of the robot. When the robot is under the action of an external force, an excessively large inclination angle may be produced, causing it to lose its balance. A linear inverted pendulum with a flywheel model is employed to estimate a capture point where the robot should step to maintain its balance. In addition, the central pattern generators (CPGs) with a sinusoidal function are adopted to plan the stepping trajectories. Some experimental results are presented to illustrate that the proposed real-time balance control method can effectively enable the robot to keep its balance to avoid falling down.
... This work explores the potential of pseudospectral optimization for solving whole-body models robustly and fast. Unlike the absolute majority of studies (13) (14) (15) , our approach satisfies Pontryagin's Principle, and solves the problem faster than the only other study found that also satisfies it (16) . Our objective is to improve the ability to reach truly optimal solutions, improve robustness of solutions removing the modelling errors from simplified models and to improve computational speed towards online optimal gait generation of whole-body models. ...
Due to the high computational cost of evaluating whole-body dynamics of systems with many degrees of freedom, simplified dynamics models are widely used in optimization of robotic bipedal walking. Such models generate solutions fast, but cause modelling errors and increase imprecision in the solution. This study presents the optimization of energy consumption in the walking of a planar robot model using its whole-body dynamics, generating solutions without modelling errors and satisfying Pontryagin's Principle. The model is a nonlinear planar robot model with 6 degrees of freedom. Solutions are achieved in about 20s. Pseudospectral optimization is applied for the first time in optimized trajectory generation of a biped walker using its whole-body dynamics. The chosen method is implemented in the commercial software DIDO.
... These methods have unique advantages and disadvantages. Recently, newer gait-planning methods [9][10][11][12][13] have been developed that do not rely on the ZMP and have indeed produced marked improvements in humanoid robot walking. ...
The most important feature of this paper is to transform the complex motion of robot turning into a simple translational motion, thus simplifying the dynamic model. Compared with the method that generates a center of mass (COM) trajectory directly by the inverted pendulum model, this method is more precise. The non-inertial reference is introduced in the turning walk. This method can translate the turning walk into a straight-line walk when the inertial forces act on the robot. The dynamics of the robot model, called linear inverted pendulum (LIP), are changed and improved dynamics are derived to make them apply to the turning walk model. Then, we expend the new LIP model and control the zero moment point (ZMP) to guarantee the stability of the unstable parts of this model in order to generate a stable COM trajectory. We present simulation results for the improved LIP dynamics and verify the stability of the robot turning.
... Hu and Mombaur present an optimal control-based approach for the humanoid robot iCub that allows to generate optimized walking motions using a precise whole-body dynamic model of the robot. The optimal control problem is formulated to minimize a set of desired objective functions with respect to physical constraints of the robot and contact constraints of the walking phases [2]. Jiang et al. propose a jumping control scheme for a bipedal robot to perform a high jump taking into account the motion during the launching phase. ...
... Not only would the joint angle limits, velocities, and torque limits need to be taken into account, but also precise descriptions of contact dynamics, i.e. contact phases and contact points, need to be taken care of. In Hu and Mombaur (2018), we used the optimal control with whole-body dynamic model approach to generate a level ground walking sequence consisting of three steps for the HeiCub robot; the multiple phases optimization process required an average of 15 hours of computation time. For any of the scenarios described in this work, the computation time would be much greater and far beyond the time used to carry out the whole workflow described in this paper, which is of the order of seconds, given that the optimal control with the template model needs some seconds to compute a new solution and the inverse kinematics also needs just a few seconds to obtain the final whole-body joint trajectories. ...
In this paper, we present an inverse optimal control-based transfer of motions from human experiments to humanoid robots and apply it to walking in constrained environments. To this end, we introduce a 3D template model, which describes motion on the basis of center-of-mass trajectory, foot trajectories, upper-body orientation, and phase duration. Despite its abstract architecture, with prismatic joints combined with damped series elastic actuators instead of knees, the model (including dynamics and constraints) is suitable for describing both human and humanoid locomotion with appropriate parameters. We present and apply an inverse optimal control approach to identify optimality criteria based on human motion capture experiments. The identified optimal strategy is then transferred to a humanoid robot template model for gait generation by solving an optimal control problem, which takes into account the properties of the robot and differences in the environment. The results of this step are the center-of-mass trajectory, the foot trajectories, the torso orientation, and the single and double support phase durations for a sequence of steps, allowing the humanoid robot to walk within a new environment. In a previous paper, we have already presented one computational cycle (from motion capture data to an optimized robot template motion) for the example of walking over irregular stepping stones with the aim of transferring the motion to two very different humanoid robots (iCub@Heidelberg and HRP-2@LAAS). This study represents an extension, containing an entirely new part on the transfer of the optimized template motion to the iCub robot by means of inverse kinematics in a dynamic simulation environment and also on the real robot.
Shoe manufacturing technology is advancing faster than new shoe designs can viably be evaluated in human subject trials. To aid in the design process, this paper presents a model for estimating how new shoe properties will affect runner performance. This model assumes runners choose their gaits to optimize an intrinsic, unknown objective function. To learn this objective function, a simple two-dimensional mechanical model of runners was used to predict their gaits under different objectives, and the resulting gaits were compared to data from real running trials. The most realistic model gaits, i.e., the ones that best matched the data, were obtained when the model runners minimized the impulse they experience from the ground as well as the mechanical work done by their leg muscles. Using this objective function, the gait and thus performance of running under different shoe conditions can be predicted. The simple model is sufficiently sensitive to predict the difference in performance of shoes with disruptive designs but cannot distinguish between existing shoes whose properties are fairly similar. This model therefore is a viable tool for coarse-grain exploration of the design space and identifying promising behaviors of truly novel shoe materials and designs.
