Parametrically Excited Dynamic Bipedal Walking

This paper proposes a stabilization method for dynamic walking of a bipedal robot with real-time optimization of capture point trajectories. We used the capture point trajectories to generate the control input, which is the desired zero moment point (ZMP) with a sliding-mode ZMP controller to follow the desired ZMP. This method enables the robot to implement various dynamic walking commands, such as forward stride, lateral stride, walking direction, single support time, and double support time. We also adopted enhanced dynamics with the three mass linear inverted pendulum model (3M-LIPM). First, the compensated ZMP is calculated by both walking commands and kinematic configuration of the robot in closed form. Then, the walking pattern is obtained by using initial and boundary conditions of the 3M-LIPM, which satisfies the walking commands. The capture point (CP) trajectory is optimized in real time to control the walking stability and a capture point tracking controller is used for tracking the optimized CP trajectory, which generates an optimal control input that is near the center of the support polygon. The performance of the proposed stabilization method was verified by a dynamics simulator, Webots, and comparison with the original capture point controller-based walking algorithm is presented.

Many legged robots have been developed, and some of them have already achieved dynamically stable bipedal and quadruped locomotion. Common characteristics of these conventional legged robots are that their motions are precisely manipulated by their control (i.e., gait and balance). In the case of ASIMO (Hirose et al., 2001) and HRP2 (Kaneko et al., 2004), their dynamically stable bipedal locomotion is based primarily on the Zero Moment Point (ZMP) concept, which was originally developed by Vukobratovic (Vukobratovic, 1969): the ZMP is a fictitious point on the ground plane, where the torques around the (horizontal) x and y axes (generated by ground reaction forces, inertial forces, and torques) are equal to zero. These robots dynamically stabilize their balance by manipulating the ZMP to remain within the support polygon area defined by their square feet, and their gait control is applied on top of the balance control. As a result, these conventional legged robots require rapid information processing and high-power drives to achieve locomotion. Therefore, they are characterized as having more complex control systems and larger energy consumption than those of biological locomotion. On the other hand, these design requirements (i.e., fast information processing and highpower drives) are regarded as disadvantageous in biologically inspired robotics. In the field, the specific behaviors and/or structures of biological organisms are imitated (Vogel, 1998) (Alexander, 2002) with robotic technology. As a result, the robot systems have simple design requirements compared to conventional robot systems, and achieve complex tasks. Thus, the design effort tends to focus on control complexity and energy requirements. A representative instance is the Passive Dynamic Walker (PDW), which was originally developed by McGeer (McGeer, 1990). The PDW has no controller (i.e., no sensor and no motor), and the structure is based on the physical characteristics of human walking: passive hip joints, latch knee joints, and curved feet. The PDW walks down a slope, and its structure exploits gravity as the driving force. This design principle has been applied to the Cornell Biped (Collins & Ruina, 2005) and Denise at TU Delft (Wisse, 2003), and the powered passive dynamic walkers achieved passive dynamic walking on a flat plane. In a summary of these two approaches, conventional robots require expensive design components to achieve dynamically stable locomotion. On the other hand, biologically inspired robots require physical characteristics that exploit their own dynamics to achieve

The hardware compatibility of legged locomotion is often illustrated by Zero Moment Point (ZMP) that has been extensively studied for decades. One of the most popular models for computing the ZMP is the linear inverted pendulum (LIP) model that expresses ZMP as a linear function of the center of mass(COM) and its acceleration. In the real world, however, it may not accurately predict the true ZMP of hardware due to various reasons such as unmodeled dynamics and differences between simulation model and hardware. In this paper, we aim to improve the theoretical ZMP model by learning the real hardware dynamics from experimental data. We first optimize the motion plan using the theoretical ZMP model and collect COP data by executing the motion on a force plate. We then train a new ZMP model that maps the motion plan variable to the actual ZMP and use the learned model for finding a new hardware-compatible motion plan. Through various locomotion tasks of a quadruped, we demonstrate that motions planned for the learned ZMP model are compatible on hardware when those for the theoretical ZMP model are not. Furthermore, experiments using ZMP models with different complexities reveal that overly complex models may suffer from over-fitting even though they can potentially represent more complex, unmodeled dynamics.

This paper investigates what effect semicircular feet had on dynamic bipedal walking. It has been clarified in paper (I) that underactuated virtual passive dynamic walking can be realized by using the rolling effect, which acts as the ankle-joint torque virtually. This paper (II) then studies the rolling effect on the gait performances in detail through theoretical investigations and numerical analyses. We first perform parameter studies, and confirm that a suitable choice of the foot radius increases walking speed effectively. It is then theoretically clarified that increasing walking speed is achieved not by the rolling effect during the single support phase but by reducing the energy-dissipation due to heel-strike collisions. It is also confirmed that mass of the leg is necessary to drive the robot's center of mass in the walking direction. Furthermore, extension to redundant models with knee-joints is discussed.

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> The control of a biped humanoid is a challenging task due to the hard-to-stabilize dynamics. Walking reference trajectory generation is a key problem. Linear Inverted Pendulum Model (LIPM) and Zero Moment Point (ZMP) Criterion-based approaches in stable walking reference generation are reported. In these methods, generally, the ZMP reference during a stepping motion is kept fixed in the middle of the supporting foot sole. This kind of reference generation lacks naturalness, in that the ZMP in the human walk does not stay fixed, but it moves forward under the supporting foot. This paper proposes a reference generation algorithm based on the LIPM and moving support foot ZMP references. The application of Fourier series approximation simplifies the solution, and it generates a smooth ZMP reference. A simple inverse kinematics-based joint space controller is used for the tests of the developed reference trajectory through full-dynamics 3-D simulation. A 12-DOF biped robot model is used in the simulations. Simulation studies suggest that the moving ZMP references are more energy efficient than the ones with fixed ZMP under the supporting foot. The results are promising for implementations. </para>

The authors have been using the ZMP (zero moment point) as a criterion to distinguish the stability of walking for a biped walking robot which has a trunk. The authors introduce a control method of dynamic biped walking for a biped walking robot to compensate for the three-axis (pitch, roll and yaw-axis) moment on an arbitrary planned ZMP by trunk motion. The authors developed a biped walking robot and performed a walking experiment with the robot using the control method. The result was a fast dynamic biped walking at the walking speed of 0.54 s/step with a 0.3 m step on a flat floor. This walking speed is about 50% faster than that with the robot which compensates for only the two-axis (pitch and roll-axis) moment by trunk motion.

Linear Inverted Pendulum model (LIPM) is a classical theory of biped robot gait planning. However, the fixed Zero Moment Point (ZMP) of each step does not conform to the laws of human motion. In this paper, we proposed a VSP-based gait planning algorithm to make the ZMP trajectory comply the law of Heel to Toe and smooth the velocity of the center of mass (CoM). We found it remarkable that combining Virtual Supporting Point (VSP) with classical LIPM increases gait stability. The simulation and hardware experimental results on THU-Walker Platform strongly verified our algorithm.

This paper focuses on real-time walking pattern generation for humanoid robots with linear inverted pendulum model (LIPM). In general, there are many issues in generating proper walking patterns of center of mass and zero moment point (ZMP) with the LIPM since the LIPM has two drawbacks such as instability and non-minimum phase property. For resolving these difficulties, the paper proposes a new real-time approach by combining a feedback and a feedforward controller. The feedback controller employs a pole placement method which shifts the poles of the LIPM in order to improve system stability. The feedforward controller utilizes advanced pole-zero cancelation by series approximation method (APZCSA) for reducing non-minimum phase property which occurs by an unstable zero and is not able to be dealt with by the feedback controller. In addition, the APZCSA improves the tracking error induced by finite series approximation. Using the two controllers, the proposed method makes the transfer function of overall walking pattern generation system approximately unity and consequently generates a stable walking pattern which follows a desired ZMP according to walking path. The efficiency of the proposed method is verified by walking pattern planning examples and experiments with the humanoid robot MAHRU-R.

Many biped robots that achieve high-efficient walking have been proposed in the recent years. Many researchers have thus developed biped robots that achieve active walking based on passive dynamic walking (PDW). Although these biped robots achieve energy-efficient active walking, their walking speed is generally slow. We have proposed a flat-footed PDW robot with the ankle springs and inerters. Using the ankle inerters, this robot achieves more high-speed PDW than PDW of conventional flat-footed robots. We thus developed a flat-footed biped robot achieving active walking on level ground by mimicking the flat-footed PDW. By simulations and experiments, we show that our biped robot achieves more highspeed and energy-efficient active walking than conventional biped robots.

This research presents a simple mapping methodology for gait biomechanics of a human being into joint angles of a 10 degrees of freedom (DOF) biped robot. The joint angles are mapped by considering the zero moment point (ZMP) criterion. The walking control of the robot is performed by an optimal state feedback controller. The walking trajectories are planned in the sagittal plane, and they are generated in compliance with the ZMP of the robot – keeping the robot within the support polygon – by dividing the control process in two stages: unique support and double support. A linear inverted pendulum model (LIPM) is used as an approximate single mass model of the robot during gait. Results of this research include simulation-based analysis and real-time implementation results, which show accurate robot movements with limited robustness under slippery platforms.

To achieve bipedal walking in real human environments, a bipedal robot should be capable of modifiable walking both on uneven terrain with different heights and on flat terrain. In this paper, a novel walking pattern generator based on a 3-D linear inverted pendulum model (LIPM) is proposed to achieve this objective. By adopting a zero moment point (ZMP) variation scheme in real time, it is possible to change the center-of-mass (COM) position and the velocity of the 3-D LIPM throughout the single support phase. Consequently, the proposed method offers the ability to generate a modifiable pattern for walking on uneven terrain without the necessity for any extra footsteps to adjust the COM motion. In addition, a control strategy for bipedal walking on uneven terrain with unknown height is developed. The torques and ground reaction force are measured through force-sensing resisters (FSRs) on each foot and the foot of the robot is modeled as three virtual spring-damper models for the disturbance compensation. The methods for generating the foot and vertical COM of 3-D LIPM trajectories are proposed to achieve modifiable bipedal walking on uneven terrain without any information regarding the height of the terrain. The effectiveness of the proposed method is confirmed through dynamic simulations.

The presented work is motivated to implement the principles of passive dynamic walking (PDW) on the powered bipeds. The Virtual Slope Walking (VSW), as one PWD control approach, is chosen to be mathematically analyzed and realized to achieve fast dynamic walking. Our method of formulating the fixed point of the limit cycle gait is presented, and the walking pattern generation is designed in the polar coordinate for the better consistency with the model. The algorithms are implemented on a planar robot Stepper-2D, which achieved the walking speed from 0.69m/s upto 1.03m/s. The self-stabilized walking has the best robustness at the speed of 0.81m/s, where the robot was able to walk on the level ground, step over random obstacles smaller than 4% of the leg length, and step over a big obstacle of 7.2% of the leg length.

We report on a novel ZMP feedback control strategy which regulates the center of mass trajectory of a robot based on the measured zero-moment point (ZMP). Based on the trajectory generated by preview control and the linear inverted pendulum model, the controller has an extra model-based control input which is driven by the center-of-mass (CoM) acceleration, which provides sensitive response to disturbance. Thus, the robot motion is balanced between the long-term nominal CoM/ZMP trajectory tracking and the short-term trajectory modification for disturbance rejection. In addition, the controller has two layers of adjustment strategy. When the disturbance is small, the robot is regulated by merely adjusting its CoM trajectory. In contrast, when the disturbance is large, the foot position of the robot is simultaneously changed to provide a more effective response with characteristics of larger intrinsic stability. The performance of the control strategy is also experimentally evaluated using a child-size biped robot.

This paper proposes a model predictive based approach for dynamic balance of a NAO H25 Humanoid robot. Due to the inherent mechanical properties of a humanoid robot, it has a huge potential to lose its stability during its movements. Using an integrated dynamic model leads to control three potential situations a humanoid robot may encounter, namely under-actuation, over-actuation, and fully-actuation. In this paper, by resorting to the so-called Zero-Momentum Point (ZMP) criterion for robots stability, an investigation is carried out on the stability equations and movement patterns in their relevant state space regarding to choosing ZMP as a sustainability criterion, to the end of achieving an appropriate path for robot's center of mass and two legs. In order to apply the concept of the Model-Predictive Control (MPC) based walking pattern, the ZMP state space equations are distributed along the horizon length, besides defining a specific cost function, which covers control input and ZMP path. Some linear constraints are applied to ZMP implementation and robot steps, which leads to optimize the cost function. This procedure is followed by evaluating how the weight of different parts of the cost function influences its performance. Implementation and simulation in MATLAB, Python, and ADAMS estimates and verifies the MPC performance.

In this paper, we analysis gaits of 3D quasi-passive walker with flat feet driven by an antagonistic pneumatic artificial muscle. Many bipedal robots have been proposed to realize the high energy efficiency walking. The passive dynamic walking does not require control input. Generally, a foot shape of passive dynamic walking robot is an arc foot. It is intended to establish a control method and walking mechanism to achieve energy efficient and stable gait. Therefore, we developed 3D quasi-passive walker with flat feet. An antagonistic mechanism is constituted by a pair of McKibben muscle. And an antagonistic pneumatic system is used as joint actuators of linkage mechanisms which control the torque, joint stiffness and position simultaneously. This paper shows that the 3D quasi-passive walking in the level ground can realize by the simple swinging (sine) input of the frontal direction, and the stride of the robot is proportional to lateral-plane input. Finally, we examine the structure of an ankle, the control method of an antagonistic pneumatic system and these gaits analysis using COP (Center Of Pressure) and ROS (Roll-Over Shapes).