Time optimal control for quadruped walking robots

A time optimal control method for quadruped walking robots is developed and installed into a practical robot system. The fastest straight walking has already been obtained by our previous study. Though that is only straight walking, moving range is restricted. Therefore, the purpose of this study is theory construction of curved walking for quadruped robots because of increasing moving range and earlier arriving at target. By means of deriving curved path stroke of robot body, placing position of legs, and dynamic equation of robot body, we obtain it.

This paper proposes an innovative current control strategy for permanent magnet synchronous motor (PMSM) drives with both isotropic and anisotropic rotors, based on Pontryagin's time optimal control (TOC) theory. The control strategy assures the fastest current response after any demanded torque reference variation. It can be demonstrated that such a control exhibits faster response than hysteresis control, even if the latter is often considered the fastest control scheme. The theoretical principle of TOC is at first illustrated. It is then applied to PMSM drives and verified by simulations. Comparison to other control strategies is included in the paper. At last, an experimental validation is carried out and measured results are presented and discussed.

We have proposed the sway compensation trajectory of robot body to realize dynamically stable walking for a quadruped walking robot. This method uses the lateral motion of the robot body to keep a zero momentum point (ZMP) on a diagonal line between the support legs. However, energy consumption of the sway compensation trajectory is larger than the case without trajectory control because periodical acceleration of the body has to be generated to realize sway motion. In this paper, we propose a new sway compensation trajectory for a quadruped walking robot named 3D sway compensation trajectory. This method is the extension of conventional sway compensation trajectory toward three dimensional motion, and the position of ZMP is controlled not only by lateral body motion but also by longitudinal and vertical body motion. Next, we demonstrate that energy consumption for walking can be reduced by combining these motions through theoretical analysis and computer simulation, and an optimum 3D sway compensation trajectory that minimizes energy consumption is proposed. Optimum trajectory is applied to a quadruped walking robot and the validity of the proposed trajectory is verified using the dynamic motion simulator, ADAMS.

In this paper, a new development in the time optimal control theory in sliding mode control systems for multi-quadrant buck converters with a variable load is presented. In general, the closed-loop time optimal control system is applied to multi-quadrant buck converters for output regulation, so that an optimal switching surface is obtained. Moreover, an adjusted optimal sliding mode controller is suggested which adjusts the controller parameters to give an optimal switching surface. In addition, a description of the transient response of the closed-loop system is proposed and used to damp any output or input disturbances in minimum time. Numerical simulations and experimental results are employed to demonstrate that the output regulation time and transient performances of dc/dc converters using the proposed technique are improved effectively when compared to the classical sliding mode control method.

In this paper, in order to meet the requirements of stable foot support and balanced walking of quadruped hydraulic robot, a new control method is proposed. The "hard" contact between the feet of the quadruped hydraulic robot and the ground are changed into "semi flexible" contact to reduce the impact force produced by the contact and collision of the two and reduce the unbalance of the force. The designed "semi flexible" scheme was to coat the "semi flexible" modified polytetrafluoroethylene (PTFE) material on the surface of the bottom material of the robot's foot. The modification of PTFE was optimized by SRIM, and the optimization scheme was selected for experimental study to obtain the Al2O3/PTFE material for which the abrasive resistance met the requirement of robot motion. It was also used in robot force imbalance control experiment research optimization. The study clearly revealed that it can effectively reduce the power of the impact force caused by the contact and collision of the feet of the quadruped robot and the ground, reduce the swing and the unbalance of the motion force of the quadruped robot, and meet the requirements of the precise positioning and the force balance walking of the motion of quadruped hydraulic robot to a certain extent.

For an easier and natural teleoperation of the quadruped walking robots, a user interface is developed with a steering wheel and an assembly of pedals. In order to change the speed and the direction of the quadruped walking robots according to the operation of the steering wheel and the pedal, an online gait transition method is developed. Using this human-machine interface and online gait transition, the operator can control the quadruped walking robots almost in the same way as a driver drives a car. A quadruped walking robot named JROB-2 is used in the experiments. As the steering wheel is also equipped with various control buttons, it is possible to assign various single command tasks, like stand up and sit down etc., to these buttons and hence control the robot for these tasks without even touching the keyboard or mouse. The paper presents the details of this interface.

This paper proposes a quadruped walking robot that has high performance as a working machine. This robot is needed for various tasks controlled by tele-operation, especially for humanitarian mine detection and removal. Since there are numerous personnel landmines that are still in place from many wars, it is desirable to provide a safe and inexpensive tool that civilians can use to remove those mines. The authors have been working on the concept of the humanitarian demining robot systems for 4 years and have performed basic experiments with the first prototype VK-I using the modified quadruped walking robot, TITAN-VIII. After those experiments, it was possible to refine some concepts and now the new robot has a tool (end-effector) changing system on its back, so that by utilizing the legs as manipulation arms and connecting various tools to the foot, it can perform mine detection and removal tasks. Toaccomplish these tasks, we developed various end-effectors that can be attached to the working leg. In this paper we will discuss the mechanical design of the new walking robot called TITAN-IX to be applied to the new system VK-II.

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

Reduction of the energy consumption is one of the most important problems to utilize quadruped walking robots for various works on rugged terrain. The authors have studied basic strategy to achieve high energy efficiency when the quadruped walking robot do the motion essentially requires positive power by the analysis of body rising motion. This paper discusses the energy efficiency of the slope walking motion by the quadruped walking robot. First, we investigate the walking posture in consideration of ideal actuator characteristics where the robot consumes few negative powers at each joint which causes the main energy loss of the walking robot. Then, we investigate optimal walking posture in consideration of DC motor characteristics by the full search of three gait parameters which define the crawl gait. Furthermore, we derive the optimal walking motion by the optimization of three gait parameters which are kept constant during one cycle gait and instantaneous parameters such as body velocity and supporting forces changed at each moment simultaneously.

A new method, inspired by the gait of horses, for regulating the ground reaction force of a quadruped walking robot, is proposed. The method uses the mechanical stiffness switch with a dual‐spring structure inspired by an equine distal forelimb. This includes the use of a low‐stiffness spring activated at the moment of touchdown and a high‐stiffness spring for the support phase. This method has been applied to Jinpoong, a hydraulic quadruped walking robot, and has contributed to the robot's stable gait on uneven terrain.

This paper presents a computational method for generating time-optimal control (TOC) trajectories for robotic point-to-point motions. The algorithm is composed of two phases: initialization and refinement. In the initialization phase, a two-point boundary value problem (TPBVP) resulting from the perturbed TOC problem is solved for an appropriately chosen perturbation parameter. The solution obtained from the initialization phase is then refined by solving a set of initial value problems (IVPs) sequentially and/or in parallel until the desired solution is achieved. The proposed two-phase method is computationally efficient since the resulting TPBVP is solved only one time and the remaining problem becomes solutions to a set of IVP subproblems. The algorithm was used to investigate the effects of robot parameters on the TOC structure through computer simulations on an example robot system with different motion configurations, including both regular and singular trajectories.

Biological observations on tetrapods locomotion deduce that anti-phase synchronization (APS) between fore and rear parts is beneficial for achieving a high-speed walking. On the other hand, theoretical analysis and experimental studies on quadruped robots suggest that a flexible spine potentially improves the gait efficiency and adaptability via smoothing the ground collisions. However, these two mechanisms have never been placed together by a comprehensive investigation in terms of their synergetic effect. Namely, an advanced principle is still lacking in combining the APS and the spine flexibility for quadruped walking robots. To address this issue, we construct a mathematical model for a quadruped dynamic walker under different spine conditions. First, the APS effect is generated via entrainment-based control method under a rigid spine condition. Then, flexible spines realized by three kinds of springs are compared with the rigid one via theoretical analysis. The results suggest that the APS mechanism and the flexible spine can be synergized via an appropriate deformation control. The theoretical findings not only uncover locomotion control mechanisms for quadruped walking robots, but also provide additional understandings of tetrapods dynamic walking from a mechanical engineering point of view.

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.

A robotic system can be viewed as a collection of lower-dimensional systems that are coupled via reaction forces (Lagrange multipliers) enforcing holonomic constraints. Inspired by this viewpoint, this letter presents a novel formulation for nonlinear control systems that are subject to coupling constraints via virtual “coupling” inputs that abstractly play the role of Lagrange multipliers. The main contribution of this letter is a process - mirroring solving for Lagrange multipliers in robotic systems - wherein we isolate subsystems free of coupling constraints that provably encode the full-order dynamics of the coupled control system from which it was derived. This dimension reduction is leveraged in the formulation of a nonlinear optimization problem for the isolated subsystem that yields periodic orbits for the full-order coupled system. We consider the application of these ideas to robotic systems, which can be decomposed into subsystems. Specifically, we view a quadruped as a coupled control system consisting of two bipedal robots, wherein applying the framework developed allows for gaits (periodic orbits) to be generated for the individual biped yielding a gait for the full-order quadrupedal dynamics. This is demonstrated on a quadrupedal robot through simulation and walking experiments on rough terrains.

Dynamic balance has to be maintained during motions of biped systems when their feet are in contact with the ground. As a necessary condition, this indicates that the calculated zero moment point (ZMP) position should be within the specified foot support region throughout the entire motion. A critical term in the ZMP formulation is the rate of angular momentum (RAM) for each link, which should be evaluated accurately and efficiently in motion planning and simulations. In this study, we propose a recursive Lagrangian method based on Denavit–Hartenberg convention to calculate the RAM for each link and the corresponding sensitivity. This method allows the evaluation of each link’s dynamic contribution to the ZMP position. The effectiveness of the proposed approach is demonstrated by simulating bipedal motions of walking and running along with their comparison against existing approaches (direct method and global force method). The accurate RAM calculation in ZMP based on the proposed approach resulted in the improved motion trajectories and reliable ground reaction forces for high-speed bipedal motion predictions.