Planar Leg Research Articles

Optimal planar leg geometry in robots and crabs for idealized rocky terrain

Natural terrain is uneven so it may be beneficial to grasp onto the depressions or ‘valleys’ between obstacles when walking over such a surface. To examine how leg geometry influences walking across obstacles with valleys, we (1) modeled the performance of a two-linkage leg with parallel axis ‘hip’ and ‘knee’ joints to determine how relative segment lengths influence stepping across rocks of varying diameter, and (2) measured the walking limbs in two species of intertidal crabs, Hemigrapsus nudus and Pachygrapsus crassipes, which live on rocky shores and granular terrains. We idealized uneven terrains as adjacent rigid hemispherical ‘rocks’ with valleys between them and calculated kinematic factors such as workspace, limb angles with respect to the ground, and body configurations needed to step over rocks. We first find that the simulated foot tip radius relative to the rock radius is limited by friction and material failure. To enable force closure for grasping, and assuming that friction coefficients above 0.5 are unrealistic, the foot tip radius must be at least 10 times smaller than that of the rocks. However, ratios above 15 are at risk of fracture. Second, we find the theoretical optimal leg geometry for robots is, with the distal segment 0.63 of the total length, which enables the traversal of rocks with a diameter that is 37% of the total leg length. Surprisingly, the intertidal crabs’ walking limbs cluster around the same limb ratio of 0.63, showing deviations for limbs less specialized for walking. Our results can be applied broadly when designing segment lengths and foot shapes for legged robots on uneven terrain, as demonstrated here using a hexapod crab-inspired robot. Furthermore, these findings can inform our understanding of the evolutionary patterns in leg anatomy associated with adapting to rocky terrain.

Stable periodic motion of a controlled segmented leg model of pedal locomotion with inelastic ground-foot collision

Human and robotic legged locomotion can be described with complex multi-degree-of-freedom dynamic models, whose bifurcation or parameter analysis may explain some features of typical patterns during motion. In this paper, we focus on the effect of kinematic parameters and foot placement techniques on the ground-foot impact intensity. The work is based on a multibody dynamic model of a segmented leg, which possesses some fundamental characteristics of locomotion systems: (a) distinct topology in the flight and ground phases; (b) kinetic energy absorption due to partially/fully inelastic ground-foot collision; (c) an active control strategy for maintaining a prescribed mechanical energy level; and (d) different control strategies for the flight and ground phases. We obtain a quantitative measure for the foot collision intensity by analytic calculations. The pre-impact velocity conditions are obtained from a hopping three-segmented planar leg model that imitates pedal locomotion. The single-legged model contains the foot, the shank, the thigh and a reaction wheel attached to the hip, which models the effect of the upper body. The existence of stable periodic motion associated with hopping is shown for a wide range of parameters by means of finding suitable control torques in the ankle, the knee and the hip joint. The parameters of the linear feedback controller are tuned to optimize different cost functions, such as running speed, energy efficiency and impact intensity. We also investigate how the stability of the periodic motion depends on the control gains.

The Influence of Non-planar (Spatial) Links in the Static Characteristics Behavior of Planar Parallel Manipulator

<span>Conventional planar manipulators have all their links in a single plane. Increasing payload at the end-effecter/mobile platform can induce high stress in the links due to cantilever nature of links. Thus it limits the total vertical load that can be applied on the mobile platform. In contrast to the links in conventional planar parallel mechanisms, non-planar links are proposed in this paper, i.e., links are made inclined to the horizontal plane and non planar legs are constructed. Although the links are made non-planar, the rotary (or prismatic) joints axes remain perpendicular (or parallel) to the plane of the base platform, which retains the planar motion of the end-effecter. For studying the application of such non planar links in planar manipulators, new models of inertia, stiffness and leg dynamics have to be developed. In this article, these models are developed and with the developed models, the static analysis is done on the planar manipulators with non-planar links and the performance is compared with the corresponding conventional planar manipulators.</span>

A novel approach to gait synchronization and transition for reconfigurable walking platforms

Legged robots based on one degree-of-freedom reconfigurable planar leg mechanisms, that are capable of generating multiple useful gaits, are highly desired due to the possibility of handling environments and tasks of high complexity while maintaining simple control schemes. An essential consideration in these reconfigurable legged robots is to attain stability in motion, at rest as well as while transforming from one configuration to another with the minimum number of legs as long as the full range of their walking patterns, resulting from the different gait cycles of their legs, is achieved. To this end, in this paper, we present a method for the generation of input joint trajectories to properly synchronize the movement of quadruped robots with reconfigurable legs. The approach is exemplified in a four-legged robot with reconfigurable Jansen legs capable of generating up to six useful different gait cycles. The proposed technique is validated through simulated results that show the platform׳s stability across its six feasible walking patterns and during gait transition phases, thus considerably extending the capabilities of the non-reconfigurable design.

Optimal control of a half-circular compliant legged monopod

This paper investigates an optimal control strategy for the dynamic locomotion of a simplified planar compliant half-circular legged monopod model. We first present a novel planar leg model which incorporates rolling kinematics and a new compliance model, motivated by the use of similar leg designs on existing platforms. Two locomotion tasks, moving at a prescribed horizontal velocity and a one-shot jump to maximum possible height or length, are then investigated within this model. The designs of two high-level optimal controllers for these tasks are then described to investigate the performance and stability of resulting behaviors, based on the optimization of trajectory parameters for a closed-loop low-level Proportional-Derivative controller at the hip. Our results show that with these optimized parameters the robot can achieve stable locomotion at a desired horizontal velocity and can successfully jump over and across a specified height and length. Finally, the composition of objective functions capturing multiple criteria is also investigated for the one-shot jump task.

Kineto Static Design and Stiffness Analysis of a New Kind of 3-RRR Planar Parallel Manipulator

This paper presents a new design of a 3-RRR planar manipulator with non-planar legs. In contrast to the conventional 3-RRR planar parallel mechanism, the links are not planar. They are elevated above the X-Y plane and non planar legs are constructed. The kinematics of this model is realized on a common projected plane and traced back to its elevated position. The moment of inertia for the inclined links is computed. A stiffness model is established for the proposed design of 3-RRR manipulator and compared with a conventional 3-RRR planar manipulator. The analysis shows how the proposed design has better stiffness along all the three directions of motion.

Joint forces in dynamics of the 3-RRR planar parallel robot

Recursive matrix relations for the complete dynamics of a 3-RRR planar parallel robot are established in this paper. Three identical planar legs connecting to the moving platform are located in the same vertical plane. Knowing the motion of the platform, we develop first the inverse kinematics problem and determine the positions, velocities and accelerations of the robot. Further, the inverse dynamic problem is solved using an approach based on the principle of virtual work. Finally, compact matrix equations and graphs of simulation for the input torques of three actuators, the external and internal joint forces are obtained.

Inverse dynamics of the 3-PRR planar parallel robot

Recursive modelling for the kinematics and dynamics of the known 3- PRR planar parallel robot is established in this paper. Three identical planar legs connecting to the moving platform are located in a vertical plane. Knowing the motion of the platform, we develop first the inverse kinematics and determine the positions, velocities and accelerations of the robot. Further, the principle of virtual work is used in the inverse dynamics problem. Several matrix equations offer iterative expressions and graphs for the power requirement comparison of each of three actuators in two different actuation schemes: prismatic actuators and revolute actuators. For the same evolution of the moving platform in the vertical plane, the power distribution upon the three actuators depends on the actuating configuration, but the total power absorbed by the set of three actuators is the same, at any instant, for both driving systems. The study of the dynamics of the parallel mechanisms is done mainly to solve successfully the control of the motion of such robotic systems.

Power requirement comparison in the 3-RPR planar parallel robot dynamics

Recursive matrix relations for kinematics and dynamics of the commonly known 3-RPR planar parallel robots are established in this paper. Three identical planar legs connecting to the moving platform are located in the same plane. Knowing the motion of the platform, we develop first the inverse kinematical problem and determine the positions, velocities and accelerations of the robot. Further, the principle of virtual work is used in the inverse dynamic problem. Several matrix equations offer iterative expressions and graphs for the power requirement comparison of each of three actuators in two different actuation schemes: revolute actuators and prismatic actuators. For the same evolution of the moving platform in the vertical plane, the power distribution upon the three actuators depends on the actuating configuration, but the total power absorbed by the set of three actuators is the same, at any instant, for both driving systems.

Optimal leg design in a hexapod walker

Leg performance during walking was examined with a computer model of a planar leg built of three segments. The service zone, where the leg tip may reach, is non-homogenous with respect to leg performance efficiency. The subzones in the anteromedial, posteromedial and mediolateral parts of the service zone provide approximately straight leg tracks and minimal rocking under the simplest control (i.e. there is proportional change of all the joint angles from their start values to their finish values). Legs of real insects are arranged in these three subzones, or at least in two of them. In the central subzone, the rocking can be large and the particular control system which controls the straight leg tracks may be necessary. The control system must activate some muscles twice for each stepping cycle. The skew alignment of the proximal leg articulation is necessary for the adjustment of various stance widths on the plane and on the narrow ground.

A three-dimensional model for leg and foot kinematical studies

Legged robots based on one degree-of-freedom reconfigurable planar leg mechanisms, that are capable of generating multiple useful gaits, are highly desired due to the possibility of handling environments and tasks of high complexity while maintaining simple control schemes. An essential consideration in these reconfigurable legged robots is to attain stability in motion, at rest as well as while transforming from one configuration to another with the minimum number of legs as long as the full range of their walking patterns, resulting from the different gait cycles of their legs, is achieved. To this end, in this paper, we present a method for the generation of input joint trajectories to properly synchronize the movement of quadruped robots with reconfigurable legs. The approach is exemplified in a four-legged robot with reconfigurable Jansen legs capable of generating up to six useful different gait cycles. The proposed technique is validated through simulated results that show the platform׳s stability across its six feasible walking patterns and during gait transition phases, thus considerably extending the capabilities of the non-reconfigurable design.